numbers

The Surprises Never Eend: The Ulam Spiral of Primes

goodmath — Tue, 22 Jun 2010 09:58:52 +0000

The Surprises Never Eend: The Ulam Spiral of Primes

One of the things that's endlessly fascinating to me about math and
science is the way that, no matter how much we know, we're constantly
discovering more things that we don't know. Even in simple, fundamental
areas, there's always a surprise waiting just around the corner.

A great example of this is something called the Ulam spiral,
named after Stanislaw Ulam, who first noticed it. Take a sheet of graph paper.
Put "1" in some square. Then, spiral out from there, putting one number in
each square. Then circle each of the prime numbers. Like the following:

If you do that for a while - and zoom out, so that you can't see the numbers,
but just dots for each circled number, what you'll get will look something like
this:

That's the Ulam spiral filling a 200x200 grid. Look at how many diagonal
line segments you get! And look how many diagonal line segments occur along
the same lines! Why do the prime numbers tend to occur in clusters
along the diagonals of this spiral? I don't have a clue. Nor, to my knowledge,
does anyone else!

And it gets even a bit more surprising: you don't need to start
the spiral with one. You can start it with one hundred, or seventeen thousand. If
you draw the spiral, you'll find primes along diagonals.

Intuitions about it are almost certainly wrong. For example, when I first
thought about it, I tried to find a numerical pattern around the diagonals.
There are lots of patterns. For example, one of the simplest ones is
that an awful lot of primes occur along the set of lines
f(n) = 4n²+n+c, for a variety of values of b and c. But what does
that tell you? Alas, not much. Why do so many primes occur along
those families of lines?

You can make the effect even more prominent by making the spiral
a bit more regular. Instead of graph paper, draw an archimedean spiral - that
is, the classic circular spiral path. Each revolution around the circle, evenly
distribute the numbers up to the next perfect square. So the first spiral will have 2, 3, 4;
the next will have 5, 6, 7, 8, 9. And so on. What you'll wind up with is
called the Sack's spiral, which looks like this:

This has been cited by some religious folks as being a proof of the
existence of God. Personally, I think that that's silly; my personal
belief is that even a deity can't change the way the numbers work: the
nature of the numbers and how they behave in inescapable. Even a deity who
could create the universe couldn't make 4 a prime number.

Even just working with simple integers, and as simple a concept of
the prime numbers, there are still surprises waiting for us.

goodmath Tue, 06/22/2010 - 05:58

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It'd be interesting to see these graphs with a color gradient varying with number of divisors instead of just black for two divisors (prime) and white for more than two. I wonder what kind of patterns would emerge.

We did some Ulam spiral visualizations recently. Check them out at:

http://blog.morphism.com/2010/05/building-numbers.html

I assume the fact that the squares all extend on two diagonals coming out from center is completely unsurprising given that squares are represented by x[n] = x[n-1] + 2n - 1 -- or also x[n] = x[n-2] + 4(n-1)

I also did a quick check to see if it was simple pareidolia by generating a 200x200 grid with 4203 random dots in it (or rather, each of the 40000 pixels having a 4203/40000 chance of having a dot) - because according to Wolfram Alpha there are 4203 primes less than 40000 (http://www.wolframalpha.com/input/?i=primes+less+than+40000).

But several runs provided nothing like those hard diagonal lines. It all just looked (unsurprisingly) like noise.

At least you got me thinking...

Is the clustering all that surprising? The primes will need to be on diagonals because that is where you find the odd numbers. What happens when you randomly distribute a similar number of points on the odd numbers using a distribution that mimics the density of the primes (approximate Ï(x))?

Well, the Sack's spiral looks like the Death Star, perhaps this is a proof of the existence of the Dark Side of the Force??

The intuition that occurred to Jeff @4 also occurred to me. Do interesting patterns show up if you plot only odd numbers?

I've seen the Ulam spiral before and the first time I saw it the first thing I asked was whether the diagonal lines were the level you'd expect from a reasonably random model. I've never got an answer, at some point when I have free time I intend to do the obvious thing and make some spirals which have points colored in with probability 1/ log n and see how visually similar they look. If there's a real phenomenon going on here they should look different in a very obvious way.

I wrote a program a few days ago that made Ulam spirals, they make pretty desktop backgrounds. :3

This is my current: http://img529.imageshack.us/img529/615/ulam3.png

You could possibly also try plotting on a fibbonacci type of spiral

@nick and @jeff coloring in all odd numbers would only give you a uniform grey shading. Not a spiral.

Don't believe me? Try it.

This is a surprisingly robust process. You can try a lot of naive approaches - like shading in random numbers with the same probability distribution as the primes; or shading various odds - you'll either get an extremely strong pattern (like a uniform grey for all odd numbers, or a cross-hatch far various determinstic subsets of the odds), but this scattershot process with a high-level uniformity? It's really hard to design some kind of set that produces a pattern like this. That's part of what makes it so damned fascinating. You can create things with much more structure, and you can create things with much less structure - but finding something that's got this kind of structure to it is really surprisingly difficult.

if you draw primes in circles an the distribution of primes is somehow related with pi, some kinds of patterns should occur, right? But you're: It's fascinating. Again and again and again. If you love mathematics, you'll never be bored;-)
Thanx for this nice post.

@Maht #10: I wasn't suggesting coloring all the odd numbers - I was suggesting plotting a Ulam spiral that only includes odd numbers, then coloring the primes.

@Nick #13 Just coded that, exhibits similar diagonal patterns to the normal Ulam spiral.

...now I want to plot it on a hex grid...

#14, and others:

If you produce any interesting variants, I'd love to link to them. If you don't have a place to host the image, just email it to me, and I'll host it at SB.

@Thasc #14 Nice! Thanks for trying it out.

This has been cited by some religious folks as being a proof of the existence of God. Personally, I think that that's silly; my personal belief is that even a deity can't change the way the numbers work: the nature of the numbers and how they behave in inescapable. Even a deity who could create the universe couldn't make 4 a prime number.

Additionally, it's not actually an argument for God's existence so much as somebody going "Oooh, something neat was discovered! Quick, attribute it to God!"

Oh, good.. Someone already has...

http://mathworld.wolfram.com/PrimeSpiral.html

A month back I made a big PDF of the Ulam spiral that one can download at http://imprompt.us/2010/ulam-spiral-take-2/

It prints out quite nicely on a 24" roll-fed printer.

#2
Did you try only having random odd numbers generated (dots in a checkerboard pattern)?

Also, there's a typo in the title.

I am wondering how it would be extended to higher dimensions. Instead of following a curves, we may define a way to fill a surface.

It's actually the Sacks spiral, not Sack's. It was discovered by Robert Sacks.
I came across Ulam's spiral when I was looking up some things about Ulam's cellular automata, but I hadn't seen Sacks' variant. Pretty neat.

Something similar that I find even more intriguing is the following construction: For some natural numbers D and N, take all polynomials of degree less than or equal to D, with nonzero integer coefficients from the range -N to N, inclusive. Find all complex roots for the set of polynomials, then plot the density of roots on the complex plane. The resulting plot will look roughly similar for most choices of D and N, with increasing detail for larger values of D. Any guess what it ends up looking like?

Well, here's what you get for D = 24, N = 1. Anyone got an explanation for that?

For further details (and proper credit for the previous image), see this page about plots of polynomial roots.

Plotted on a triangle:

1
2 3
4 5 6
7 8 9 10 ...

http://rob.tarrfamily.com/share/Triangle.gif

How about this:

1) Consider a series of squares formed by a series of uniform steps that doesn't cross a diagonal. The nth square will have a value determined by a quadratic polynomial in n.

2) IIRC, for an odd prime p, a quadratic polynomial with integer coefficents whose n^2 coefficient does not vanish mod p will have (p+1)/2 residues mod p and (p-1)/2 nonresidues mod p.

3) Diagonal sequences will be all even or all odd.

In short, polynomial sequences will tend to be streakier with respect to primes.

For example, Euler's prime-generating polynomial n^2+n+41 is never divisible by any prime less than 41, and probably has average luck with respect to the higher primes.

By contrast, n^2+n+43 is average wrt. 3, and is worse than average wrt. 5, 7, and 11 (2 out of 5 are divisible by 5, 2 out of 7 are divisible by 7, and 2 out of 11 are divisible by 11).

This may not show that God exists, but I'm starting to wonder about the devil ;-)

Elaboration on #26:

Define the luck of a sequence (mod p), where p is prime, as:
- "perfect luck" if no member is divisible by p
- "average luck" if 1/p members are divisible by p
- "poor luck" if 2/p members are divisible by p
- "no luck" if every member is divisible by p

(Ignoring the edge case of when p itself appears in the sequence (which will happen at most finitely many times for polynomial sequences), unlucky sequences have fewer primes than usual, and lucky sequences have more primes than usual.)

The luck of a quadratic sequence a_n := Ann + Bn + C mod p will be:

: For p = 2
:: A != B (mod 2) -> average luck
:: A == B (mod 2), C == 0 (mod 2) -> no luck
:: A == B (mod 2), C == 1 (mod 2) -> perfect luck

: For p = 2k+1
:: A != 0 (mod p)
::: Set d := C % p.
::: For 1 value of d, the series has average luck; for k values of d, the series has poor luck; for k values of d, the series has perfect luck.
:: A == 0 (mod p), B != 0 (mod p) -> average luck
:: A == 0 (mod p), B == 0 (mod p), C == 0 (mod p) -> perfect luck

(Well established results in number theory; search for quadratic residues if you want proofs.)

Fix A and B with A+B even.

There will be 1 way to choose C (mod 2) so that the sequence has perfect luck mod 2, 1 way to choose C (mod 3) so it has perfect luck mod 3, 2 ways to choose C (mod 5) so it has perfect luck mod 5, and so on. (Exceptions when p is an odd prime dividing A.)

By the Chinese remainder theorem, it is possible to choose C so that the sequence has perfect luck for an arbitrary number of primes.

If a sequence has perfect luck for a number of small primes, members are significantly more likely to be prime, even if the sequence has poor luck with later primes. A random integer will 12% chance of having no prime factor < 100. A random integer with perfect luck for primes 2 through 37, but poor & independent luck for primes 41 through 97 has a 65% chance of having no prime factor < 100.

And as it happens, diagonals on the spiral are generated by 4NN + 2mN + C, which will have the desired feast/famine properties.

I think you found your book jacket illustration, Mark.

I'd like to see an animation of the sieve of erasothenes running against the numbers plotted out like this.

Also - as I understand it, the density of the primes is approximately n/ln(n), right? So to plot out a spiral that has an even density of primes across its area, the radius at n should be not sqrt(n) - which gives you an even density for all nunmbers - but ... ?

Actually, all things considered, sqrt(n) will be pretty good, because the rate at which the rate changes is much less than sqrt(n).

Hmm. I feel an applet coming on ...

#20,

I totally was not expecting that PDF to dynamically generate the spiral. As the PDF loaded in the browser, I saw it growing from the center of the spiral out...

Here's an idea, just an idea... In the diagram, lines of slope 2, 3 (or 1/2, 1/3) and higher (or lower) don't have as many points on them, period (much less primes). So any cluster of primes on those lines wouldn't be as dense as on the slope-1 and slope-minus 1 lines, so they wouldn't be as noticeable. -- I'm not sure that's the reason, but it did occur to me.

Well of course there are still surprises waiting for us. I mean you can interpret pretty much all mathematics as statements about the primes.

Hell, we know that even the question of which Diophantine equations are solvable is non-computable and hence can't ever be fully characterized in a predicatively useful way.

Working with the first 100K primes.

To plot prime number n, theta = sqrt(n)*2pi, and r = sqrt(n) to give an even distribution of numbers across the disc, or r = sqrt(n / log(n)) to give an even distribution of primes across the disc.

@32: Skewing the picture by a linear transformation would expose those lines, and it doesn't: if you use the transformation matrix ((1 1) (0 1)), you see dotted vertical lines and even hints of lines of slope 1/2, but no diagonals.

Dang, spoke too soon. Lines with slope 3 do actually show quite a bit, and even slopes can be seen if one wants to see them. I would guess that this just corresponds to forms 4n^2+2bn+c differing only by a constant from 2n(2n+b), which is likely to be divisible by all kinds of primes, so they "have a smaller chance" to be composite themselves. The same pattern is probably true for numbers of form 2n(n+1)(n+2)+c as well, it is just that they do not form an easily recognizable pattern on the spiral.

hi . your math blog is very good . thanks .
luck .
my math blog is persian , whoes I link its translate .

I wonder what happens when you do a Sacks spiral with one revolution every odd perfect square? ie, 2 3 4 5 6 7 8 9 in the first circuit, 10-25 in the next circuit, etc. That would more closely match the Ulam spiral.

Multiples of many (all?) numbers form diagonal patterns. The spaces left will surely also form occasional diagonals, am i missing something or can it be that simple?

Move a bishop around a chessboard a few times and record squares that haven't been crossed and you should see something similar.

As Paul Murray said (30) I think an animation of the sieve would make this a little more clearer to see.

After seeing the Ulam and the Sacks spirals I had been playing with spirals and primes myself a while ago and came up with this:
http://incubator.quasimondo.com/flash/wheel_of_primes.php

I fear it's probably just a pretty picture that I got after exploring various spiral factors; still I was quite surprised when I hit on it.

Interesting post.

For example, one of the simplest ones is that an awful lot of primes occur along the set of lines f(n) = 4n2+n+c, for a variety of values of b and c.

Should that be f(n) = 4n2+bn+c?

Reminds me of the plots of the large scale structure of the universe with boxes along lines and holes.

I wonder how you could generalize the concept to 3D to get that kind of plot.

this is fascinating stuff. those patterns--so close, so tantalizingly close to looking like something predictable. amazing, thanks for sharing.

I don't know whether you read your comment threads in their entirety, but I'd be very interested to hear your thoughts about the following. You said: "Personally, I think that that's silly; my personal belief is that even a deity can't change the way the numbers work: the nature of the numbers and how they behave is inescapable. Even a deity who could create the universe couldn't make 4 a prime number." I agree that simply creating the physical universe doesn't mean you can change the behavior of numbers, which are "inescapable" in the sense that the laws of logic would not allow them to behave in any other way. In other words, the behavior of numbers are fixed as soon as the laws of logic are.

But what if God created logic itself? It seems counterintuitive, but what if even the most basic axioms of logic, like "P implies P", were actually contingent upon God's will? In essence, what I'm claiming is that God "wrote" truth, in the sense that he chose which propositions are true and which aren't. And he wrote logic, in the sense that he chose how propositions logically followed from each other. So yes, I think God designed Sack's spiral.

Remember, if your only response is that my world view, in which logic and mathematics could have been different from what they are, is logically impossible, then you've missed the point!

@44:

Logic isn't a single thing. It's a formal system, defined from abstract rules. First order predicate logic, which is the main logic we use in math, isn't part of the universe. It's an abstraction, defined by a system of rules.

There are some things which are inevitably true. And no one and nothing can possibly get around that.

God *can't* create a universe where "A ∧ ¬ A" is true in FOPL. Not because of any limit on the power of god, but because it's meaningless. FOPL is something specific, which has a set of rules that define it. You *can't* create a universe where those rules don't work - because they don't depend on the universe; they're pure abstraction. Within their system, the results that you can get from them are absolutely inevitable.

Creating a "universe" where A ∧ ¬ A is true in FOPL is like creating a Universe
where the universe doesn't exist. What does it mean to create a universe where the universe doesn't exist? It doesn't mean anything. It's a non-sensical phrase.

I agree that simply creating the physical universe doesn't mean you can change the behavior of numbers, which are "inescapable" in the sense that the laws of logic would not allow them to behave in any other way. In other words, the behavior of numbers are fixed as soon as the laws of logic are.

@46:

Close, but you're still missing one absolutely crucial bit: the laws of logic are arbitrary. Logic isn't something that's defined by the physical properties of the universe. Logic is just a system for describing statements and inference rules.

FOPL seems very natural to us - but that's mainly just because we've grown up using it. Even before we knew what formal logic was, we were learning things in terms of FOPL. But there's no reason that anyone has to use FOPL.

To give an example of a different approach, many people argue that intuitionistic logic is preferable. But in first order intuitionist predicate logic, A&lor;¬A is not necessarily true!

Even a world creating deity couldn't make numbers work differently in FOPL.

a math animation blog

@45 I'm sorry, but I don't think I was sufficiently clear. When I said that God created logic, I didn't just mean that God created formal logical systems, like First Order Logic or the Propositional Calculus. I meant that he created logical consequence itself. A formal system is just a set of axioms, which give you some theorems to start off with, and a set of inference rules, which tell you how to produce new theorems from old ones. But given the specification of a particular formal system, how do you actually conclude that something is a theorem?

An axiom may be of the form A, and a rule of inference may be of the form "If A is a theorem, then B is a theorem." But then how do you actually conclude that B is a theorem? You need the logical principle that if P is true and P implies Q, then Q is true. Where does this logic come from? Of course, you can use an axiomatization of logic like the Propositional Calculus, but that too is a formal system, so you're just kicking the problem somewhere else. (If you want to read more about this, check out this dialog by Lewis Carrol.)

My point is that there is something pre-formal called Logic. You can formalize things like logical consequence, but ultimately to even use formal systems you need the original pre-formal God-given notion. To respond to your example, it is indeed possible that God could have made "A & not A" a theorem of first order logic, since he has full control over that pre-formal Logic we rely upon.

Also, I don't understand why you think that "pure abstractions" are out of reach of God. I guess the issue boils down to what your philosophical views are. If you believe that truth, in particular mathematical truth, is just a subjective thing produced by the human brain, then all of this would be irrelevant, because the primeness of 4 would just be a matter of opinion. If, on the other hand, you believe as I do that mathematical truth is independent of human beliefs, then the question arises: where does mathematical truth come from? I think the only truly satisfactory answer is that something as absolute as Truth can only come from something as absolute as God.

Did anybody notice my posts 26 & 28? I gave an explanation why some diagonals are prime-rich. (Short version: Given a set S of primes and fixed A, B, with A+B even, it is possible to choose C so that An^2 + Bn + C is never divisible by any prime in S.)

@50

I think your explanation gets to the heart of the matter. There is actually a long-standing conjecture about the density of primes in sequences generated by quadratic polynomials going back to Hardy and Littlewood from which one would be led to expect that some diagonals should have higher density than others. This is a generalization of their conjecture about the number of primes of the form n^2+1, and was in turn generalized to polynomials of arbitrary degree and to collections of polynomials by Bateman and Horn. I think the heuristic argument given by Bateman and Horn for their conjecture is quite similar to yours.

It seems to be "common knowledge" that the the sharp diagonal lines in the Ulam spiral are an unexpected phenomenon. Where this "common knowledge" came from, I'm not sure, but I suspect that professional analytic number theorists do not share the surprise. I would be interested to hear opinions about this, one way or the other.

@49 "To respond to your example, it is indeed possible that God could have made "A & not A" a theorem of first order logic, since he has full control over that pre-formal Logic we rely upon."

