Writing this blog, I get lots of email. One of the things that I get over and over again is a particular kind of cluelessness about the idea of infinity. I get the same basic kind of stupid flames in a lot of different forms: arguments about Cantor's diagonalization; arguments about

calculus (which I've never even written about!); arguments about

surreal numbers; and worst of all, arguments about nullity.

There's a fundamental bit of foolishness that underlies all of the flames. *Infinity is not a number*. It's a mathematical concept

related to numbers, but it is *not*, **not** a number.

The most recent version of this is an email from last week, titled "Nullity and concerning your ignorance of it". It's pretty typical of the basic confusion that follows from treating infinity as a number.

I was searching the internet for stuff on Nullity, this new number that I noticed you think is crank. Since internet arrogance and ignorance are at an all time high, I thought I might share some explanation of it as I'm not sure if my response was presented on your site.

First off, 1/0 is infinity because if you were to divide 1 into zero pieces, you would need to divide an infinite number of times. Same goes for -1/0 .

Now, what is interesting about infinity is that when you add or subtract a number from it, infinity remains. When you subtract infinity from infinity, something else happens entirely: the two concepts stack up with each other. Thus, [1/0 - 1/0] gives you 0/0 as the denominators need not change, obviously. For simplicity's sake, just think of the positive and negative signs as cardinal directions when dealing with infinity.

Now, what this means is that both the positive and negative directional infinities are being represented, and that is the totalization of the catesian plane, or nullity. You can think of it as the limit to the cartesian plane, or all of the cartesian plane, or none of the cartesian plane... whatever suits your ship. Nullity is the summation of all sets of infinity.

Hope this helped.

You can see the confusion right away: "1/0 is infinity because ...". Bzzt. No. 1/0 is *not* infinity. 1/0 is *nothing*. 1/0 isn't defined in our number systems: it's not a number. In fact, it's not just not a number, it's *nothing*. It's a meaningless expression. Asking what 1/0 is is like asking "What's the square root of a nice juicy plum?". Or what predicate makes the logical statement "∀x: P(x)∧¬P(x)" true?

If you treat infinity as a number, you fundamentally break everything that makes arithmetic work. For example, the most basic definition of numbers that I know of is Peano arithmetic. Peano arithmetic is a set of axioms that defines how the natural numbers work. It's the set of axioms that are typically used as the fundamental basis of a formal definition of numbers. One of the Peano axioms says that for every natural number, there is *exactly one* natural number that is its successor; and every natural number *except* zero is the successor of *exactly one* natural number.

What's the successor of infinity? Or to phrase it a slightly different way (by using the Peano definition of addition), what's ∞+1? As my clueless correspondent says, "what is interesting about infinity is that when you add or subtract a number from it, infinity remains."

So, ∞+1 = ∞. And ∞+1+1 = ∞. And ∞+1+1+1 = ∞.

And there