It's simply not possible for God or any being to have "control" over basic "laws" of logic (or rather, the referents thereof as the formalization is a human construction as Mark notes). Consider: it would have been impossible for God to instantiate the referent of the law of non-contradiction for that would necessitate that before He did so, He both existed and didn't exist. It would also necessitate that He could at any point revert reality to it's "pre" LoNC state. Which would of its own necessitate that He would both be able to do so and NOT be able to do so.

TL;DR - grounding logic in God's will yields incoherency and chaos and is thus impossible.

@52 I'll refer you to the last sentence of my first comment: "Remember, if your only response is that my world view, in which logic and mathematics could have been different from what they are, is logically impossible, then you've missed the point!" Of course it is logically absurd for logic to be different than what it is, just as it is logically absurd for "A & not A" to be a theorem of first order logic. But the key word is "logically". It is useless to reason logically about a situation in which logic itself is different.

@53: "It is useless to reason logically about a situation in which logic itself is different."

Ah...hm...well, then you might as well shorten that sentence to read, "It is useless to reason." I hope you at least realize that your own argument is self-contradictory: if you're right, you're also wrong. A bit like arguing for solipsism; sure you might be right, but if you are you're talking to yourself...why argue the point?

Damn this lack of an edit feature!

@53: Besides which, it's not in any way a case of "mathematics and logic being different. It's a case of them not existing at all. If the state of affairs of existence were to be such that the law of non-contradiction were not to be a veridical representation of it, there could be no logic or mathematics at all; there would be only eternal, immutable chaos. Without a means of determining the state of reality, there would be no way to formalize (or even conceptualize) logic, not to mention the total inability to form any abstractions or concepts whatsoever. Without distinction of differences, there could be no numbers and no way to conceive or formalize mathematics.

@53:

You're basically falling back to an un-falsifiable position, while challenging us to falsify it:

You're asserting that there is some sort of pre-logic, without which logic as we understand it couldn't exist. But, you cannot describe or define that pre-logic in any meaningful way.

Naturally, any attempt to argue against that position is doomed to fail, for the same reasons as any non-falsifiable argument: since your argument is fundamentally based on something undefined, no matter what someone puts forward as a counter-argument, you can always say "No, that's not what I'm talking about".

Logic is a complete, self-contained formal system of abstract rules. Those rules have inevitable consequences. Those consequences aren't dependent on any "pre-formal" system. In fact, they aren't dependent on any notion of semantics or meaning. Logic is purely mechanical, syntactic inference.

You don't need any agent to tell you that if "A" is true, and "A ⇒ B" is true, then "B" is true. That's not dependent on meaning. It's not dependent on a pre-formal logic.

You're also pulling out one of the standard, rubbish traps of theistic arguments: that if there's no god, then there's no truth. Again, that's crap. The universe exists. We can see that. That's true. Whether or not God exists, I can observe that I exist. That is true, regardless of whether or not there is a God.

If you define numbers axiomatically (i.e., using logic), then the prime numbers exist. They're just a simple consequence of definition. God can't make 4 prime, because 4 is 2*2.

God can't create a universe where first order predicate logic doesn't work - first order predicate logic doesn't even require a universe to exist at all: it's a pure abstraction. An all-powerful deity could create a universe where FOPL doesn't accurately describe anything about the behavior of the universe. For example, blow away cause and effect, and FOPL ceases to be a particularly useful tool. But that doesn't change the fundamental validity of
FOPL as a self-contained formal system.

I can define a logical system which is valid, but which doesn't do anything particularly useful. I can use FOPL to define a number system which works in bizarre ways. (In fact, John Conway did exactly that with the Nimbers, which are incredibly strange.) But the existence of an alternate number system doesn't change the fact that if you take FOPL and the Peano axioms, you'll get the natural numbers. And no matter what anyone does, no matter what form the universe takes, no matter what the deity wills, if you use FOPL and Peano to define the natural numbers, 4 will not be a prime.

As I've said plenty of times around here: I'm not an atheist; I'm a religious Jew. But I find this kind of argument for theism to be pathetically shallow and foolish. I don't understand the need to create these ridiculous assertions like "If there's no god, then you could just say 4 is prime". That's doesn't support your argument: it just shows that you don't understand what you're talking about.

Hello.

These spiral of primes are your 2D thought pattern produced onto a 2d sheet. You have to make quantum leaps from point to point based on an ever expanding and then contracting personal math pi. It is your personal thought being projected onto one inner side of perfect cube box that you alone inhabit. Inside, at each moment your wave function completely collapses and then returns depending on your WILL).

You can see the fully completed equations between our Matrix/Avatar/Inception "real life" that leads in thought and logic to classical theory which finally leads to quantum theory here:

http://www.wix.com/hyperstig/hyperstig

All of existence is built on these 3 principles constantly extrapolating from the original zero => [The Big Bang(i.e. The centre super-massive black hole at the centre of our galaxy) that is attempting to pull us(STARS back in)].

http://www.wix.com/hyperstig/simple

There will be a total collapse of our sector of the universe. It is a gravity wave; the same kind that wiped out Atlantis; the dinosaurs and other major civilizations.

Unless we apply the simple visual principle to keep re-building the "big box" the outcome is an escalating probability towards TRUE. We will either make it or we won't.

If not, our "powerful" civilization will be wiped out in a day leaving nothing but small radiations from the edge of the black hole as hawking described.

We are nothing but imagination and memories and WILL anyhow. I wonder if we'll make it this time??? I guess it's a better shot as we have better communications this time.

The universe is digital. It always has been. It's nothing more than mathematical logic that sprang from the fear of NOT existing.

A B C easy as 1 2 3 easy as DO RAY ME ABC 1 2 3 BABY YOU AND ME GIRL.

I hope you do TOO. I guess you will seeing as you're fractal just like me.

It just that I found the bottom, made sure everything was al-right and then start to look around and find you people.

You are a singular STAR with a perfectly cubed black hole at the centre of you. How do you think we're communicating?

The internet???

Do you think that's air you're breathing???

That's just alternating fission and fusion.

Pick a random length of pie(your stars diameter) and STOP and look around(360 degrees in all directions).

Start making connections in a non linear way. You simply are a box and are in a box.

Time does not exist now. We don't need it.

It is just a construct that either points you in a direction(old school watch) or gives you a position in a perfectly cubed grid that tells you x,y,z where you are.(digital). You are who, what and where you are.

Start making connections with your "family".

Superposition holds true. You are everywhere and no-where. That's why no-one in the world knows what we need to do.

You're in a box that surrounds the milky way. It's the first quantum computer and each of you are a bit in that system. I couldn't show the ALPHA release until now for reasons I'll keep to myself.

Just know that it is a self organising system that will work with you; not against. Karma rings TRUE fully in this system. I just thought I'd let you know that.

Finally; the math PROVES that re-incarnation is TRUE.

It doesn't say what you will become; but your memory and WILL lives forever. Once you have been "born" you CANNOT die. You might as well face whatever you have done and start your LIFE again. What's the point in waiting for what comes after???

Enjoy.

Ste.

@#57: Let me be first to say: LOLWUT?!

Hello people,

I just went through the document. I also tried similar spirals with numbers divisible by other numbers. I am sharing the images in http://punarvak.blogspot.com/2010/07/ulam-spirals-and-its-variants.html. Have a look at them.

The spiral for Fibonacci numbers have very scarce details. Because occurance of fibonacci numbers is only scarce.

Check the image in my blog www.punarvak.blogspot.com.

I have a number of short publications and presentations, and hundreds of pages of notes, on the Semiprime Spiral.

A semiprime, also called a 2-almost prime, biprime (Conway et al. 2008), or pq-number, is a composite number that is the product of two (possibly equal) primes. The first few are 4, 6, 9, 10, 14, 15, 21, 22, ...

For an example, see:
http://www.research.att.com/~njas/sequences/A113688

A113688 Isolated semiprimes in the semiprime spiral.

65, 74, 249, 295, 309, 355, 422, 511, 545, 667, 669, 721, 723, 749, 758 (list; graph; listen)

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1,1

COMMENT
Write the integers 1, 2, 3, 4, ... in a counterclockwise square spiral. Analogous to Ulam coloring in the primes in the spiral and discovering unexpectedly many connected diagonals, we construct a semiprime spiral by coloring in all semiprimes (A001358). Each integer has 8 adjacent integers in the spiral, horizontally, vertically and diagonally. Curious extended clumps coagulate, slightly denser towards the origin, of semiprimes connected by adjacency. This sequence gives isolated semiprimes in the semiprime spiral, namely those semiprimes none of whose adjacent integers in the spiral are semiprimes. A113688 gives an enumeration of the number of semiprimes in clumps of size >1 through n^2.

REFERENCES

Stein, M. and Ulam, S. M. "An Observation on the Distribution of Primes." Amer. Math. Monthly 74, 43-44, 1967.

Stein, M. L.; Ulam, S. M.; and Wells, M. B. "A Visual Display of Some Properties of the Distribution of Primes." Amer. Math. Monthly 71, 516-520, 1964.

S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.

LINKS
Eric Weisstein's World of Mathematics, "Prime Spiral".

Eric Weisstein's World of Mathematics, "Semiprime.".

EXAMPLE

......................

... 17 16 15 14 13 ...

... 18 5 4 3 12 ...

... 19 6 1 2 11 ...

... 20 7 8 9 10 ...

... 21 22 23 24 25 ...

CROSSREFS

Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, A033988, A033989-A033991, A033996, A063826.

KEYWORD
easy,nonn

AUTHOR
Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 05 2005

Take a sheet of graph paper. Put "1" in some square. Then put "2" in the square to the right and continue on from there, putting one number in each square. Then circle each of the prime numbers.

If you do that for a while - and zoom out, so that you can't see the numbers, but just dots for each circled number, what you'll get will look something like this:

. . . ...... .. . . . . .. . .

Oh my gawd - notice how the prime numbers form a completely straight line!!!

I call that Ankers line. It's amazing how were still discovering underlying principles of math and the universe.

@50 and @51 seem to explain why there are diagonals. But why does the Ulam figure as a whole seem to have a half-rotation symmetry? Notice that the NE-SW prime-rich lines in the NW quadrant seem to match up with those in the SE. But while NW-SE prime-rich diagonals do exist, they don't have the half-rotation symmetry.
Or has someone already answered that?

My method really divides things into N, S, E, W quadrants.

Produce the polynomials for some of the NE-SW lines in ? quadrant and the corresponding ? quadrant and we'll see if there's a relationship.

This is a real effect. The line joining the NW to the SE corner of the figure divides the square into two triangles. The lines lying above the division and heading in the NE direction have equations of the form 4x^2-2x+c; the lines lying below the division and heading in the SW direction have equations of the form 4x^2+2x+c. For a given c value, these two lines are expected to contain the same density of primes as x=0,1,2,... In fact, if x is allowed to take negative values, these two lines are seen to be the same. That is, they connect to each other, after spiraling around a bit in the center. To be more precise, the density of primes lying on a quadratic is conjectured to be sensitive to the discriminant, which equals 4-16c^2 for both types of line. This is the reason for the symmetry you observed.

Why isn't there a similar symmetry between the NW and SE lines? In this case we divide the square into two triangles by drawing the line joining the NE and SW corners. The lines lying above the division and heading in the NW direction have the form 4x^2+c, while the lines lying below the division and heading in the SE direction have the form 4x^2+4x+c = (2x+1)^2 + (c-1). Notice that these two types of lines are not related to each other by negating x as was the case before. (For both types, the "negative" half of a line coincides with its positive half.) The discriminant of the NW lines is -16c, while the discriminant of the SE lines is 16-16c. It might appear that for a given NW line 4x^2+c, the SE line 4x^2+4x+(c+1) would have the same discriminant, and therefore the same density of primes. The reason this doesn't happen is that if c is odd then c+1 is even, and 4x^2+4x+(c+1) contains no primes at all! Likewise if c is even, 4x^2+c contains no primes, while 4x^2+4x+(c+1) is expected to contain infinitely many of them.

It is important to note that nobody has any idea how to prove that even a single quadratic ax^2+bx+c contains infinitely many primes, although this is widely believed to be the case as long as the quadratic doesn't fail for one of the obvious reasons, the "obvious" reasons being that

1) a, b, c have a common factor;
2) (a+b) and c are both even;
3) the polynomial is reducible.

The conjecture for the density of primes on a quadratic obviously depends on there being infinitely many of them, and is therefore even farther from rigorous proof.

Thx, guys. The Ulam spiral no longer seems magical. So I'm not sure whether I have gained or lost.

I used to come across similar patterns when I made attempts at pixel art when I would overlay different types of cross-hatching. Removing the multiples of a prime, even in a spiral, should approximately cross hatch and so this pattern makes sense.

I'm not sure this sort of thing has much value. It strikes me as falling prey to the same kind of thing that religious use of gematria and Bible Codes do - there are an infinitude of ways you can arrange things. Evolution gave you eyes and a mind geared to finding patterns where there aren't any.

Dorky Poll: Numbers

drorzel — Wed, 07 Apr 2010 09:29:54 +0000

Dorky Poll: Numbers

I'm still getting things squared away after my blogging break, but as a step on the way back toward normal programming, here's a Dorky Poll: What kind of numbers do you most like to work with?

You can only choose a single answer, which I'm sure will come as a disappointment to many of those favoring the later options. You could always vote a second time from a different computer, though...

drorzel Wed, 04/07/2010 - 05:29

Tags

Triangular, specifically, but I also preferred Real Analysis to most other things in general.

Double precision floating point.

I always liked NaN myself. Division by zero, anyone?

I don't see primes listed... Yes, I know they're a subset of another option, but they're special enough to deserve their own mention.

Numbers that make sense ;). Today's computer kid generation just lost all sense of significant digits, like using an instrument that's calibrated to 0.1% accuracy and then presenting a report discussing the variations in the 5 - 8th digit. Just because that's what the computer spits out after an A-D conversion, so that's the precision of the measurement, and the randomness has to be explained away. Even worse if they "recognize" the discrete values an A-D converter produces as "quantization" of the phenomenon.

Ordinals.

Missing options: algebraic and rational numbers.

The brightly colored ones, please...

Don't know about yours, but my classical measurement apparatus does not work with real numbers. It likes rational ones, instead.

Oh, life would be so easy if the guy behind the apparatus were so rational as the numbers he reads!

While I agree with Eric and Don, it is a daily challenge working with a finite model of the reals which is not reflexive and trichotomy does not hold, i.e. x != x may be true, and none of x < 0, x == 0, or x > 0 can be true.

ones that don't require floating point to represent approximate. integers, rationals and fixed-point decimals, that is. bonus points if they don't require bignums either; arbitrary-size integers are cool and all, but limited-size ones are so much faster to work with.

(poll? there's a poll, like with radio buttons and whatnot? it must be getting filtered out by my ad blocking, if so.)

I'm a mathematician, of the algebraic persuation, so I voted 'complex numbers'.

HOWEVER: Numbers are for pussies, it's the mathematical structures built on the basal number systems that are interesting. Algebraists care about (among other things) solutions of polynomials, which makes structures over the complex numbers interesting. Differential geometers only care about the differentiability of their mathematical gadgets, so they work with stuff over the reals. Analysts work with both real and complex analysis. Topologists don't care about numbers, they work in complete generality. Combinatorists work with integers (or natural numbers, strictly speaking). Mathematical logicians work with arbitrary ordinals and cardinals. No one works with pure imaginary numbers.

Also: it's 'octoniOns', complex numbers are not messy (at least not in the 'grey areas' sense), and Ron @10: there is no finite model for the reals, and the rest of your comment is nonsense, so I assume your are pulling an educated trolling.

FWIW, whenever I am asked what my favourite number is, I answer (truthfully), 'n'.

I like round numbers, because sometimes people get a little overly precise. Not long ago I was asked if I was picking the depth to a coal seam (1-2 feet thick*) at the top or bottom of the seam. The seam was more than 1000 feet below land surface and, in the hilly coal country of southwestern Virginia, I'd guess the surface elevation surveyed for the well had an precision of at best +/- 5 feet. I thought to myself, 'what does it matter?'

*American coal and petroleum geologists will go metric when Hell freezes over

The world actually appears to be complex. QED operates over the complex numbers. Strings moving through time sweep out world-sheets which live in Teichmuller Spaces which have a natural complex structure. We may only be able to imagine rationals, but that's our shortcoming.

and Ron @10: there is no finite model for the reals, and the rest of your comment is nonsense, so I assume your are pulling an educated trolling.

Ron's comment makes perfect sense; it's about the problems inherent in using floating-point numbers on computers.

I used quaternians once in my life. it was pretty fun actually. once i got my mind wrapped around it...

My comment was garbled by html parsing. I was referring to IEEE-754 floating point arithmetic, where x != x may hold for NaN (not a number) values of x, and for which all of x is less than 0, x is equal to 0, and x is greater than 0 are false at the same time, since any comparison to a NaN value is false.

Bah, I'm with Kronecker. Give me the integers and thus, the Rationals.

Mathematical Nature?

grrlscientist — Thu, 25 Mar 2010 05:59:23 +0000

Mathematical Nature?

tags: nature, numbers, geometry, mathematics, Fibonacci sequence, Golden Ratio, Angle Ratio, Delaunay Triangulation, Voronoi Tessellations, filmmaking, animation, Cristobal Vila, Nature by Numbers, streaming video

In this beautiful video, "Nature by Numbers," filmmaker Cristobal Vila presents a series of animations illustrating various mathematic principles, beginning with a breathtaking animation of the Fibonacci Sequence before moving on to the Golden Ratio, the Angle Ratio, the Delaunay Triangulation and Voronoi Tessellations. The words are scary-sounding, but the math is beautiful and the film serves to remind us of the intimate relationship between nature and math.

This movie was inspired by numbers, geometry and nature. You can learn more about CristÃ³bal Vila at etereaestudios. That website has more information about the film, including a fascinating step-by-step explanation of the theory behind the film (English is side-by-side con EspaÃ±ol), along with stills, screenshots, tutorials and workshops.

NOTE: after I'd scheduled this entry, I realized that my colleague, Bioephemera already showed it to her Sb readers. Even though I don't think our readership overlaps, I rescheduled this to pop up a few days later anyway. In the meanwhile, I've been enjoying this beautiful video every day while I waited to share it with you.

grrlscientist Thu, 03/25/2010 - 01:59

I'm sure our readers overlap somewhat - besides, it's viral now - a professor up here showed it to his class on Wednesday. :) Isn't it awesome?

yeah, it's great! some of the best dreams i ever had were when i was taking calculus, and i dreamt about the formulae and proofs i was studying. watching this video makes me want to take more mathematics classes.

I always love seeing visual artists reveal the inherent beauty, symmetry, and elegance of mathematics - especially for someone like me who cannot see or understand it by looking at symbols but only when translated into pictures and movement - then it becomes clear. That's just how my brain works, part of being a high-functioning autistic I guess.

Single cells in the monkey brain encode abstract mathematical concepts

neurophilosophy — Thu, 21 Jan 2010 10:50:10 +0000

Single cells in the monkey brain encode abstract mathematical concepts

OUR ability to use and manipulate numbers is integral to everyday life - we use them to label, rank, count and measure almost everything we encounter. It was long thought that numerical competence is dependent on language and, therefore, that numerosity is restricted to our species. Although the symbolic representation of numbers, using numerals and words, is indeed unique to humans, we now know that animals are also capable of manipulating numerical information.

One study published in 1998, for example, showed that rhesus monkeys can form spontaneous representations of small numbers and use them to choose containers with more pieces of fruit. More recently, it was found that monkeys can perform basic arithmetic on a par with college students. Now, German researchers report that not only do rhesus monkeys understand simple mathematical rules, but also that these rules are encoded by single neurons in the rhesus prefrontal.

Animal experiments and neuroimaging studies performed with humans have implicated the prefrontal cortex (PFC) in the processing and execution of numerical operations. In humans, this part of the brain is engaged during tasks involving mathematical rules, and it has long been known that damage to the PFC can lead to impaired quantitative reasoning. Sylvia Bongard and Andreas Nieder of the Institute of Neurobiology at the University of Tubingen therefore hypothesized that PFC neurons are involved in encoding aspects of numerosity, and designed a numerical task based on simple numerical rules to test this.

Two rhesus monkeys were shown pairs of visual stimuli consisting of sets of dots and trained to compare them by applying two simple mathematical rules. In each trial, they were shown a sample set of dots followed, after a short delay, by a test set with a different number of dots. The 'greater than' rule required the monkeys to release a lever if the test set contained more dots than the sample set, whereas the 'less than' rule required them to release the lever if it contained fewer dots. During the interval between each pair of stimuli, a cue was presented, indicating which of the two rules should be applied.

While the monkeys performed this task, microelectrodes were used to record the activity of approximately 500 individual and randomly selected PFC neurons. The response of each cell was determined during four different time periods in each trial: the time during which the sample set of dots was displayed, the delay between the sample and the cue indicating which rule to apply, the time during which thecue was displayed, and the delay between presentation of the rule-related cue and the monkeys' response to it.

Significantly, the monkeys immediately applied the mathematical rules to all the stimuli pairs they were shown, even when the sample sets contained numerosities that had not been previously presented. Selective responses were recorded during the interval between the cue and the response. 90 rule-selective neurons (~19% of the total from which recordings were made) were detected, which fired independently of the number of dots presented or the sensory properties of the rule-related cue. Of these, 50 fired exclusively when the monkeys produced 'greater than' responses, and the remaining 40 fired exclusively when they produced 'less than' responses. Rule selectivity was not encoded immediately, but emerged in the cells after a short period of time.

Across hundreds of trials, the monkeys had a minimum success rate of 83%. The researchers compared the neuronal responses of individual rule-selective neurons during trials in which the monkeys gave correct responses with trials in which they made errors. The firing rates were found to decrease significantly when the monkeys made the wrong choices. The selectivity of the responses also enabled the reearchers to predict which rule the monkeys were applying during each trial, from the cellular activity they recorded.

Thus, single neurons in the lateral PFC of the rhesus monkey can flexibly encode abstract mathematical rules which guide greater than/ less than decisions. Each session involved large numbers of unique trials, so it was impossible for the monkeys to solve the task by learning. Instead, they were required to understand relationships between numerosities in each pair of stimuli, and to apply these principles to make their decisions. These findings are consistent with a model which proposes that the PFC contains a network of distinct rule-coding neuron clusters, each of which receives input from a corresponding internal memory cluster and sends its output to a dedicated downstream cluster. The findings also add to a body of evidence suggesting that humans and other primates process numbers using common cognitive skills with a shared evolutionary origin.

Bongard, S. & Nieder, A. (2010). Basic mathematical rules are encoded by primate prefrontal cortex neurons Proc. Nat. Acad. Sci. DOI: 10.1073/pnas.0909180107.

Cantlon, J. F. & Brannon, E. M. (2006). Basic math in monkeys and college students [Full text]

Hauser M. D., et al. (2000). Spontaneous number representation in semifree-ranging rhesus monkeys. Proc. R. Soc. Lond. B Biol. Sci. 267:829-33 [PDF]

neurophilosophy Thu, 01/21/2010 - 05:50

Tags

Sorry, Denise - but God didn't make numbers

goodmath — Mon, 19 Oct 2009 13:54:52 +0000

Sorry, Denise - but God didn't make numbers

I was planning on ignoring this one, but tons of readers have been writing
to me about the latest inanity spouting from the keyboard of Discovery
Institute's flunky, Denise O'Leary.

Here's what she had to say:

Even though I am not a creationist by any reasonable definition,
I sometimes get pegged as the local gap tooth creationist moron. (But then I
don't have gaps in my teeth either. Check unretouched photos.)

As the best gap tooth they could come up with, a local TV station interviewed
me about "superstition" the other day.

The issue turned out to be superstition related to numbers. Were they hoping
I'd fall in?

The skinny: Some local people want their house numbers changed because they
feel the current number assignment is "unlucky."

Look, guys, numbers here are assigned on a strict directional rota. If the
number bugs you so much, move.

Don't mess up the street directory for everyone else. Paramedics, fire chiefs,
police chiefs, et cetera, might need a directory they can make sense of. You
might be glad for that yourself one day.

Anyway, I didn't get a chance to say this on the program so I will now: No
numbers are evil or unlucky. All numbers are - in my view - created by God to
march in a strict series or else a discoverable* series, and that is what
makes mathematics possible. And mathematics is evidence for design, not
superstition.

The interview may never have aired. I tend to flub the gap-tooth creationist
moron role, so interviews with me are often not aired.

* I am thinking here of numbers like pi, that just go on and on and never
shut up, but you can work with them anyway.(You just decide where you want
to cut the mike.)

It's such concentrated stupidity, it's hard to know quite where to start. So
how about we start at the beginning?

Denise O'Leary claims not to be a creationist by "any reasonable
definition"? Yeesh. No point even trying to argue with that. She's just playing
the usual ID'ers games with the definition of "creationist".

Then, very rapidly, we get the usual victimization rant. Poor, poor
Denise. Such an unfortunate soul, so looked down on. I mean, she spews
non-stop nonsense, and all she gets for it is a nice salary, lots of attention,
a publishing contract, and some television interviews. Those IDers sure are
put upon, aren't they?

Then - shock! - she gets something right. The subject of
the interview was goofy people who want their house numbers changed, because
they think that they got unlucky numbers. Yeah, that's pretty stupid.
Absolutely.

It brings to mind an interesting story. Back when I was in college, my
family had to move. My parents had taken out a ten-year renegotiable mortgage,
and they couldn't afford the increased payments while also making the tuition
bills for me and my brother. They ended up selling the house very quickly. But
it was really strange. The people who bought it were Chinese, and they hated
just about everything about the house. They hated the landscaping.
They hated the kitchen. They hated the tiling. They hated the slate foyer.
They hated the windows. They hated the parquet wood floors. They thought it
was too big. Honestly, if there was anything that they actually liked
about the house, I don't know what it was. But they bought it. Because it
faced in the right direction, and it was the only house facing exactly
that direction on the market. Their feng shui master had told them that they
must have a house that faced in that direction - that anything else
would bring them terrible luck. So they bought it.

People believe all sorts of strange things. There are all sorts of peculiar
superstitions, about numbers, names, shapes, colors, directions. It's all silly.
And it's amazing how many of us still hold on to those odd ideas, or at least
the behaviors that they imply. To get personal, I know perfectly well that
nothing I say is going to cause the world to turn on me and make something
bad happen. But European Jews have a lot of superstitions about drawing attention
to themselves, and I never say things like "Well, things couldn't
possibly get any worse", or "Things are so great, I can't imagine how they could
get better". Those are both statements that "draw attention". I know how stupid
it is, but that doesn't change the feeling I get in the pit of my stomach when
someone says something like that.

So yeah, superstitions like that are silly, and they do deserve to be
mocked. Mine included. But I'll bet you dollars to donuts that Denise wouldn't
buy a house where a satanist had performed his phony rituals without getting
it purified by a priest with holy water, and that she wouldn't see anything
remotely silly about it. She sees her superstitions as legitimate,
but others as mockable.

ANyway, enough of that. Let's get to the good part.

She says "No numbers are evil or unlucky. All numbers are - in my view -
created by God to march in a strict series or else a discoverable* series, and
that is what makes mathematics possible. And mathematics is evidence for
design, not superstition."

Oy, oy, oy.

Numbers were not created by a supernatural being. No deity, no matter how
powerful, could have created a universe where numbers didn't exist, or didn't
work.

This is a surprisingly difficult and subtle point. But numbers, in some
sense, aren't real. They're purely conceptual. There's no such thing
in the real universe as the number 2. There are plenty of examples of "two
objects", but the number 2 doesn't exist. Far worse, there is absolutely no
way of claiming that π really exists. There are no perfect circles in the
universe. And the only sense in which π can possibly exist in the real
universe is as a measurement.

Numbers are an artifact of reasoning. They don't exist out there in the
void, waiting for someone to find them. They're a consequence of a simple set
of rules. And those rules must work. There's no way that God can
change the nature of an abstraction that doesn't really exist. He could make
it impossible for us to conceive of those rules. But the rules would
still work. Even if there was no universe at all, those
rules could still be said to exist, and therefore, that the numbers still
exist.

It comes down to a deceptively simple question: "What is a number?". And
there is no single answer to that question! I can define numbers informally,
by counting. I can formalize that a bit, and get two different kinds of
numbers: ordinals and cardinals. I can formalize differently, and get surreal
numbers. Still another way, I can start with different rules, and get Piano
numbers. Or another way, and get computable numbers. I can define real
numbers, complex numbers, vectors, quaternions. Those are all perfectly valid
concepts - and they're all different. Which one really
defines numbers? All of them. None of them. Take your pick. Numbers are
what you want them to be. They don't exist outside of your mind. They're a tool
that we use to understand the universe - but they don't have any real,
objective reality.

But Denise's stupidity doesn't end there. She needs to qualify things - the
numbers "all proceed in a strict series, or else a discoverable series".

Bzzzt. Wrong.

You can look at that statement in two ways. One way of looking at it - which
I think is the one she meant - is just completely, utterly, wrong. The other way,
which you could reasonably argue is the correct interpretation, is totally
fouled up by that qualification.

Interpretation one:

"The numbers all proceed in a strict series". My initial reading
of this is that "series" implies a listing or enumeration of one number after
another.

The problem with this is that you can't put the real numbers into
that kind of series. The real numbers are an uncountable set: you can't
enumerate the elements of an uncountable set. So they can't possibly be
put in a series.

You could weasel out of that problem, by saying that the
qualification solves the problem: you can enumerate the rational
numbers: you can put them into a kind of series. Since she explicitly mentions
numbers like π as being exceptions, you could argue that she meant
that the rational numbers could be put into a series, and that the "discoverable
series" qualification was meant to cover the irrational numbers.

Alas, that doesn't work either. First, from her wording and description, I
really don't think that when she said the numbers are in a strict series, that
she had in mind an ordering where, for example, 2 comes before 1/3, and
1/3 comes before 1/100. But you can't enumerate the rationals in
anything like comparison order, which is what I think she was trying
to say.

In addition to that point, I'd say that there's something seriously wrong
with a definition where the exception covers the overwhelming
majority of cases. Most numbers are irrational - but her phrasing implies that
the irrationals are sort-of strange exceptions.

But I left the worse for last. As I've mentioned before, href="http://scienceblogs.com/goodmath/2009/05/you_cant_write_that_number_in.php">most
numbers are undescribable. You can't discover them. You can't
describe them. You can't name them. You can't point at them. And yet, by the
definition of real numbers, they must exist. So even forgetting about
the whole ordering issue, the idea of all numbers being discoverable, is just
totally wrong. They're not. Numbers are much stranger, much less
rational, less intuitively comprehensible, less well-behaved than her naive
understanding.

Interpretation Two

The second interpretation is that "the numbers all proceed in a strict series" is a poorly
phrased way of saying that the real numbers are totally ordered. That is a fact:
given any two distinct real numbers X and Y, either X<Y or Y<X. That's correct. But if
that's what she meant, then she blew herself out of the water with the qualification: because
irrational numbers like π are still part of the total ordering of the real numbers.
Pulling them out by that qualifier implies that she doesn't believe that they're part of
the series - which in this interpretation means that you can't always compare them. But even given
two irrational numbers, they're always comparable. Even the undescribables.

And the qualification still fails exactly the same way it did in case one: most
numbers aren't discoverable, describable, nameable, identifyable, or enumerable.

So again, she fails miserably.

The takeaway point here is that numbers are both less real, much stranger,
and frankly a whole lot more interesting than Denise O'Leary imagines. As
usual for Creationists (and yes, Denise, you are a creationist!),
she's taken a simplistic understanding of something, mistaken her simplistic
understanding for a deep comprehension of it, and then argued that on the
basis of its alleged simplicity that it must have been designed by her deity.

Her version of numbers can't account for undescribable numbers. It can't
account for much of the beautiful strangeness of numbers. It can't account for
logical wierdness like Gödel's incompleteness theorem, which relies on the
logical structure of numbers. It can't account for some of the magnificent strangeness
that people like Greg Chaitin have studied. As is all too common, she's so satisfied
with her simplifications that she's completely missed both the pathology and the beauty
of numbers. It's sad.

It should be obvious, looking at this blog, that I'm deeply in love with
mathematics. Math is beautiful, and fascinating, and frustrating, and strange.
People like Denise O'Leary try to sap out everything that makes it wonderful
in order to be able to say that they understand it, and that their personal
deity created it. God didn't create math. Math is a collection of formalisms
that we created from the basic rules of logic - and those rules
must hold, no matter what the universe is like. Because they aren't
rules about the universe - they're self-contained rules about concepts that
they describe.

If you're religious like me, you might believe that there is some deity that
created the Universe. Or you might not. But whether there is a God or not has nothing
to do with whether A∧¬A == false.

goodmath Mon, 10/19/2009 - 09:54

Tags

Debunking Creationism

numbers

This reminds me of your post over a year ago and 'i' or 'j' or square root of -1. My math prof would go on an extended rant about the term 'real' numbers versus 'imaginary'. He did it so often I swear I've almost got it memorized.

There is no such thing as a Real number he'd insist. Go out in your back yard and dig up a 2. and so on and so on ...

I thought she meant the integers, myself...

re Interpretation 1: Reals are at least orderable; complex numbers are just as valid as reals and they can't even be ordered.

More like only "Natural" numbers, right? ;)

Re: rmp

I figure as long as you're going to count the real numbers as "numbers" there's really no reason to leave out the complex numbers. I mean, for most practical purposes you only really need the rationals anyway (you can approximate pi as closely as you want with rationals). If you are going to count the reals, then why stop there? The complex numbers lack only one property that the reals have, which is an ordering that plays nice with the field axioms, and they have many more properties that are highly useful, such as being algebraically closed. Not to mention that the complex numbers are themselves capable of describing certain real-world phenomena.

Of course, I'm not saying we should stop at the complex numbers either. My general feeling is that if you want to define "numbers", you need to decide first of all which axioms, structures, or properties you'd like "numbers" to have and go from there.

@2:

She couldn't have meant integers: she explicitly mentions π. So she's got to be including rationals and irrationals.

But the point is, she thinks that there's a collection of things called numbers, and that they've got some kind of objective existence. But in reality, numbers are what we say they are; and we typically use the word "numbers" to mean lots of different things - and depending on what we mean, we're talking about totally different things. Are we talking about cardinal numbers? Are we talking about complex numbers? Computable numbers? Surreal numbers? Real numbers? Peano numbers?

She's probably talking about Humpty Dumpty numbers. They have whatever properties the creationist in question wants them to have, no more, no less.

Why would you consider someone stupid for not believing your interpretation of a "a surprisingly difficult and subtle point"?

I actually feel that the only good evidence for the universe being anything more than the observable facts is the fact that our ape brains have conceived of this system we call 'mathematics' which appears to describe the universe extremely well at scales which our brain never evolved to comprehend (conceived without us intentionally trying for this end - the theory of ODEs et cetera predates by far general relativity and quantum theory). That's not what she said, but it's in the same kind of vein.

I thought the standard line was "God created the natural numbers, all the rest is the work of man."

(Not proposed as a contention with anything MCC says, it just seems obligatory to mention it in this discussion.)

I think it's unnecessary to rant at her lack of understanding of number theory; the secondary point she was making (which is far more interesting to rant about) is that she believes that mathematics the concept is necessarily a product of design, not "superstition".

Based on your response I think you'd agree with her there, in that mathematics is not defined by the universe, even as it makes use of the same. Math is just a natural consequence of logic, and it is a "designed" construction. Of course Denise thinks that an invisible old man did the designing and intelligent people think that various animal intelligences do the designing (at different levels of proficiency and formalization).

The universe doesn't anthropomorphically need a definition of real numbers in order to operate but we humans find that a definition of real numbers is key to understanding the universe and describing its operation.

Anyway, I'm not sure that Denise was wildly off the wall here. Of course her mathematics is faulty, but she's not a mathematician and the specifics weren't crucial to her point.

As for her claiming she wouldn't fall into a superstition trap... well, who wants to bet she wouldn't accept a hip replacement with serial number 666?

Obviously she doesn't know what she's talking about, but as Leopold Kronecker said: "God made the integers; all else is the work of man" (not that I happen to agree, since I don't believe in God)

@8:

I thought I was very clear about that.

She's got an incredibly shallow and simplistic concept of numbers. But that shallow and simplistic understanding fits what she *wants* to believe - and so she doesn't look any farther.

In particular, she wrote her babble it as part of a rant about how people stupid were for attaching superstitious meaning to numbers, while simultaneous attaching her own superstitious meaning to numbers, and proclaiming that her superstitious meaning is a demonstration of how smart she is, compared to the gap-toothed morons.

I am thinking here of numbers like pi, that just go on and on and never shut up, but you can work with them anyway.(You just decide where you want to cut the mike.)

Speaking of the number 2 I think that this is the age of her target audience judging by her description of an irrational number.

Because that same "subtle point" holds for the existence of any human thought, especially mathematical concepts.

Just because they are encoded in the firing patterns of neurons, and in the patterns of language, doesn't mean that they exist.

I kept thinking about this quote: "God made the integers; all else is the work of man" by Leopold Kronecker. If you take out the last paragraph concerning Pi, then it reads more plausible. No?

Interesting. You espouse a strong antiplatonist stance. But even Kronecker said that God created the integers.

Mathematicians treat the natural numbers as if they exist independently from humans in the sense that the naturals have properties that humans can discover, but humans cannot define or change the properties.

All other numbers and the operations performed on those numbers are defined by humans.

I think mathematicians often say that "God" created the numbers more as a shorthand way of saying that human being didn't create them; that they have an existence independent of us. (Denyse probably intended the stronger statement, though.)

Mark, I think that her sentence makes marginally better sense if you just insert a comma in the middle: "All numbers are - in my view - created by God to march in a strict series[,] or else a discoverable* series." I read the first half as referring to the integers, and then the second half brings in the rationals and irrationals, such as pi.

I'm not sure I agree with your take here. I agree that numbers aren't the work of a deity that the descriptor given by O'Leary is once again wrong. But it isn't an egregiously wrong position. Indeed, it isn't clear to me what it would mean for a universe to exist with different numbers. But that's connected to the issue that I can't conceive of a universe where basic logic worked differently. But that's not to say that it couldn't exist in some sense. The general human inability to conceive of something is not a priori a reason to assert its non-existence.

A lot of very serious mathematicians, past and present, would disagree with you strongly. (Among them Paul ErdÃ¶s, if Wikipedia can be believed.)

You seem to have stepped into one of the longest-running debates in the philosophy of mathematics without even realizing it (?).

I get that argument all the time, that numbers have some kind of existence by themselves. The people arguing can't seem to grasp that the number 2 would not exist without people. There would be "2" objects, but unless there was other intelligent life to put words to the concept of "x number of objects," there wouldn't be any number 2. The same with arguments for other concepts. I think it usually comes down to some Platonic ideal or else the idea of "well, God exists, therefore His mind has the number 2 in it, so it has an existence" (or something - the argument never made sense to me, so I'm probably not giving it properly).

Nice article, small nit though. I'm guessing you didn't mean "piano numbers" but "peano numbers".

Also, isn't the woman's name Denyse O'Leary?

While I have no problems with you making fun of Denise, Nemo is right. You've stepped into a serious debate and taken sides, and you've represented your side as obvious truth without even a mention of those who disagree with you (such as Godel, for example). Whether or not numbers exist is a philosophical question most mathematicians don't care about.

The use of numbers for counting is a human abstraction. But the natural numbers had properties that existed before humans discovered their use for counting.

For example:

1 + 2 = 2 + 1

is a property of positive integers that humans discovered, humans did not invent it.

That's what they mean by "independent existence."

On the other hand, negative numbers did not have properties before humans defined them. There is nothing in nature to tell us:

-1 X -1 = +1

That is a rule of a human-defined operation that does not exist independent of humans.

Of course it is crazy to say that Ï exists as a physical object in nature, any more than buildings or bees or blogs. None of these things are terms in the fundamental laws of physics. (Well, except Ï.)

Yeah man, but the equivalence class of things that can be put in one-to-one correspondence with the set {1,2} always existed before any human beings or other intelligent animals came along and thought about it or made such correspondences themselves.

I've enjoyed your blog for quite some time, but the past few posts have been nothing more than bile. I completely agree with your position, but I miss the "Good Math" half of the title. Best of luck.

Are you ever going to write again about things we actually care about, or are you going to rename the site to "Good Religion, Bad Religion?"

Um, @26, I always thought that 1 + 2 = 2 + 1 because the operation '+' over the set of 'integers' is defined that way. By humans. It's an axiom, not a discovery. And that -1 x -1 = +1 is also axiomatic. As in: define binary operator '-' by the set of relations {.x.=., .x-=-, -x.=-, -x-=+} where '.' is the absence of - (sometimes for convenience we use '+' instead of '.', but with the same absence meaning).

Associating the set of integers ordered by the '<' operator with some arbitrary set of real world objects is also a human invention. A very old one too, since it made the difference between being rich or poor back in the very old days. Though the '<' way of describing those sets is relatively new. Again, though, naming the members of such a set required the human invention of names for human abstractions of thingness as distinct from specific bricks. 'two-ness' does not exist outside of human invention, and is never a property of real objects, only of human to human communications about real objects.

In fact, I might be able to make a case that all mathematics is about communication, be it between humans or between myself and a future version of myself. It is about describing a certain class of concepts, and what is description if not communication. And communication of abstractions between humans is obviously (I hope obviously) a human invention.

@31

I'm trying to follow Courant and Robbins "What is Mathematics?"

The lowest level of abstraction is to start with natural numbers representing concrete objects and combine them by addition and multiplication the way you would take objects in boxes and combine them physically. That's the level where you make discoveries such as the commutative laws.

My understanding is that this level represents the first part of the famous Kronecker quote. When you go beyond natural numbers, and the addition and multiplication of them as if they were concrete objects, is when you get to Kronecker's "everything else."

I am corrently reading Lee Smolin - The Life of the Cosmos in which he contends that even the most basic logic is contingent on the universe having enough stucture in it to make a distinction between thing and not thing, let alone have entities capable of making such distinctions.

At first that seems to lead to a circular argument that (simplified) complex structure needs logic needs complex structure. The conclusion is that the universe *doesn't* need logic, maths, physics to work. It just works, and logic is totally man made. I think ErdÃ¶s, Kroenecker et al lose this round.

@ 18,

Mathematicians treat the natural numbers as if they exist independently from humans in the sense that the naturals have properties that humans can discover, but humans cannot define or change the properties.

All other numbers and the operations performed on those numbers are defined by humans.

No, really, we* don't. Mostly, we don't think about it too much at all, but if we do, the natural numbers are just a particular algebraic structure. You can assert them as primary, or you can choose different foundations (sets, functions, successor operations, categories, etc). Really, whatever comes in handy at the time.

Clearly, the counting numbers have a special place in all our hearts, but a lot of that is just that counting numbers are the ones which we learn first as children, are embedded in our everyday language, and have especially simple physical analogs.

Put another way: we might think natural numbers are special, but math doesn't**.

* full disclosure: not yet a licensed mathematician, in training.
** illustrative metaphor: not approved for literal use.
---

@29,

What's your problem with Mark's enchiladas?

"But whether there is a God or not has nothing to do with whether Aâ§Â¬A == false."

To be honest, the only way I would be able to imagine an all-powerful god is if he were somehow able to set Aâ§Â¬A == true. If he were somehow more powerful than the laws of logic. Sometimes I think that's what most religious people think too (if they thought hard enough about it) considering the huge amount of contradictions they believe in. God would be truly all powerful if he were able to bend the rules of logic in such a way that numbers, such as we think of them, come out to be a logical result from the axioms we chose.

Of course, if one believes god is more powerful than logic, no amount of reasoning is going to have any effect.

Aâ§Â¬A == false

Did God create logic? If you're denying Platonism, why not go even further and look into paraconsistent logic?

You might not like it, but I guess a large percentage of mathematicians has certain Platonist beliefs (though maybe secretly or unconsciously). This has probably partly to do with the fact that many of the really abstract objects that were invented (discovered?) throughout the centuries, first based on things happening in nature, but later purely based on other mathematics, turn out to be useful in describing nature.

In Carl Sagan's novel "Contact", the extra-terrestial civilizations have a belief in a god that is able to define pi with a sequence of digits (in base 2, but I don't think that is explicitly mentioned), millions of digits into the expansion, that form a rectangle (with primes as the lengths of its sides, of course) of mostly zeros, with a few 1 bits, which when looked at form a circle! The atheistic heroine of the book eventually finds this sequence, and that's how the book ends.

I like this idea, and I seem to remember, that if you look long enough at the binary/decimal expansion of pi, that eventually, you will see any given sequence of digits you desire. Is that true? Or is it an unanswerable question?

"Far worse, there is absolutely no way of claiming that Ï really exists. There are no perfect circles in the universe. And the only sense in which Ï can possibly exist in the real universe is as a measurement."

If I can be allowed a little nit-picking here:
I disagree... not because there are no perfect circles, but because there are no perfect measurements. Measure the length of a circle (with radius = 1) with the best instruments, and all you'll get is a number with many decimals, perhaps, but not an infinity of them. It will only be a rational approximation of Ï. I'd say that "the only sense in which Ï can possibly exist in the real universe" is as a limit: if you can improve indefinitely the quality of your measurements, you'll get ever closer to it. You're right, it's a concept and nothing else.

Some people really are clueless about maths. (And biology, too...)

#38: damn, i just wanted to mention it (i mean 'Contact' by Sagan). btw, it ruined the book for me, and i cannot understand, what Sagan wanted with it. i'm pretty sure that pi is not something for which you can just decide a value, and then create a world, where pi has that specific value. pi "is the ratio of any circle's circumference to its diameter in Euclidean space" (from the wiki), and it is so, no matter what kind of world you are living in.

of course i'm open to any proof to the contrary, but i won't hold my breath.

Yes, as Scott asked, what's with the Piano numbers? Are there only 88 of them, counted in base 13? :-)

More on topic, Denyse is seriously confused about a lot of things. This just adds to the list.

@39:

Where are you going to find a perfect circle to measure?

In our universe, there's no such thing as a perfect circle. Even if you had infinitely precise measuring tools, there's nothing that you could measure that would produce exactly π.
It exists only as a mathematical formalism. It's an valuable mathematical formalism which describes lots of interesting and important things. But nowhere in the universe can you find anything whose value is π.

@38:

I really, really hated the end of Contact.

There are tons of constants in the universe that are, in some sense, variable. They're properties of the shape or structure of the universe. If Sagan wanted to have a message from God encoded in the universe, it would be easy enough to do it - the ratio of masses of the fundamental particles, or the value of the cosmological constant for two examples. But instead, he chose a value which can't be changed or shaped.

Hell, he could even have used the value of a measured π - if space isn't completely flat (and it isn't), then the measured value of π would vary slightly from the value of the theoretical π.

But being a physicist, he knew that even with the best imaginable instruments, measurements have a small number of significant digits. You just can't get the thousands and thousands of digits that he needed for his story.

So rather that doing bad physics, he did bad math. It pissed me off.

There is nothing in nature to tell us:

-1 X -1 = +1

Really? If I reverse the operation of taking one object away (do "negative one," "negative one" times), isn't that adding one object ("positive one")?

Mark: Denise wouldn't buy a house where a satanist had performed his phony rituals without getting it purified by a priest with holy water, and...she wouldn't see anything remotely silly about it. She sees her superstitions as legitimate, but others as mockable.

So how do you feel about mezuzot? Legitimate or mockable? Legitimate for you, OK to mock for others?

I wrote: So how do you feel about mezuzot? Legitimate or mockable? Legitimate for you, OK to mock for others?

That reads on the page as more hostile and challenging than I intended it to be. I am actually sincerely curious as to the answer, if you'd care to share it.

"I like this idea, and I seem to remember, that if you look long enough at the binary/decimal expansion of pi, that eventually, you will see any given sequence of digits you desire. Is that true? Or is it an unanswerable question?"

It's an open question. Most mathematicians think it's true for a number of good reasons, but so far no dice in terms of proof. The formal question is whether pi is a normal number.

@45:

How do I feel about mezuzot? That depends on what the person who hangs them thinks.

Some people hang them because they believe that they're lucky. I think that's stupid and mockable. Thinking that hanging a little scroll of paper in your doorway is going to make good things happen to you, and neglecting to hang one is going to make bad things happen? That's every bit as silly as insisting on buying a house that faces 5 degrees west of north, or refusing to buy a house whose address is #13.

Some people hang mezuzot because it's a reminder. Jews believe that there are rules that you should follow - not because following the rules is going to make anything good happen, but because following the rules is the right thing to do. Mezuzot are a visible reminder of those rules - and you see that reminder every time you enter or leave a home with the mezuzah on the doorpost.

I'm in the second group. I see the mezuzah as something to make me think about what's right, and what's wrong, and what I should be doing.

@38

I think you can take some convergent infinite series , you can reorder the terms of that infinite series to converge to another set value. So if I have an infinite series which converges to Pi, I could take the numbers which define the series and reorder them to instead converge to e (or 1 or 2 or any number).

I think the criteria for that reordering is if it is an infinite convergent alternating series... though that might be too strict a condition. I think the question is whether it is still the same series after you reorder the terms which depends on whether or not its uniformly convergent. I could be wrong, I'm going off of memory.

Sorry if that's off topic, that's what I thought of when I saw your post.

Some of these comments are going off the deep end into metaphysics and similar wishy-washiness. An abstract formalism of numbers is just that: abstract. Abstract concepts don't have a physical, tangible existence. However, there are some physical, tangible objects that exist in the universe whose behavior can be modeled - by humans - with numbers. We found that our beefed up system of logic can be used to do nifty things, like keep track of how many cattle we're accumulating or to predict how cold it will feel tomorrow. But cattle don't reproduce dependent on a definition of natural numbers, and the temperature is a function of particle motion, not the real number line.

Mark's "no such thing as exactly pi" doesn't even enter the question; many numbers do appear in our models exactly, be they natural numbers or not. We can describe abstractly various universes with various rules and then use numbers to build a model with which to predict grander things about them. The universe is "built" around the rules, not the model.

@48

The criteria for a conditionally convergent series are that the series must have an infinite number of both positive and negative terms, and the series formed by taking the absolute values of the terms in the original series must diverge. (Or equivalently, the positive terms taken alone diverge to positive infinity and the negative terms taken alone diverge to negative infinity.)

Such a series can be re-ordered to converge to any (without loss of generality, positive) target value but alternately summing positive terms until the target is overshot, and then summing negative terms until the target is undershot.

I wish more believers were as honest as you are... I wish more unbelievers were as honest as you are... :)

You get 100 cool pts!

@50

Yes, I remembered that the process involved over and under shooting which is why I said alternating series, but that's definitely not general enough. As you said, infinite number of positive and negative terms while the absolute value of the series converges. Thanks for reminding me.

Erp, strike aboslute value converges - replace with diverges. I'll just be quiet now. :)

The ending of Sagan's book Â»ContactÂ« bugs me too.

Â»The Argus computer had gone deeper into Ï, deeper than anyone on Earth, ...Â« but the Â»anomaly showed up most starkly in Base 11 arithmetic, where it could be written out entirely as zeros and onesÂ«. (I have the paperback version in front of me.)

Further: Â»The program reassembled the digits into a square raster, an equal number across and down. The first line was an uninterrupted file of zeros, left to right. After a few more lines, an unmistakable arc had formed, composed of onesÂ« and so on.

Having read it again after 10 years, I'll try to code a circle search for Ï -- well, if you can call those jagged paths of ones a circle. In the simplest form, the bit sequence

0 0 0 0 0
0 0 1 0 0
0 1 0 1 0
0 0 1 0 0
0 0 0 0 0

would satisfy Sagan's description for a representation of a circle with radius 1. Occurances with larger radii would no doubt be less frequent, but still somwhere Â»insideÂ« Ï.

Unclear to me: Are the Â»circlesÂ« supposed to be hollow or filled with ones? And: is garbage between each scanline allowed? This search resembles more and more the infamous Bible Code search, alas with a Â»scriptureÂ« text of infinite length and variation.

Just wanted to add the obvious: while finding patterns in Ï like those Sagan described would be be cool, and all the awesomer, the larger the radii are; it wouldn't mean that some intelligence put it there.

It's just the law of large numbers combined with the fact that Ï is an algebraically independent transcendental number (if not normal, which still is an open question, yet probable and plausible, as I understand).

I really enjoyed the parts deconstructing the "marching in series" statement. I think it would be even stronger with less meta-discussion.

I also read that sentence in a way that makes a perfect sense to me, when I replace MY "all numbers" by Denise's "all numbers" - or maybe by her demographics, tribe, what have you. No, they don't deal with all the wonderful weirdness of numbers, ever. All that subtlety and complexity is completely invisible, practically non-existent to them. These people deal with natural, rational, and a few select irrational numbers like Pi, and that's probably it. It reminds me of the ethnographic studies of some tribes that count, "One, two, many."

To me, there is an issue of whole cultures or demographics having their own alternative or limited math worlds. Do we mock them? Do we mandate that they learn all the "number weirdness" we know and love? Do we say it's "cultural differences" and celebrate the diversity?

This is another philosophical question to add to the growing list. What should we do about groups of people effectively living in different math worlds - say, the world where most numbers they actually meet are rational?

This is a really entertaining and informative discussion! I think some have read too much into Ms. O'Leary's comments on number, surely not made to posit a formal mathematical postulate! It seems to me rather to be saying that order and number in the universe are mutually dependant, that this relationship is discoverable, and it is far from unreasonable to infer the existence of a transcendant, powerful, thinking,
planning God. In fact mathematicians have been known to give praise to the Creator for the number embedded in nature. Such statements in themselves did not render their theories true or false, testing is the only real way. Conversely, a series of errors on the path to a true solution would not render that mathematician's belief in God false or insane.

Moreover, Mr. Chu-Carroll, where is your proof that God didn't "make" numbers? Are you entitled to it by virtue of your mathematics prowess? Is this a problem so self evident it requires no proof; like 1+1.!

You misspelled your adversary's name, so it is plausible other simple realities elude you! ;)

"Numbers were not created by a supernatural being. No deity, no matter how powerful, could have created a universe where numbers didn't exist, or didn't work."

Also, the meaning of this statement eludes me. Is the second supposed to follow from the first?

I'll try: "The most powerful conceivable deity could not create a universe which did not contain and function according to numbers. Therefore conclude that our universe, which contains numbers which work, was not made by any conceivable deity"

Re #s 57 & 58 - I haven't had any math practice since high school calculus nearly 40 years ago, but if I can take a crack at what I understand Mark to be saying (surely he and/or others will correct as necessary), it is that there are no such things as numbers in the Universe. Numbers are abstract concepts that can be used to describe real physical phenomena, but are not themselves to be confused with the phenomena they can be used to describe.

In fact, not all numbers we know of have any sort of correspondence to real physical phenomena; and beyond that, there are infinitely many numbers we haven't even found, let alone used to describe anything, in any category of what we define as "numbers." So it isn't a question of whether a God would have the power to create something, it's that there's no "something" to create.

Much better than my lame attempt above, read the Wikipedia articles on real and complex numbers, particularly with reference to Ms. O'Leary's description of numbers as being in a series, and also regarding correspondence (or lack of same) with any physical reality.

A number of people have said, in effect, "Numbers are abstracta, not existents." Is there a false dilemma lurking here?

"God created the integers" is the first part of a quote. "the rest is the work of mankind" finishes it.

Many have woundered about the unreasonable effectiveness of mathematics. Penrose found his solution in the Platonic forms.

Also some, such as Hardy, were proud of the fact that their math had no practical applications. After he died practical applications were found.

"Is God a Mathematician" is a good book on the subject.

rmp @ 1:

That reminds me of a time when a physics professor was explaining that the operators for quantum mechanical observables had to have real eigenvalues.

"You won't ever measure a complex position or energy. They aren't real."

The class burst into laughter, as he realized what he had said.

And yes, I know that was only tangentially related (maybe in that the prof came close to making the incorrect implication that real numbers are more real than imaginary ones), but I thought it was funny enough to share anyway.

While I agree with your post in general, there are a couple of clear mistakes you make, so it's pretty rough to mock people the way you do without having all the facts yourself. First of all, you say:

"Numbers were not created by a supernatural being. No deity, no matter how powerful, could have created a universe where numbers didn't exist, or didn't work. "

Let me first say that I don't believe in God. But the problem with your sentence is you don't comment on whether God exits or not, but you say that if God existed, and no matter how powerful he was, he couldn't have created a universe where numbers didn't exist. This is not true. A simple example would be just a completely empty universe, with nothing inside. Surely an all-powerful deity could create that! I don't see how numbers would exist there. A more non-trivial example would be a universe where everything was just an amorphous blob, and even if there were sentient beings, you could imagine beings whose consciousnesses are mixed with each other and intermingled, so that there is no way of talking about separate entities. If there is no way of separating things from each other, or talking about one thing vs another, then how could numbers exist in that world or work? And if I can think in 5 minutes of universes with no numbers, why couldn't an all powerful deity create them? The only difference is that I don't believe that deities exist.

Now the second mistake is you say that there are no perfect circles in the universe. This is also not true. The horizon of a black hole can be in a state that is a perfect circle. Another example would be a state of a closed string - these can also be perfect circles.

M Dj: Even a completely empty universe would constitute 1 (one) existence, one spacetime and two types of symmetry: isotropy (directional symmetry) and homogeny (translational symmetry). Either way, there would be a countable amount of minima and maxima of something, e.g. of zero-point energy or space-time curvature, due to virtual particles.

The same is true for the second model you're proposing, the amorphous blob universe (I like that concept). What's more, the blob has to be made out of something. Ãther? (-;

Yeah, superstition brings bad luck.

@ Marko:
The empty Universe can be compact and doesn't have to contain any symmetry. There would be 1 spacetime, but you see you're just applying the concepts of our Universe to the description of another (hypothetical) Universe. But in that other Universe there is no mechanism by which the concept of numbers emerges, since there are no quantities to be counted or compared, or anyone to compare them.

I've got to pull you up on the comment:

"Numbers were not created by a supernatural being. No deity, no matter how powerful, could have created a universe where numbers didn't exist, or didn't work."

The second sentence seems to have no grounding as far as I can see. True, we can't imagine a universe in which if you were placed there as a probe numbers would not exist. I'd simply claim that because we can't conceive of a universe where numbers don't exist, we also can't conceive of a deity creating such a universe. This seems to be a fault of our own imagination and not of the power of an all-powerful creator.

In fact the first of the above sentences seems to be based on a bias that in a universe which was created by a deity there are external things of which it had no control. This seems simply to be a particular point of view. If you're going to dismiss someone's opinions you need to look at your own biases for belief.

It can't account for logical wierdness like GÃ¶del's incompleteness theorem, which relies on the logical structure of numbers.

fwiw, GÃ¶del (and arguably Cantor) believed firmly that the human mind can, in principle, resolve any rational question that it can pose. In his view, his incompleteness results only demonstrated the poverty of the formalizations we have, and there should be no reason we could not develop stronger methods which could solve all problems we could pose. ("Stronger" in a more fundamental sense than e.g. strengthening the Peano axioms with ad hoc axioms which solve whatever particular problem we might consider.) This is fairly surprising from the point of view of the popular conception of GÃ¶del, but he devoted much of his life to such considerations. His method of attack was apparently based in Husserl's phenomenology.

There is a good paper on this very topic I happen to be (slowly) reading right now in the Bulletin of Symbolic Logic, I believe it's posted online. The title is "GÃ¶delâs Program revisited".

I'd simply claim that because we can't conceive of a universe where numbers don't exist, we also can't conceive of a deity creating such a universe.

Isn't inconceivability supposed to be an attribute of God? (c.f. Job)

This would be a great post even with no mention of O'Leary silliness, but poor Denise gets PWND anyway, making this a double treat!

I can't believe there are still people who believe in Cantor's theory. It's complete silliness and makes the people who believe in superstition look like the smart ones by comparison. Cantor was well known to treat infinity as a finite number and that's all he's doing with his theory. It doesn't mean anything.

Yes, her name is Denyse O'Leary - her blog post that this was quoted from is here: http://post-darwinist.blogspot.com/2009/10/mathematics-gap-tooth-creati… . I'm also pretty sure that, while the Discovery Institute does reference her "work", she's not officially associated with them.

Now, I do agree that her statement is entirely ignorant about mathematics, but as creationists go, she at least tries to exist on the reasonable side of things. I have to respect her a little bit, for, in other works, trying to come to terms with the multiverse theory in terms of religious belief (which I haven't really seen too many try to do). As an ardent atheist, I think she's wrong and very ignorant of fundamental physics, but I at least respect her for being a tiny bit more consistent than most.

O'Leary's viewpoint (despite her not being familiar with the total ordering or well-ordering of the reals, which is really hard to fault a lay person for) is not all that dissimilar to that of a mathematical Platonist. Platonists believe that all mathematics, all numbers, truly exist in the universe, and we are only discovering them as we go, not inventing. As a formalist, I think this is silly, but it's still a widely held belief, by members of the mathematical community even. Mathematical constructivists even exist, as practising mathematicians, where the only numbers (or functions, or well-orderings, etc.) that exist are those that we can describe. The philosophical discussion to completely dismiss mathematical Platonism and Constructivism are more subtle than you might want to admit (and I say this as someone who disagrees with both of those points of view).

@73: Whether one "believes in" Cantor's theory or not hasn't much to do with the practice of mathematics. Fascinating work continues to build on it and will probably continue indefinitely. There is a book summarizing work from this century called "The Higher Infinite." You might find it interesting whether you regard it as "real" or mathematical "science fiction", either way, it's still fun stuff.

As far as whether it "means anything" outside of itself, that is a subtle issue and a hard case to make either way, even for the integers, as the debate on this thread demonstrates.

Umm, assuming choice every set can be well ordered. So you can "list out" the reals in a certain sense.

1) God created the universe.
2) The universe contains the concept "numbers".
Therefore ... God created "number".

Such a judgmental lot. It kills me how this group seems to pride itself on how "smart" it is simply because you are able to blather on about something someone else (probably a professor or text) taught you about numbers. We are all equally stupid and no matter what you are an expert on you are being called stupid in a multitude of parallel universes. For instance, here you mock people for their 'infantile' understanding of numbers and number theory while in a parallel blog people are mocking your use of the English language or your ignorance of philosophy or religion. Or they are making fun of how much energy of your lives you have wasted on a topic that doesn't change anything for 99.99% of the world.
By the way God must have made numbers because had anyone checked the bible they would know that Numbers is between Exodus and Deuteronomy. (that's a joke -- please go on wasting your life with blogs about how stupid people seemingly don't know the same things you do)

An ignorance about nuances of the English language and a widespread ignorance of basic underlying issues of philosophy or religion are probably not reasons for alarm. Basic innumeracy is, particularly on the part of the decision makers of our society We live in a universe that is increasingly dominated by mathematical facts of life.

The title of the Blog should have given you a clue that involved a critique of the use of mathematics. The value of blogs like this is that they provide some education along with their critiques. And if there is some one-upmanship in the comments, that has to do with being human. Consider disagreements a chance to do at least a Google search to further your understanding, or go read blogs that irritate you less.

"Anyway, I didn't get a chance to say this on the program so I will now ... All numbers are - in my view - created by God ... I tend to flub the gap-tooth creationist moron role..." -Denise O'Leary.

Denise didn't stand up for her beliefs in this interview. She had an idea of what they expected of her and withheld those views. I wonder why she would do such a thing--given how much more acceptable it is to be theistic than atheistic in America. I can only conclude that she knows how silly of a notion it is so she saved it for preaching to her choir. At least she didn't subject her metro area to that unfounded propaganda.

You can't write that number; in fact, you can't write most numbers.

goodmath — Fri, 15 May 2009 08:41:47 +0000

You can't write that number; in fact, you can't write most numbers.

In my Dembski rant, I used a metaphor involving the undescribable numbers. An interesting confusion came up in the comments about just what that meant. Instead of answering it with a comment, I decided that it justified a post of its own. It's a fascinating topic which is incredibly counter-intuitive. To me, it's one of the great examples of how utterly wrong our
intuitions can be.

Numbers are, obviously, very important. And so, over the ages, we've invented lots of notations that allow us to write those numbers down: the familiar arabic notation, roman numerals, fractions, decimals, continued fractions, algebraic series, etc. I could easily spend months on this blog just writing about different notations that we use to write numbers, and the benefits and weaknesses of each notation.

But the fact is, the vast, overwhelming majority of numbers cannot be written
down in any form.

That statement seems bizarre at best. But it does actually make sense. But for it to
make sense, we have to start at the very beginning: What does it mean for a number to be describable?

A describable number is a number for which there is some finite representation. An
indescribable number is a number for which there is no finite notation. To be clear,
things like repeating decimals are not indescribable: a repeating decimal has a finite
notation. (It can be represented as a rational number; it can be represented in decimal notation
by adding extra symbols to the representation to denote repetition.) Irrational
numbers like π, which can be computed by an algorithm, are not indescribable. By
indescribable, I mean that they really have no finite representation.

As a computer science guy, I naturally come at this from a computational
perspective. One way of defining a describable number is to say that there is
some finite computer program which will generate the representation of
the number in some form. In other words, a number is describable if you can
describe how to generate its representation using a finite description. It
doesn't matter what notation the program generates it in, as long as the
end result is uniquely identifiable as that one specific number. So you could use
programs that generate decimal expansions; you could use programs that generate either
fractions or decimal expansions, but in the latter case, you'd need the program to
identify the notation that it was generating.

So - if you can write a finite program that will generate a representation
of the number, it's describable. It doesn't matter whether that program ever finishes
or not - so if it takes it an infinite amount of time to compute the number,
that's fine - so long as the program is finite. So π is describable: it's
notation in decimal form is infinite, but the program to generate that representation is finite.

An indescribable number is, therefore, a number for which there is no notation,
and no algorithm which can uniquely identify that number in a finite amount of space. In theory, any number can be represented by a summation series of rational numbers - the indescribable ones
are numbers for which not only is the length of that series of rational numbers
infinite, but given the first K numbers in that series, there is no algorithm
that can tell you the value of the K+1th rational.

So, take an arbitrary computing device, φ, where φ(x) denotes the result of
running φ on program x. The total number of describable numbers can be no larger than
the size of the set of programs x that can be run using φ. The number of programs
for any effective computing device is countably infinite - so there are, at most,
a countably infinite number of describable numbers. But there are uncountably many
real numbers - so the set of numbers that can't be generated by any finite program
is uncountably large.

Most numbers cannot be described in a finite amount of space. We can't compute with
them, we can't describe them, we can't identify them. We know that they're there; we can
prove that they're there. All sorts of things that we count on as properties of
real numbers wouldn't work if the indescribable numbers weren't there. But they're
totally inaccessible.

goodmath Fri, 05/15/2009 - 04:41

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Yes that has always been awesome (in the literal sense) to me. All the numbers humanity ever could describe, ever, even to infinite time, are just a sprinkle on the number line.

"The number of programs for any effective computing device is countably infinite..." Prove it. OK, I'm not that lazy, I looked it up. Basically, all you need to know is that "finite computer programs can always be represented as a finite text file." Order the programs alphabetically... 1:1 mapping with the natural numbers... blah blah blah...

What if the program describing the number *isnt* finite, but can be described itself by a finite program? Like a genetic algorithm, for example.

Would that number count as undescribable, or would it simply open up a new class of describable numbers, or would they already somehow be included in your definition?

when explaining this to people i often like using a geometric example:
draw a line segment on a piece of paper, consider the length of that segment to be 1. if you put a point on that line, the length from that point to the end of the line will virtually always be an indescribable number.

What about numbers which are describable, but we don't know what the description is?

Consider the smallest even number >2 which is not the sum of two prime numbers, if there is such a number, or else 2. We know that 2 is describable, and we know that all even numbers are describable, so, in either case, that number is describable.

Re #3:

That's not an escape. It comes back to basic computation theory: if you could write a program that generates a program that does X, you can write a program that does X.

If you could write a program that could generate an infinitely long program that generated a number, you could bypass the middle step and just write a program that generated the number.

Re #5:

Doesn't matter. Knowing which numbers are describable, and knowing how many numbers are describable are two different problems. In terms of computation, it comes back to the good old halting program; you know that some programs will halt, and some won't - but you can't say which ones. But you can reason about properties of various subsets of programs, even if you can't always tell which programs are in which subsets.

I'm confused about pi. Where is the finite program that can compute the infinite digits of pi? Does it matter that such a program might need infinite memory to run or you don't consider memory a part of the program? All programs to compute pi that I know have some upper limit on the number of digits they can compute.

This isn't the best definition of describable to use. There are more general definitions. For example, I can describe in an intuitive sense a number in the following way:
List all possible Turing machines in some order (we can do that explicitly). Construct a number a as a having a 1 in the nth digit iff the nth Turing machine halts on the blank tape. This number is intuitively describable but is not describable under your definition.

A slightly more general definition which includes all of yours but requires more detail to be more precise is that the number can be uniquely specified in ZFC using a finite statement. The basic result still holds in this larger setting.

TomS, all *integers* are describable in the sense Mark's talking about here. (Whether or not we know what they are.) In particular, e.g., if you pick a formalization of number theory and a way to encode proofs therein as integers, then the integer corresponding to the shortest proof of Fermat's last theorem is describable, even though we'll never know what that is.

But a *real number* that there's no way to compute isn't describable in Mark's sense, although of course the fact that we don't know a way to compute a number doesn't mean that there isn't one.

Re #8:

There are a variety of ways to compute pi, mostly based on algebraic series. For example, see:

http://mathworld.wolfram.com/PiFormulas.html

Those formulations are all built on an infinite process, where each step gets you closer to the actual value of pi. Any of them can be implemented as a program that outputs a series of progressively better approximations of pi; further, any of them can be implement in a form which outputs individual digits of pi in sequence after they're sure that a given digit won't be changed by further steps.

Of course, we are talking about theoretical programs. You can't "output" an infinitely long representation without having infinite storage available. In the typical formulations of theoretical computers, you do have infinite storage available.

Re #9:

I *think* that we're still looking at the same basic definition. For the example that you used, we can generate that as the limit of the result of a computation. Take a turing machine whose first approximation is 0. Then it runs one step from program 0. IF that's "halt", it writes a 1 in position 0. Then it runs one from program one. Then one step from 0, one step from 1, one step from 2. Then one from 0, one from 1, one from 2, one from three. The usual diagonal structure used for iterating over all programs. As it adds machines to the sequence that it's iterating over, it sets their initial value to 0; if/when they halt, it writes a 1 in that position. The result of that process in the limit is the number you're describing. You'll never get it exactly, and you'll never know when a given bit has stabilized; but if it runs forever, it will generate the number. So the number is describable by a non-terminating finite program.

Or am I missing something?

'It doesn't matter whether that program ever finishes or not - so if it takes it an infinite amount of time to compute the number, that's fine - so long as the program is finite.'
That doesn't really make sense, because if the program doesn't terminate, it doesn't produce any description. I think it might be better to say that the program produces progressively better approximations, i.e. it is a computable number.

Another important distinction that should be made is the difference between computable and describable. E.g. your favourite number, Chaitin's Omega is definitely describable (i.e. provably there is such a unique number, under appropriate premises), but not computable.

What intrigues me about this whole thing is that all of mathematics really condenses into a finite set (at any given time there is only finite, if expanding, space for mathematical scripture), even though conventional theories contain all these uncountable sets.

#8: Try google: "program that computes pi". A kid taking computer programming for the first time can write such a program easily.

#4 brings up an interesting conundrum. Whereas by drawing a finitely long line on a piece of paper, and whereas you assume it to be of length one, and whereas you place a point randomly on said line, therefore the distance from the point to the end of the line is likely to be an indescribable number. Yet, by undertaking this procedure, you've just described it, symbolically and in finite space, by drawing a line with a dot on it.

True, the representation doesn't consist of decimal figures with which you can readily perform arithmetic with. Nonetheless, the space taken to represent the undescribable number is itself finite and expressed visually. In other words, it has a description.

Is "finite amount of space" the right criteria? This is very informal, but I could imagine a decimal expansion where each digit necessarily takes a larger and larger amount of space to generate, so the space requirements are effectively unbounded, but the amount of space needed to calculate the *next* digit is always bounded.

The key criteria in my thinking is the program "writing" the number (I can't help but think in terms of decimal expansions of pi, it seems) has to be productive - it has to always be able to generate some more of the number in a finite amount of space and time, but the amount of space required as the decimal expansion streams out could be arbitrarily large.

#15 I don't see this as a conundrum, but I could be wrong. What you describe is not a finite procedure for generating a particular number, it's a finite procedure for finding numbers in a range of numbers. In other words, it doesn't uniquely characterize any single number - the criteria for some number being describable is there is a finite procedure that allows for calculating that number.

But I don't know the theory behind this deeply, so this is, at best, informed speculation.

Finite amount of space refers to the length of the program, not the amount of memory it allocates.

Just use computable math and intuitionistic logic, leave all the other numbers and classical logic to the mathematicians.

No infinity, no problem.

#18 Aaah yes, must have misread that part of the post.

I think that I agree with #15 that we're ignoring the whole set of geometrical descriptions of numbers. For example, the geometric descriptions of Ï and â2 are rather trivial even though we can't get them exact without the ability to draw perfect circles or perfect right angles (much like the infinite algorithm that would be needed to express these numbers as complete decimals).

@Maya Incaand: Hear, hear. Too bad that the classic theories tend to be simpler. Though I've half a mind restarting Hilbert's program from Conway's surreal numbers.

#15, your idea reminds me of the story of an alien visiting Earth. The alien wants to take as much of Earth's culture and knowledge back with it as it can, but its spaceship is pretty small. The alien's solution? It takes all of the information that it's gathered, joins it into a huge binary file, then interprets it as a binary decimal expansion: 0.1011010110110001...

The alien then takes a carefully calibrated rod from its spaceship and trims it down so that it's a fraction of its old length equal to the number just calculated. The alien can now return home with all of Earth's knowledge.

Of course, this would quickly run into problems with physics. Assuming the rod was 1 m long to start with, the alien could record about 116 bits using this method before requiring sub-Planck length resolution. This also assumes that the universe isn't discrete. If there's an uncountable number of positions in space, just how many are there? Aleph_1? More than aleph_1? Beth_1? To quote Scott Aaronson, "We don't want the answers to "physics" questions to depend on the axioms of set theory!"

@mds
I think we don't want the answers to *any* questions to depend on the axioms of set (ZFC) theory!

Mark, yes you are correct that in this case they are the same. I'm not convinced that the definitions are the same. For example, if I had said instead of halts on the blank tape I had said "halts on all input tapes" then there's no obvious way to generalize to that question. Moreover, even with an oracle that answers the halting problem one cannot answer whether a given machine will halt on all inputs. So I suspect that this formulation does in fact give a more general result.

"It doesn't matter whether that program ever finishes or not"

The program doesn't have to finish, but it has to eventually generate the number to any particular accuracy.

I wrote about this topic a while ago: http://igoro.com/archive/numbers-that-cannot-be-computed/

So, an example of a undescribbible number is infinite sequence of digits generated by a true quantum generator of digits.

Re #27:

Yes, exactly. And per information theory, the vast majority of numbers are, basically, something that could be generated by a true random digit generator.

The way that I actually like to think of it is in terms of compressibility. I know that it's a strange way of approaching it to most folks, which is why I didn't discuss it in the main post. But you can think of a program that generates a representation of a number as a sort of compressed form of the number. And as the standard proofs about compression show, most strings are random and non-compressible. Same holds for numbers: most numbers don't have enough structure to allow us to describe them by anything less than their full infinite representation.

Just wanted to pop in and thank you for this blog Mark. Your writing is clear and engaging and the topics you explain are always thought-provoking. Thanks so much!

Well I guess it gives an whole new meaning to the Friday Random Ten theme.

Of course, this is talking about the set of Reals; that is to say, where every upper-bounded sequence of numbers has a least upper bound. Each real number is actually the limit of some bounded sequence of rationals. Two reals are considered equal if for any arbitrary rational Îµ, there is always some natural number of terms N past which the two sequences are within Îµ of each other. Computable real numbers may be represented by the program which recognizes a term in that bounded sequence by halt-accepting. (However, Turing's halting problem means that there are sequences for which there can be no program, no matter what particular Ï you use. On the other hand, Ï can be upgraded with halting oracles if need be... at least, if you're a mathematician.)

I think the Reals requires having the Axiom of the Power Set, but I'm not sure. If so, and if you are willing to take Refutation of the Power set, accepting that there may not be a Power Set for infinite sets (such as the Integers)... there's only one Cardinality of Infinity, and no Reals. At that point, you may take your computing device Ï, the "set" of Ordinals, and denote any "computable" number by the (finite) representation of what level hyper-Ï computing device is required and the (finite) program recognizing the terms of the upper-bounded limit sequence generating the number.

However, most mathematicians like having the Reals instead of just the Countable-Ordinal Computables.

I think it would be more correct to say that the quantity of numbers that we cannot write is not only "most" numbers, but would in fact be an infinite quantity of numbers. So, how many numbers can we write? (1/infinity)?

Re #4:

By creating a physical representation (a line and a point) you have made the number describable. You could count the number of molecules it takes to create the line (with a few assumptions), then count the number of molecules that it takes to go from one end to the point. So not only is the number describable, it's rational.

It's like Feynman's argument with topologists from "Surely You're Joking, Mr. Feynman" about how they could rearrange the pieces of an orange peel, and make it bigger than the sun. Feynman said, "But you said an orange! You can't cut the orange peel any thinner than the atoms."

http://www.physicsforums.com/archive/index.php/t-182942.html

A describable number is a number for which there is some finite representation. An indescribable number is a number for which there is no finite notation.

An "indescribable number", however, does have a form.. it's simply an infinite form, contrary to this statement:

But the fact is, the vast, overwhelming majority of numbers cannot be written down in any form.

Perhaps I am picking nits, but when you say cannot be written in any form, to me that sounds like it has neither a finite nor an infinite representation. That is, it must have no representation at all. I do not believe that any number has no representation in this sense.

Indeed, any number (real or complex) could be realized (or represented if you prefer) as a limit point of a sequence. (#31 alludes to this). To me that is perfectly acceptable from... especially since those numbers you define as describable form a set of measure zero.

Erick: two different kinds of infinity there.

The infinity of the natural numbers is referred to as "Aleph 0", a "countable" infinity. The real numbers have a larger infinity, which the naturals have no bijection (one-to-one and onto) mapping. We can write a countable infinity of numbers, but that still leaves an uncountable infinity of inexpressible Reals left.

Weirdly, the inexpressable reals DO have a one-to-one and onto mapping to the whole of the reals. (Encountering this sort of weirdness for the first time often results in English majors giving up on higher math.)

Paul: Indeed, any number (real or complex) could be realized (or represented if you prefer) as a limit point of a sequence.

The catch is, not every infinite sequence of rational numbers can be finitely represented. And if it can't be finitely represented, you can't write it.

However, if you allow for infinite representations, yes; all real numbers can be expressed by some infinite sequence of digits after the decimal point.

Would it not be easier to just look at the notion of representation without going on about computability. I think that the "computability approach" makes the picture to fuzzy.

If we look at how we can write numbers down we just start by looking at how many different symbols we have at our disposal. We can agree that there is a finite set of symbols but just in case we assume that there are a countable infinite amount of symbols.

Now we piece our symbols together to strings. We can agree that a string can at most have a countably infinite amount of symbols.

So how many numbers can we represent by such a string?

The answer is countable infinite and therefor we can not represent all existing numbers since there are an uncountably infinite amount of real numbers.

So my argument above would show that even if we allow an infinite amount of decimals we still could not represent every number. So there is a more profound result than that there are numbers representable by any FINITE string of symbols.

Paul (34):

That's all well and good, but it's kind-of circular:

Finite representations give us a 'bridge' between the concrete and the abstract - they can be encoded in a computer's memory, or a human mind (modulo some annoying philosophical gymnastics), and then 'operated on'.

On the other hand, an infinite representation of, say, an infinite set isn't any more accessible than just the set itself.

(I suppose one response to all this 'finitude chauvinism' would be to imagine a universe where memories were infinite, and computers/minds could do computations requiring infinitely many steps and return a result. Perhaps time in this universe would look like the "long line" in topology.)

I remember when I first realized that the set of numbers which could be described is countable. It was very disturbing just how big this whole uncountability concept makes things.

I imagined some neo-Lovecraftian story involving a horror from the beyond driving mathematicians mad by pressing an undescribable number in all of it's detail into their minds.

Arne (38): That isn't so surprising. After all, if I count each possible word as a "letter" then I have a countably infinite alphabet.

But whether I classify them as words or letters doesn't change the number of possible sentences.

Moment, please:
As we had the pleasure to learn reals in school by nested
intervals from rationals, can I construct Omega by it ?
And what about Dedekind cuts and decimal expansions ?

I find it a bit furtive to imply that Omega certainly belong to real numbers without mentioning how the reals
in question were constructed.

"We know that they're there; we can prove that they're there."

So we could inductively prove their existence, but not deductively, the limits to computational power could also be indicative of the constraints to notational systems in cognitive situations.

Just use computable math and intuitionistic logic, leave all the other numbers and classical logic to the mathematicians.

No infinity, no problem.

i = 0;
while (1) {
++i;
}

I think this is somewhat addressed by Interval Arithmetic....
http://en.wikipedia.org/wiki/Interval_arithmetic

#13, a program does not have to finish in order to have output. This Perl program never halts, but has plenty of output: print "hello world\n" while 1;

This is the same proof that most numbers are transcendental because there are only as many rational polynomial equations as integers. All computer programs can be reduced to Turing machine programs and there are only as many Turing machine programs as integers. If you count programs that halt or don't halt or blink the little green lights on your Turing machine in a certain pattern, it doesn't matter. At most, you have as many programs as integers.

There are just a whole lot more real numbers than integers.

In fact, there are so many more real numbers than integers that you can do almost all of real number mathematics without the algebraic (non-transcendental) numbers and without the rationals and without the representables.

Now, are unrepresentable numbers interesting? Consider the smallest unrepresentable number. That number is clearly interesting ...

Re #47:

Reminds me of an old UNIX fortune:

Consider the set of all numbers that haven't been considered before ... wait, they're all gone!

What I was trying to get at with my post is that the entire discussion above is that there are numbers that are not finitely representable. It is even given in MarkCC's post:

"A describable number is a number for which there is some finite representation. An indescribable number is a number for which there is no finite notation."

I think it misses the point to talk about numbers that have no finite representation. The interesting numbers in this context are those that have no representation at all, finite or infinite.

Also since I in my proof above allowed for an countable infinite amount of symbols and specified no method of representation we find that most numbers are not describable by decimal expansion. But not only by decimal expansion. They are not describable by any equation or computer program, even IF we allow for the equation / computer program to be infinitely long.

From original post:
"In theory, any number can be represented by a summation series of rational numbers."

It can not both be that this is true and that my argument is correct so I'm going to ask MarkCC to elucidate on this a little.

Arne:

I see the error now.

If your representations include infinitely long strings of digits then in fact the number of possible representations is *uncountably* infinite. This is what Cantor's diagonal argument establishes.

Also, as I said in (38): The whole point about finite representations is that they can be stored and manipulated here in the physical world (by minds and computers). On the other hand, an infinite representation is just as 'remote' from us as the thing it represents.

47: Surely there isn't such a number? There are non-describable numbers in any nontrivial interval of the reals, so the infimum of all non-describables is -infinity (and therefore not a number at all), and that of all positive non-describables is zero (and therefore describable). Which means neither set has a minimum... neither does any subset of the non-describables that you can finitely describe, because if the infimum (or supremum, or essential infimum/supremum, or just about anything else you can ask for) of such a set is a real number, it's obviously describable.

What is a notation system?

I take it that the English language doesn't count? If it did, it would seem like you can describe at least one indescribable number as follows: "The first positive indescribable number."

Though I just realized that description might not uniquely describe any number if, given one indescribable number, there's always another one between it and zero.

And no similar description would pick out an indescribable number uniquely if, given any pair D and I where D is describable and I is indescribable, there is a B which is indescribable and which is between D and I.

Which seems plausible.

Never mind then.

Kris:

Replace "smallest indescribable real number" with "smallest indescribable ordinal" and you have your paradox.

Its probably already been pointed out (I didn't read all the comments thus far) that this concept was the focus of a paper written by Alan Turing in 1936. In the process of inventing the computer, he already realized the shortcomings of computers in that they can only deal in "computable" reals. He recognized that even irrational numbers, although having infinitely many digits, they can possibly have a finite representation, and are thus computable. Unfortunately it can be shown via Borel measures that when randomly selecting a real number from a unit measure it is infinitely improbable that the selected number would be a "computable" real. There is an excellent lecture by Gregory Chaitin on these very ideas here:

http://www.youtube.com/view_play_list?p=6A6833CBFA87B5C8

Great blog, Mark. Keep it up.

Neil - what you described requires the Axiom of Choice, if I'm not mistaken.

You state, in your definition of an 'indescribable' number, that 'given the first K numbers in that series, there is no algorithm that can tell you the value of the K+1th rational. '

You can write a very simple algorithm which, given the first K values in the decimal expansion, writes down all 10 possible K+1 values. How this basic counter-example could have been overlooked is baffling.

The very fact that a decimal expansion exists means that the number is computable, so your argument makes no sense.

You need to include in your definition that the program has to halt. If it acts like you say, that 'It doesn't matter whether that program ever finishes or not', then the program I described above satisfies your conditions.

Re #66,67:

The "program" you describe doesn't output a specific number. The definition of a describable number (or at least, the definition that I'm using for this post) is that there exists a program which generates that number, and only that number - that is, that there is a program which is a description of the number.

You can write a program that generates all of the finite rational sequences - but it will never generate an infinite one. And there's no program that uniquely specifies or describes an indescribable number.

Or, to put it another way: almost all numbers are boring.

I suspect a clearer definition is one in terms of limits. The definition of a computable number x is one for which you can produce an function of N and some N such that for any Îµ, f(N)-x < Îµ. With an indescribable number there is no such f. There may be particular Îµ that you can find an f and an N, but no general case.

Something I wonder in relation to the genetic algorithm comment is whether you can produce a program that will produce a correct limit for a given Îµ with some probability p that is better than guessing. i.e. if you have a value for Îµ, can you produce a value for 0.1Îµ with p > 0.1.

Interestingly, there is a famous paradox lurking in the vicinity of the point in this post - take the sentence "The smallest positive integer not definable in under eleven words." Well, there is such an integer, isn't it? But I just described it using less than eleven words.

Many people, on encountering Berry's paradox, think it's kinda silly (just as they do with the liar which trades on the same semantic knot). What's utterly awesome, however, is the way Boolos used this one to give the until now by far shortest proof of GÃ¶del's incompleteness. The idea behind the proof is to hold a proposition that holds of x if x = n for some natural number n to be a definition for n, and then show that the set {(n, k): n has a definition that is k symbols long} is representable with GÃ¶del numbers. "m is the first number not definable in less than k symbols" can thus be formalized and shown to be a definition.

Ok. I'm not sure this is related too closely to the discussion at hand (but it is related to the title of the post), but wanted to share it anyway.

TSK: As we had the pleasure to learn reals in school by nested intervals from rationals, can I construct Omega by it ? And what about Dedekind cuts and decimal expansions ?

Yes, you can construct Omega that way... once you figure out how to construct the set of all Turing Machine programs that halt. Which, finitely, you can only do by axiom. Which acts as an Oracle, acting to increase the power of your Ï to a hyper-Ï, and giving you the possibility of a hyper-Omega.

Repeat ad infinitum et nauseam.

Arne: It can not both be that this is true and that my argument is correct

Your error is in presuming that all sequences of rational numbers have a finite representation.

Stephen Jones: The definition of a computable number x is one for which you can produce an function of N and some N such that for any Îµ, f(N)-x

...where Îµ is a rational number; and |f(N)-x|, if you want computables and not merely computable Reals. The computables spill out into the Complex plane, too.

"All sorts of things that we count on as properties of real numbers wouldn't work if the indescribable numbers weren't there. But they're totally inaccessible." Sometimes it's good to remember what Cantor himself believed: that numbers have an "intrasubjective and immanent reality." Perhaps there is a reason why they chose the following words when they put up a plaque in Halle commemorating Cantor's innermost conviction that "the essence of mathematics lies in its freedom." - "No infinity, no problem?" No infinity, no cry? Cantor didn't mind shedding a few tears over the beauty of the inaccessible. These tears can be thought of as dense and indescribable as the reals.

Whether the program halts or not is not an issue. The program does not have to halt. A program that computes all the digits of pi or the square root of two never halts. There is no last digit in either case.

The problem is that there are not enough programs. A program is an integer. There are only so many integers. There are many, many more real numbers. Programs always produce the same output each time they run. That means that there are lots of numbers out there that you cannot write a program to compute.

This is just like our host's discussion of compression algorithms. There are more long strings than short strings, so any algorithm that maps lots of long strings into shorter strings will actually have to map lots of short strings into longer strings. It's just counting.

If we want to get around this, we have to define a mechanism for programming in which the programs are real numbers, not integers.

It's possible to write descriptions that always halt.

For example, say that a function f of one variable describes a real number y iff, for any x > 0, f(x) terminates and |f(x)-y| < x

If those numbers are 'undescribable' and 'inaccessible', what are they usuful for? Can we define the set of all describable reals and prove the essential theorems about reals for that set?

Re #66:

Who said that things need to be useful?

Seriously, the basic definitions that we use for real numbers
necessarily imply that those numbers exist. You lose an awful lot of basic stuff when you try to create a reduced set of real numbers. People have done work on computable numbers, which are close to the definition of describable numbers, and you *can* do it, but it's a lot of work, and it's clear that you're really constructing a subset of the numbers. The fact that they're conceptually there is pretty
much unavoidable.

You can drop the irrationals entirely, and wind up with the rational numbers. They work pretty nicely for lots of things. But algebra breaks: a polynomial of degree N no longer has N roots, because there is no rational number that can solve many of them. In a universe of rational numbers, where does Y=2-X2 cross the X axis?

The original critique of Dembski said: "Unfortunately, most real numbers are undescribable. There is no notation that accurately represents them. The numbers that we can represent in any notation are a miniscule subset of the set of all real numbers. In fact, you can prove this using NFL."

Could you clarify how this is proved using the NFL theorem?

There is a related thread now at the n-Category Cafe blog ("boundless" is in its title, something like "Bounding the Boundless") about the philosophical history of Infinity, with special attention paid to Giordano Bruno, Isaac Newton, Isaac Barrow, and Isaac Asimov.

It might be fun for some readers here to read that also, and comment there as well as here.

n-Category Cafe has blogmasters expert in Mathematical Physics, Philosophy of Mathematics, and Computation Theory.

JC wrote:

> What if the program describing the number *isnt* finite, but can be described itself by a finite program? Like a genetic algorithm, for example.

This doesn't work. Where are the random numbers coming from? The Turing Machine is deterministic, remember. There's just the symbols on the tape, and a fixed set of rules. If the random numbers aren't included in the program, then it can't do anything; if they're included then you run into the same problem as for other programs.

What???

"Re #3:

That's not an escape. It comes back to basic computation theory: if you could write a program that generates a program that does X, you can write a program that does X.

If you could write a program that could generate an infinitely long program that generated a number, you could bypass the middle step and just write a program that generated the number. "
It is at least not so simple, because math theorems You mention about are regards first order logic. So if You where able to made some validity of second ( or higher) order logic programming, things may be quite different. The most iportant thing in Your post is conception of computality, which is only just that: conception with stright and tight definition.

In a finite universe, are there undescribable integers?

Kevin @ 73 asks:
"In a finite universe, are there undescribable integers?"

Rather than quibble about whether "finite" means space, or time, or both, and whether "undescribable" applies to integers exactly as it does to reals, then YES.

If there are N possible states of the finite universe, then there are at most N descriptions of integers that can occur in any given state of the universe. Hence there are no more than N describable integers. So if I have a set of N+1 distinct integers, including each of the N describable integers, then that set must contain an undescribable integer.

That is motivational, and not crisp enough to be axiomatical, but you see why I say "YES!"

And you should also see why it gets back to "compressibility" as Marc CC has explained it.

In your proof, you state "The number of programs for any effective computing device is countably infinite...." Remembering the roots of programming in analogue computers, the voltages, resistors, potentiometers, etc. could take arbitrary values in the set of Reals. As such, I would argue that *some* computing devices have an uncountably infinite amount of possible programs.

I took a random walk through the blogosphere and found a site with a nest of posts regarding indescribable numbers, Berry's Paradox, incomputability, Turing machines and "compressibility". And I, having small mathematical ability but some acquaintance with Douglas Hofstadter, I took the "road" (read explanation of the improbable) less intuitively obvious.
Great site and comments.

plain and simple kids, if you want indescribable numbers, look for numbers which no algorithm could generate. pi can be described by geometry, hence its identity is known and can be clearly written in code. indescribable numbers can be irrational and/or infinite and rely on currently non-existing logical framework.

an interesting question: is it possible to prove all the limits of our algorithms and hence locate where all the indescribable number definitions/generators lie?

Partial randomness and dimension of recursively enumerable reals
Authors: Kohtaro Tadaki
(Submitted on 15 Jun 2009)
http://arxiv.org/abs/0906.2812

#5: pi is easy to describe. It is, as you sure know, the ratio of radius to the circumference of a circle. A perfectly valid description. MCC never said that it has to be described in the decimal system, in fact he stated the opposite.

Can you write the these numbers or represent these numbers in Mayan?

54 or 1954?

If I take a stick that is 28 inches long and I use my dividers/compass to divide it into 6 equal sections, I will be successful. I can physically divide a 28 inch long rod into 6 equal sections. But I can't divide 28 by 6 on paper. Math seems to need some improvement. Our base 10 system does not seem to work. We need some other way to talk about numbers.

But I can't divide 28 by 6 on paper.

sure you can. 28/6 = 14/3 exactly; there's nothing wrong or odd about fractions. use a smart enough calculator and it'll throw that expression into exact fractional format for you, no questions asked.

Our base 10 system does not seem to work

no worse than any other positional-value number system. one key insight is that number bases really aren't as fundamentally important as we sometimes think they are. in many ways, strings of digits with an optional decimal point is just a notational convention; it's not something that defines what numbers really are.

Indescribable numbers don't exist unless you can give an example.

@83

There are some mathematicians who work with 'constructive' mathematics whou wouold agree with you. They work with the intuitionistic variety of logic in which the law of the excluded middle ( A or Not A ) does not apply.

However most mathematicians work with 'classical' logic in which, to prove the existence of something, it is sufficient to show that a contradiction would result if it did not exist. By definition, no example of an indescribable number can be given - so you cannot constructively prove they exist, but if they did not exist, then you can prove a contradiction with the definition of the real numbers (as a complete, archimedean, ordered field).

I believe this is a direct consequence of the Borel-Cantelli lemmas. In that if x is an irrational, and p/q is rational in lowest terms, then |x - p/q|< 1/q^2, but if you want to approximate it better, then the set of p/q such that |x-p/q|<1/q^(2+e) is a null set for any e>0.

jman said:

> plain and simple kids, if you want indescribable numbers, look for numbers which no algorithm could generate.

It's certainly not so plain and simple, after all you got it wrong. In fact, Chaitin's main claim to fame is a number named after him that is describable but that no algorithm could generate.

This argument really says that Turing Machines cannot write down all numbers.

The jury is still out as to whether or not humans have "free will" and I think that, if we do, we don't know if it would be Turing Equivalent.

So it's supposedly conceivable that a human might be able write down all possible numbers -- or am I missing something?

@87:

You're missing something.

What the "undescribable" thing really means is that no notation can possibly represent the vast majority of numbers. And since any combination of a finite number of notations is still a notation, then no finite combination of notations can possibly represent the vast majority of numbers.

If you start allowing an infinite number of notations, then things start getting weird. But still, you wind up with even any combination of a countably infinite number of notations can't represent most numbers.

So you end up with the fact that to represent all numbers, you need an uncountably infinite number of notations. And if you do that, you wind up right back where you started: now you have undescribable notations.

I have a problem with the statement: "The number of programs for any effective computing device is countably infinite..."

This is true only if all programs are finite. Otherwise, it's trivially not true. Let Y be the set of all programs. Let X be the set of all finite subroutines (a library of all finite subroutines if you like). This set is at least countably infinite and also this set is a subset of all programs Y, since some programs consist of single finite subroutine.

But, the power set of P(X) (i.e. set of all subsets of X) is also subset of Y, for every combination of subroutines from X forms a (possibly infinite) program. But the power set P(X) in uncountably infinite if X is countably infinite (which it is). Since we have shown that P(X) is subset of Y, so Y must be uncountably infinite.

Mark,

You are making this more confusing than it needs to be. You defined your representations to be finite. You then say one way to approach this is to consider programs generating these representations. But if the representations are finite, this adds nothing. You can simply write programs that print out the representation. Talking about programs adds nothing. Programs only become interesting if you consider infinite representations (like decimals) because then you can generate some of those with finite programs. Every finite string is computable so talking about computers simply adds a level of indirection.

My main point is that computability is not the same as describability but you're giving the impression it is. (Consider, eg. Chaitin's constant.) In fact, I think you ought to have made this distinction clearer.

#2: To put it another way:

Natural numbers (countably infinite) â programs that run â programs that terminate

Assuming that some of the outputs of the last set of programs are unique descriptions of numbers, that can't describe all numbers, since a subset of a countably infinite set can't cover an uncountably infinite set. That still holds if each output is a description of a finite number of numbers, since the union of a finite number of countably infinite sets is itself countably infinite.

What if a sequence of symbols could denote an infinite set of numbers, though? Would the above proof still hold? Does that depend on whether the set of possible encodings is countably or uncountably infinite?

Actually, never mind that last bit. The set of possible encodings must be countable, since you can just take the (finite) set of characters currently used to encode meaning and start enumerating, a subset of those strings will describe all possible encodings.

I'm pretty sure the cross product of two countably infinite sets is still countably infinite, so the proof above still holds even given that one output string can represent more than one number (possibly an infinite number of numbers).

I find the idea that most numbers are undescribable interesting. I was looking forward to reading about how you'd quantify. Wanna write a post on that?

this is one of my favorite topics related to numbers. It makes me think that the "real" numbers are oddly misnamed.
I know that numbers aren't strictly "real", but still, if a number can never be used, can never even be talked about specifically, then what the hell is this number. how can this number even be said to BE something!

Basics: Significant Figures

goodmath — Wed, 04 Mar 2009 19:55:07 +0000

Basics: Significant Figures

After my post the other day about rounding errors, I got a ton of
requests to explain the idea of significant figures. That's
actually a very interesting topic.

The idea of significant figures is that when you're doing
experimental work, you're taking measurements - and measurements
always have a limited precision. The fact that your measurements - the
inputs to any calculation or analysis that you do - have limited
precision, means that the results of your calculations likewise have
limited precision. Significant figures (or significant digits, or just "sigfigs" for short) are a method of tracking measurement
precision, in a way that allows you to propagate your precision limits
throughout your calculation.

Before getting to the rules for sigfigs, it's helpful to show why
they matter. Suppose that you're measuring the radius of a circle, in
order to compute its area. You take a ruler, and eyeball it, and end
up with the circle's radius as about 6.2 centimeters. Now you go to
compute the area: π=3.141592653589793... So what's the area of the
circle? If you do it the straightforward way, you'll end up with a
result of 120.76282160399165 cm².

The problem is, your original measurement of the radius was
far too crude to produce a result of that precision. The real
area of the circle could easily be as high as 128, or as low as
113, assuming typical measurement errors. So claiming that your
measurements produced an area calculated to 17 digits of precision is
just ridiculous.

As I said, sigfigs are a way of describing the precision of a
measurement. In that example, the measurement of the radius as 6.2
centimeters has two digits of precision - two significant
digits. So nothing computed using that measurement can
meaningfully have more than two significant digits - anything beyond
that is in the range of roundoff errors - further digits are artifacts
of the calculation, which shouldn't be treated as meaningful.

The rules for significant figures are pretty straightforward:

Leading zeros are never significant digits. So in "0.0000024", only the "2" and the "4" could be significant; the leading
zeros aren't.
Trailing zeros are only significant if they're measured. So,
for example, if we used the radius measurement above, but expressed
it in micrometers, it would be 62,000 micrometers. I couldn't
claim that as 5 significant figures, because I really only measured
two. On the other hand, if I actually measured it as 6.20 centimeters, then I could could three significant digits.
Digits other than zero in a measurement are always significant
digits.
In multiplication and division, the number of the significant
figures in the result is the smallest of the number
of significant figures in the inputs. So, for example,
if you multiple 5 by 3.14, the result will have on significant
digit; if you multiply 1.41421 by 1.732, the result will have
four significant digits.
In addition and subtraction, you keep the number of
significant digits in the input with the smallest number of
decimal places.

That last rule is tricky. The basic idea is, write the numbers
with the decimal point lined up. The point where the last significant
digit occurs first is the last digit that can be significant in
the result. For example, let's look at 31.4159 plus 0.000254. There
are 6 significant digits in 31.3159; and there are 3 significant digits in 0.000254. Let's line them up to add:

    31.4159
  +  0.000254
-------------
    31.4162

The "9" in 31.4159 is the significant digit occuring in the
earliest decimal place - so it's the cutoff line. Nothing
smaller that 0.0001 can be significant. So we round off
0.000254 to 0.0003; the result still has 5 significant
figures.

Significant figures are a rather crude way of tracking
precision. They're largely ad-hoc. There is mathematical reasoning
behind these rules - so they do work pretty well most of the time. The
"right" way of tracking precision is error bars: every measurement has
an error range, and those error ranges propagate through your
calculations, so that you have a precise error range for every
calculated value. That's a much better way of measuring potential
errors than significant digits. But most of the time, unless we're in
a very careful, clean, laboratory environment, we don't really
know the error bars for our measurements. Significant digits
are basically a way of estimating error bars. (And in fact, the
mathematical reasoning underlying these rules is based on how
you handle error bars.)

The beauty of significant figures is that they're so incredibly
easy to understand and to use. Just look at any computation
or analysis result described anywhere, and you can easily see if
the people describing it are full of shit or not. For example, you
can see people claiming to earn 2.034523% on some bond; they're
not, unless they've invested a million dollars, and then those last
digits are pennies - and it's almost certain that the calculation
that produced that figure of 2.034523% was done based on
inputs which had a lot less that 7 significant digits.

The way that this affects the discussion of rounding is
simple. The standard rules I stated for rounding are for
rounding one significant digit. If you're doing a computation
with three significant digits, and you get a result of
2.43532123311112, anything after the 5 is noise. It doesn't
count. It's not really there. So you don't get to say "But
it's more than 2.435, so you should round up to
2.44.". It's not more: the stuff that's making you think it's
more is just computational noise. In fact, the "true" value is
probably somewhere +/-0.005 of that - so it could be slightly more
than 2.435, but it could also be slightly less. The computed
digits past the last significant digit are insignificant -
they're beyond the point at which you can say anything accurate. So
2.43532123311112 is the same as 2.4350000000000 if you're
working with three significant digits - in both cases, you round off
to 2.44 (assuming even preference). If you count the trailing digits
past the one digit after the last significant one, you're just using
noise in a way that's going to create a subtle upward bias in
your computations.

On the other hand, if you've got a measured value of 2.42532, with
six significant figures, and you need to round it to 3 significant
figures, then you can use the trailing digits in your
rounding. Those digits are real and significant.
They're a meaningful, measured quantity - and so the correct rounding
will take them into account. So even if you're working with
even preference rounding, that number should be rounded to three sigfigs as 2.43.

goodmath Wed, 03/04/2009 - 14:55

Tags

basics

numbers

sigfigs rounding numbers basics

Thanks for this explanation. I guess my objection to your last post was that I was thinking of a situation like you described in your last paragraph here, when you were describing the rounding rules for a situation akin to that described in the penultimate paragraph here.

If you're doing a computation with three significant digits, and you get a result of 2.43532123311112, anything after the 5 is noise. It doesn't count. It's not really there.

Why? I mean, isn't rounding to 2.44 more likely to yield an answer close to the true value? What makes 2.43 (assuming you favor odds in the last sig fig) a better approximation of what the error bars do than 2.44?

And for that matter, aren't error bars just an approximation for what we should "really" be using, which is probability distributions over all possible values?

Have you seen Chris Mulliss's work on this topic? He ran a bunch of monte carlo trials with different sig fig methods, and found that for mult/division and exponentials, the standard methods are less accurate than other methods. Specifically, for those operations, he recommends adding one digit onto the least significant argument of the input terms.

Here's his results for the standard method:
http://www.angelfire.com/oh/cmulliss/standard_rounding_rules_summary.htm

And for his "improved" method:
http://www.angelfire.com/oh/cmulliss/recommended_rounding_rules_summary…

Error bars and by association significant digits, have always made me quite uneasy. Maybe you can shed some light Mark. My perplexion can be summarized as follow: What well stated and correct math construct are they an approximation of?

It seems to me that the problem is a result of the fact that we are thinking at the interface between theoretical real numbers and real world measurements along with the fact that most real numbers cannot precisely be represented in an finite amount of time and space. In a theoretical setting, when not using error bars or significant digits the standard practice with real numbers seems to be to write down a bunch of digits after the decimal and then assume it represent the same thing as if we had followed by an infinite number of zeros, an assumption which also makes me quite uneasy.

But what happens in the real world where we must manipulate numbers that don't have infinite precision? We can set intervals and follow the rules but what do these intervals represent? Is it a limiting bound that "true" values are assumed never to cross? Is it a measure of variance, an interval that signals that a certain proportion of the samples are known to be within? If so, can we assume a central tendency to the distribution of the samples? And if so, wouldn't it make sense to keep more digits to know where the center of the tendency should be? Otherwise aren't we trowing out information? But then how many digits should we keep?

Furthermore, the numbers used to represent the intervals, should they have a confidence interval too? Recursively?? How many digit should we write on each numbers in this example: 3.56 +-0.56+- 0.045 +-...? What mathematical principles govern all of this?

My laymen intuition is that there is an information theoretic explanation to it all. That it might have to do with the diminishing returns on information content of extra digits in the face of rough measurements. A kind of criterion meant to save our efforts which justifies not bothering with too many digits. I am not a mathematician but this has been quite the enigma for me and I really wish someone would point me towards some insight. I really feel like I am missing a fundamental part of mathematics that is key to understanding the relation between real numbers and the real reality. Does anyone have a clue?

Good Post! I was taught these rules at least 6 years ago, in my basic high school science courses. Now as an engineering student I am constantly amazed at how few of my classmates know them. They'll leave things with 5 or 6 sig figs when they were only give a measurement with 2, and if you ask them why they did they'll give you a blank stare. I don't think this really gets taught to kids in public school, at least in AZ (I was in a good private school in HS), and I know no one ever bothered to explain it to us as freshmen.

The real area of the circle could easily be as high as 128, or as low as 113, assuming typical measurement errors.

Of course these doesn't stop some atheists from criticizing 1KI 7:23, which describes the circumference of a circle measuring 10 cubits as being 30 cubits.

They're largely ad-hoc. There is mathematical reasoning behind these rules - so they do work pretty well most of the time.

Again, I pretty much agree with you. However, you should take a look at sigma-delta modulators (SDMs). These devices can, for example, use a single bit resolution analog to digital converter to come up with a measurement that has many more bits of resolution, e.g., 12 bits. And the 1-bit A/D converter doesn't even have to be all that balanced to get good results. I have a basic grasp of SDMs, but still have difficultly explaining to others how taking a low-precision measurement repeatedly can generate high resolution estimates. I can convince myself from time to time, when I decide to look at it again, but I soon forget the details.

replace "these doesn't" with "these observations don't"
replace "measuring 10 cubits" with "having a 10 cubit diameter"

Typo: "will have on significant digit;" - should probably be "will have one significant digit;".

I agree with #3 and that's what I was always taught to do. Include the first "insignificant" digits from the imput in the calculation and then round the result to the correct number of significant ones.

I also agree with #5. I'm sick of seeing papers in peer reviewed medical journals with ridiculous degrees of spurious precision. It's something I pick up when I referee but clearly others don't.

What about the distinction between accuracy and precision? I was thaught that these are two different things. The precision is the number of digits used, regardless of whether it's justified. In Mark's example, 2.034523% would have a precision of six decimal places, but likely an accuracy much less than that (and therefore the precision used is not justified).

I agree with some of the commenters before me that significant digits (or error bars, of which "significant digits" are just one example) are a very weird approximation of the concept of uncertainty.

I would expect in most cases that the "real" value is well modeled by something like a Gaussian distribution around the measured value (with as much precision as possible, no reason to "drop digits" there). "Significant digits" or error bars suppose a uniform distribution within a given interval.

When you do your calculations using the distributions (assuming they are symmetric) your result is a distribution centered around the value you get when calculating using the centers of your initial distributions (if I'm not mistaken).

Dropping digits at any point of your calculation before obtaining the final result just arbitrarily shifts the centers of your distributions. Do you have any reason to believe that this would improve the model you're using?

If you obtain 2.43532123311112 as the center of your distribution, why would you possibly shift it to 2.435 and then suddenly claim that that's as close to 2.43 as to 2.44? It clearly is not. It may be (depending on the shape of your probability distribution) that 2.43 is almost as probable as 2.44, but arbitrarily shifting your distributions around is certainly not the best way to come to that conclusion.

On your point 4 about multiplying 5 by 3.14, I would disagree, as you are assuming that the 5 is a rounded value. It may however be exact.
Imagine for example changing a 5 dollar bill to spome currency where you got 3.14 to the dollar. Then an answer of 15.70 would make perfect sense.
I know you've tried to make the post simple, but context is very important here, and there are other times when you've over-simplified.
Writing as a statistician, I'd say that if you found the mean of about 100 integer values, it would be quite reasonable to give the answer correct to 2 decimal places, which could well be a value with 2 or more significant figures than the original values. Similar results would occur in almost any statistical computation, from Standard Deviation onwards. I would agree that there are many people who give values to a completely spurious level of accuracy; I saw a case recently when something like Â£20000 in 1870 was said to be equivalent to 1234567.89 present day pounds.

Imagine for example changing a 5 dollar bill to spome currency where you got 3.14 to the dollar. Then an answer of 15.70 would make perfect sense.

It would also make perfect cents.

If the 5 is exact, then you aren't multiplying 5 by 3.14. You are multiplying 5.0000000...(with infinite significant zeroes) by 3.14.

If the exchange rate is exactly 3.14 (e.g. if the mystery currency divides into even hundredths and this a real cash transaction, not an electronic exchange where you can have portions of the smallest unit of currency), then you are really multiplying 5.0000000... by 3.1400000000..(again with infinite sig figs). The result in this case has infinite sig figs, but would probably be listed as 15.70 because we don't need to know about fractions of the smallest unit of currency when counting out cash.

But the original statement is correct. When scientists are using sig figs, saying you multiply 5 by 3.14 means the 5 is measured to 1 sig fig.

Those of us who grew up using slide rules to do calculations understood these principles fairly well. Another skill we learned was to estimate the magnitude of our calculation first, so that we didn't put the decimal in the wrong place. It's amazing how first calculators then computer spreadsheets led to such ridiculous claims of "accuracy". I can't remember how many times I had to review this with the talented young engineers from good schools that I managed, but there's obviously something missing in the way we teach simple mathematical concepts these days.

Rounding and Bias

goodmath — Sun, 01 Mar 2009 20:32:28 +0000

Rounding and Bias

Another alert reader sent me a link to a YouTube video which is moderately interesting.
The video itself is really a deliberate joke, but it does demonstrate a worthwile point. It's about rounding.

The overwhelming majority of us were taught how to round decimals back in either elementary or middle school. (I don't even recall exactly when.) The rule that most of us were taught is:

If the first digit after the rounding point is 0, 1, 2, 3, or 4, then round the previous digit down;
If the first digit after the rounding point is 5, 6, 7, 8, or 9, then round the
previous digit up.

Here's the problem: those rules are wrong.

The problem is that if the first digit after the rounding point is zero, you're
not really rounding - that is, you're not doing anything that changes the value of the data point. But if the first digit after the rounding point is 5,
then it's exactly halfway in-between; it's not closer to the either the rounded up value or the rounded down value - it's exactly between them. Always rounding 5 up will create a bias, because it's taking the point at the middle, and shifting it as if it were closer
towards the upward side.

To demonstrate, let's try an easy example. Suppose we've got the following set
of numbers: {0, 0.5. 1, 1.5. 2, 2.5, 3, 3.5, 4, 4.5}. Let's compute the mean
of those numbers: 22.5/10 = 2.25.

Now, let's round them off: {0, 1, 1, 2, 2, 3, 3, 4, 4, 5}; and then compute the mean: 25/10 = 2.5.

With the standard rounding rule, we've biased the numbers upwards enough to create a significant error!

The correct way to round is to randomly round 5s either up or down. The standard rule, used in most scientific settings, is to pick either odd or even as the "preferred" outcome, and to always round 5s towards the preferred outcome. If we try that with our example, using
preferred even, the rounding is {0, 0, 1, 2, 2, 2, 3, 4, 4, 4}. Taking the mean of that, we get 22/10 = 2.2 - which is significantly closer to the mean of the original numbers than the
mean rounding 5s up. The practice of rounding up adds a systematic bias to the data. It's a very small systematic bias, but it's a real one.

Does it matter? Not usually. As the commentary to the video points out, over the space of a couple of years, that systematic error in rounding gas prices amounts to about a dime. For most things in our daily experience, the difference between random rounding and upward rounding for 5s is just not significant. But if you're doing statistical analysis of
large quantities of data, or you're doing computations that rely on a high degree of
precision, then it can introduce enough error to foul your results. If you're doing statistical analysis, it can do things like make an insignificant result appear to be statistically significant. If you're doing high precision computations for things like
navigation of a space probe through a gravitational slingshot, it can introduce enough error
to crash your probe.

goodmath Sun, 03/01/2009 - 15:32

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Mark,

Your explanation only works if you're removing exactly one significant digit when rounding (ie. written as real numbers, the 0 or 5 that gets chopped off is followed by an infinite string of 0's). If you assume that you are very likely to encounter a non-zero digit somewhere beyond the digit that you're rounding off, then lopping off a 0 is indeed (almost) always rounding down, and also the "rounded up" value is (almost) always closer to the "true" value than if you just lopped off the 5 and rounded down.

What #1 said.

#1 - nonsense. By what process in the world do we produce truncated numbers like you suggest? You have this odd idea that if I take some measurement, and get 2.5 as a value, then the true value is of the form 2.5xxxxx where I just don't know what the x's happen to be (i.e. the measured value is just a truncated version of the true value). But it's not. If our instruments are good, we think the value is near 2.5. Maybe a little above or a little below. There just ain't many ways to produce data where we know all the digits are true in the truncation sense.

-kevin

Re #3.

o $1.09 rounded to the nearest dollar.
o 24-bit sample rounded to 16-bits

Comments #1-3 indicate it is time for a post on significant figures. Here is the quick version:

2.5 actually means a between 2.45 and 2.55 if this is not what you want it to mean you could perhaps write 2.50 or 2.5 +/- 0.2. if you want exctly two and a half it is properly written 25 *10^1 not the lack of any decimal point makes a number exact.

WRT Gas: The pump at my station measures price to 4 sig figs and volume to 5. I think the means that it only rounds wrong one time in 200.

"If you're doing high precision computations for things like navigation of a space probe through a gravitational slingshot, it can introduce enough error to crash your probe."
In that case maybe you shouldn't be rounding.

#6, It's the basic problem of computation. You can't store infinite length numbers on a computer, except for symbolically. The second you can't store, and calculate, everything symbolically, you have to account for the need to truncate or round. And it's not even numbers you would think need special handling. 0.1 is a classic example of a number that cannot be stored exactly using IEEE754 floating point, because .1 is a non-repeating fractional number in base 2. This is why you need to use proper numeric methods to guarantee N digits of accuracy, and this post about rounding is an example of how to reduce the error of the least significant number.

Often you have more than one non-significant digit, ie, digits you want to round away. In those cases #1 is correct. It's not that you have 2.5xxxx where you don't know x, it's that you have 2.534 and you don't care about anything after the decimal point.

Also, I would say that taking 1.0000(etc) to 1 actually IS rounding, it's just rounding with a no-op, in the same way that dividing something by one is still dividing... But that's a definitions thing...

Re #4:

$1.09 rounded to the nearest dollar? $1
24-bit sample rounded to 16-bits? my argument applies.

How about this: $1.05 rounded to the nearest dollar. Mark is right. There is no single "nearest" dollar. They are both equally near. #1 tried to imply that $1.05 really stands for a true value of $1.05+delta, with delta>=0, and therefore the "nearest" dollar should more likely be $2. This is nonsense. It is almost always going to be $1.05+/-delta for any kind of real sampling or measurements.

Just to be pedantic, I assume that when you say 1.05 to the nearest dollar you mean 1.50. But the point is that 1.50, ok, is equal, but 1.5x where x>0 is NOT equal, no matter

Yeah I assume that's what #9 meant.

#8 - I think any rounding algorithm would have to loose accuracy in general.

Mark is right about rounding (for the record, so am I, although it turns out I might be wrong re. gas pump rounding, but no one really knows, because depending on who you ask the machines are either much less or much more accurate than I gave them credit for in the video).

It is kind of my lifelong dream to be deemed moderately interesting by people who like math (I went so far as to write a novel about such people), so I appreciate the link and the thoughtful commentary. -John

#12: I still maintain that you and Mark are ONLY correct about (in the case of rounding to the next integer) xxxx.5 EXACTLY. if you prefer evens, and you round 4.51 down to 4, you are doing it wrong.

I agree a post on significant digits is needed. If you measure 3.52, what you know is that what you are measuring is 3.5xxx..., where 0.0xxx... is close to 0.02. (how close depends on your tool, and should be specified.)

Well, sure, but 3.51 isn't 3.5. Obviously this is only relevant if the calculation being done ends either by 0'ing out or if the calculator in question rounds wrongly.

(Example: I was taught in third grade that 3.3345 rounded to the nearest penny would be 3.34, because you have to round up the 4 and then you round up the 3, which is totally ludicrous. But I have heard--although no confirmation from the nice people at exxon--that gas pumps regularly round this way.)

re: 15, wait, so you're saying that 3.3344444444445 gets rounded to 3.34??? that's dumb. If you were taught that in 3rd grade your 3rd grade teacher should be fired. from a cannon.

Rounding isn't a recursive process. You pick a point, and round.

If you're doing high precision computations for things like navigation of a space probe through a gravitational slingshot, it can introduce enough error to crash your probe.

Especially if readings are processed in recursive equations, where little errors can accumulate over time.

Rounding is a form of quantization. And quantization can be done in various ways (truncation, rounding, rounding toward 0, rounding toward infinity, etc.). And quantization error (quantized value - actual value) can be handled by adding noise (dither). And dither can have a Gaussian PDF, or other PDFs, e.g., triangular, depending upon the application.

Anyway, MarkCC is mostly correct, and even in the case when he is less than correct, I get his point.

Thanks for all the great information, MarkCC. I always enjoy reading your blog.

I would also like to join those asking for a post about the concepts and methods regarding significant digits.

I have tried to read material about it from NIST and others in the past, but my understanding is still very low, and I would appreciate your treatment of this subject, if it's something that would interest you.

Thanks.

My Favorite Strange Number: Ω (classic repost)

goodmath — Wed, 31 Dec 2008 07:53:39 +0000

My Favorite Strange Number: Ω (classic repost)

I'm away on vacation this week, taking my kids to Disney World. Since I'm not likely to
have time to write while I'm away, I'm taking the opportunity to re-run an old classic series
of posts on numbers, which were first posted in the summer of 2006. These posts are mildly
revised.

Ω is my own personal favorite transcendental number. Ω isn't really a specific number, but rather a family of related numbers with bizarre properties. It's the one real transcendental number that I know of that comes from the theory of computation, that is important, and that expresses meaningful fundamental mathematical properties. It's also deeply non-computable; meaning that not only is it non-computable, but even computing meta-information about it is non-computable. And yet, it's almost computable. It's just all around awfully cool.

So. What is it Ω?

It's sometimes called the halting probability. The idea of it is that it encodes the probability that a given infinitely long random bit string contains a prefix that represents a halting program.

It's a strange notion, and I'm going to take a few paragraphs to try to elaborate on what that means, before I go into detail about how the number is generated, and what sorts of bizarre properties it has.

Remember that in the theory of computation, one of the most fundamental results is the non-computability of the halting problem. The halting problem is the question "Given a program P and input I, if I run P on I, will it ever stop?" You cannot write a program that reads an arbitrary P and I and gives you the answer to the halting problem. It's impossible. And what's more, the statement that the halting problem is not computable is actually equivalent to the fundamental statement of Gödel's incompleteness theorem.

Ω is something related to the halting problem, but stranger. The fundamental question of Ω is: if you hand me a string of 0s and 1s, and I'm allowed to look at it one bit at a time, what's the probability that eventually the part that I've seen will be a program that eventually stops?

When you look at this definition, your reaction should be "Huh? Who cares?"

The catch is that this number - this probability - is a number which is easy to define; it's not computable; it's completely uncompressible; it's normal.

Let's take a moment and look at those properties:

Non-computable: no program can compute Ω. In fact, beyond a certain value N (which is non-computable!), you cannot compute the value of any bit of Ω.
Uncompressible: there is no way to represent Ω in a non-infinite way; in fact, there is no way to represent any substring of Ω in less bits than there are in that substring.
Normal: the digits of Ω are completely random and unpatterned; the value of and digit in Ω is equally likely to be a zero or a one; any selected pair of digits is equally likely to be any of the 4 possibilities 00, 01, 10, 100; and so on.

So, now we know a little bit about why Ω is so strange, but we still haven't really defined it precisely. What is Ω? How does it get these bizarre properties?

Well, as I said at the beginning, Ω isn't a single number; it's a family of numbers. The value of an Ω is based on two things: an effective (that is, turing equivalent) computing device; and a prefix-free encoding of programs for that computing device as strings of bits.

(The prefix-free bit is important, and it's also probably not familiar to most people, so let's take a moment to define it. A prefix-free encoding is an encoding where for any given string which is valid in the encoding, no prefix of that string is a valid string in the encoding. If you're familiar with data compression, Huffman codes are a common example of a prefix-free encoding.)

So let's assume we have a computing device, which we'll call φ. We'll write the result of running φ on a program encoding as the binary number p as &phi(p). And let's assume that we've set up φ so that it only accepts programs in a prefix-free encoding, which we'll call ε; and the set of programs coded in ε, we'll call Ε; and we'll write φ(p)↓ to mean φ(p) halts. Then we can define Ω as:

Ω_φ,ε = Σ_{p ∈ Ε|p↓} 2^-len(p)

So: for each program in our prefix-free encoding, if it halts, we add 2^-length(p) to Ω.

Ω is an incredibly odd number. As I said before, it's random, uncompressible, and has a fully normal distribution of digits. But where it gets fascinatingly strange is when you start considering its computability properties.

Ω is definable. We can (and have) provided a specific, precise definition of it. We've even described a procedure by which you can conceptually generate it. But despite that, it's deeply uncomputable. There are procedures where we can compute a finite prefix of it. But there's a limit: there's a point beyond which we cannot compute any digits of it. And there is no way to compute that point. So, for example, there's a very interesting paper where the authors
computed the value of Ω for a Minsky machine; they were able to compute 84 bits of it; but the last 20 are unreliable, because it's uncertain whether or not they're actually beyond the limit, and so they may be wrong. But we can't tell!

What does Ω mean? It's actually something quite meaningful. It's a number that encodes some of the very deepest information about what it's possible to compute. It gives a way to measure the probability of computability. In a very real sense, it represents the overall nature and structure of computability in terms of a discrete probability.

Ω is actually even the basis of a halting oracle - that is, if you knew the
value of Ω, then you could easily write a program which solves the halting problem!

Ω is also an amazingly dense container of information - as an infinitely long number and so thoroughly non-compressible, it contains an unmeasurable quantity of information. And we can't even figure out what most of it is!

goodmath Wed, 12/31/2008 - 02:53

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Continued Fractions (classic repost)

goodmath — Tue, 30 Dec 2008 07:05:21 +0000

Continued Fractions (classic repost)

One of the annoying things about how we write numbers is the fact that we generally write things one of two ways: as fractions, or as decimals.

You might want to ask, "Why is that annoying?" (And in fact, that's what I want you to ask, or else there's no point in my writing the rest of this!)

It's annoying because both fractions and decimals can both only describe
rational numbers - that is, numbers that are a perfect ratio of two integers.
The problem with that is that most numbers aren't rational. Our standard
notations are incapable of representing the precise values of the overwhelming majority of numbers!

But it's even more annoying than that: if you use decimals, then there are lots of
rational numbers that you can't represent exactly (i.e., 1/3); and if you use fractions, then
it's hard to express the idea that the fraction isn't exact. (How do you write π as a
fraction? 22/7 is a standard fractional approximation, but how do you say π, which is almost 22/7?)

So what do we do?

One of the answers is something called continued fractions. A continued fraction is a
very neat thing. Here's the idea: take a number where you don't know it's fractional form.
Pick the nearest simple fraction 1/n that's just a little bit too large. If you were looking at, say, 0.4, you'd take 1/2 - it's a bit bigger. Now - you've got an approximation, but it's too large. So that means that the demoninator is too small. So you add a correction to the denominator to make it a little bit bigger. And you just keep doing that - you approximate the correction to the denominator by adding a fraction to the denominator that's just a little too big, and then you add a correction to that correction.

Let's look at an example: 2.3456.

It's close to 2. So we start with 2 + (0.3456).
Now, we start approximating the fraction. The way we do that is we take the reciprocal of 0.3456 and take the integer part of it: 1/0.3456 rounded down is 2 . So we make it 2 + 1/2; and we know that the denominator is off by 0.3088.
We take the reciprocal again, and get 3, and it's off by 0.736.
We take the reciprocal again, and get 1, and it's off by 0.264.
Next we get 3, but it's off by 208/1000.
Then 4, off by 0.168.
Then 5, off by 0.16.
Then 6, off by 0.25.
Then 4, off by 0; so now we have an exact result.

So as a continued fraction, 2.3456 looks like:

As a shorthand, continued fractions are normally written using a list notation inside of square brackets: the integer part, following by a semicolon, followed by a comma-separated list of the denominators of each of the fractions. So our continued fraction for 2.3456 would be written [2; 2, 3, 1, 3, 4, 5, 6, 4].

There's a very cool visual way of understanding that algorithm. I'm not going to show it for 2.3456, because it's a bit too complicated - it would be hard to draw out the complete diagram for in a legible way. So instead, we'll look at a simpler number to make the visual work. Let's write 9/16ths as a continued fraction. Basically, we make a grid consisting of 16 squares across by 9 squares up and down, like this one.

We draw the largest square we can on that grid. The number of squares of that size that we can draw is the first digit of the continued fraction. For 9/16ths, we can draw one 9×9 square - so our first digit is one, and we're left with a 7×9 rectangle:

Now we just repeat: draw the largest square we can. That's a 7×7, and we can only draw one of it. So the second digit is one; and we're left with a 7×2:

And repeat: the largest square we can draw is a 2×2, and we can draw three of them. So the next digit is a three; and we're left with 1×2 - which is 2 1×1, so the last digit is a 2.

So the continued fraction for 9/16ths is 1/(1+1/(1+3/(1+2))), or [0; 1, 1, 3, 2].

One incredibly nifty thing about this way of writing numbers is: what's the reciprocal of 2.3456, aka [2; 2, 3, 1, 3, 4, 5, 6, 4]? It's [0; 2, 2, 3, 1, 3, 4, 5, 6, 4]. We just add a zero to the front as the integer part, and push everything else one place to the right. If it was a zero in front, then we would have removed the zero and pulled everything else one place to the left.

Using continued fractions, we can represent any rational number in a finite-length continued fraction. We still can't directly represent irrational numbers - they're going to be infinite continued fractions. But we can usually write some expression
to show how to continue the number, and for a given precision, we can approximate
irrational numbers with a much smaller representation.

So, as I just said, we represent irrational numbers using infinite continued fractions. So there's an infinite series of correction fractions. You can understand it as a series of every-improving approximations of the value of the number. And you can define it using a recurrence relation (that is, a recursive formula) for how to get to the next digit -
that recurrence relation is the real key - we can include a simple equation that describes
how, given the current finite continued fraction, to generate the next step.

For example, π = [3; 7, 15, 1, 292, 1, ...]. If we work that out, the first six places of the continued fraction for pi work out in decimal form to 3.14159265392. That's correct to the first 11 places in decimal. So 9 numerals of continued
fractions gives us 11 decimal places. In general, the conciseness savings of continued fractions is even better than that.

I'll close with a very cool property of continued fractions. The square root of most
numbers (excepting the perfect squares) is irrational. In continued fractions,
they're still irrational - that is, they're infinite continued fractions. But all irrational
square roots form repeating continued fractions. The coolest example of this? The square root of 2 in continued fractions is [1; 2, 2, 2, 2, ...].

goodmath Tue, 12/30/2008 - 02:05

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Following the procedure that you described, I get [2; 2, 1, 8, 2, 1, 1, 4]. Adding all the fractions I get 2+216/625, which is indeed 2.3456.

In particular I can't follow this sentence:

"So we make it 2 + 1/2; and we know that the denominator is off by 0.3088."

0.3088 wrt what? 0.3456 = 1/(2+x) and solving for x that's 0.893[518], not 0.3088 ... what am I missing?

Put in another way, 0.3456 = 216/625 = 1/(2+x) (yes, I know, this defeats the whole purpose of the exercise), so x = 193/216.

Also, the continued fraction you presented simplifies down to 2+8631/19555

An interesting property: it looks as though [0; 0, n1, n2, ...] = [n1; n2, ...] too, since 1/(1/n) = n for any n.

For the second example, the line 1/(1+1/(1+3/(1+2))) should be

1/(1+1/(1+1/(3+1/2)))

Hope you have a good time at Disney!

Now all we need is algorithms for doing arithmetic on 'em (including those pesky infinite recurring ones), and voila! exact computation with roots.

What about higher-degree roots - cube roots and the like?

Here's a cute one: the continued fraction expansion of
Sqrt(2 + Sqrt(3 + Sqrt(5 + Sqrt(7 + Sqrt(11 + ... + Sqrt(Prime(n))))) = 2, 9, 1, 1, 1, 7, 3, 5, 4, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 15, 1, 3, 1, 41, ...

A105548 Continued fraction expansion of prime nested radical A105546.

Actually, the algorithms for doing arithmetic on continued fractions are well known. Yes, they work on the infinite ones -- the first few terms of the answer can be determined by examining the first few terms of the inputs.

This all works well in a language with lazy lists like Haskell -- great fun!

There is a classic unpublished paper by Bill Gosper explaining how to do arithmetic and other cool things with continued fractions that I'm sure must be available somewhere on the web.

thenks****

Found it! Gosper's paper is here:
http://www.tweedledum.com/rwg/cfup.htm
A must-read for anybody serious about continued fractions.

"But we can usually write some expression to show how to continue the number"

How often is that true? Do they have to be algebraic or computable? Is there a recurrence relation for pi?

Gosper wrote several "HAKMEM" memos on continued fractions:

http://www.inwap.com/pdp10/hbaker/hakmem/cf.html

This is the same Gosper, by the way, who invented the "Glider Gun" for Conway's Game of Life.

There's a cool little theorem - the first time I saw it was in Hardy's book - that says a continued fraction is repeating if and only if it is the representation of an algebraic number. So of course all square roots of non-transcendental numbers will be repeating. Otoh, given a nonrepeating, nonterminating continued fraction, one may deduce that the number is transcendental. So if one posits that pi is represented as a nonterminating, nonrepeating continued fraction, it pops out automatically that it must be transcendental.

ScentOfViolets: That theorem must use a more expanded definition of continued fractions. Repeated continued fractions of the form described here are all irrational numbers of the form (a + sqrt(b))/c, for integers a, b, and c. When you solve the repeated fraction as a recurrence, you get a quadratic equation.

A good resource for info on continued fractions is Ron Knott's continued fraction page at: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html
There is a link there to his interactive continued fraction calculator, which allows you to convert between fraction form, decimal form, and continued fraction form and calculate the intermediate convergents.

A nice generalization of "The square root of 2 in continued fractions is [1; 2, 2, 2, 2, ...]" is:

(the square root of (1 + n^2)) - n + 1 in continued fractions is [1; 2n, 2n, 2n, 2n, ...]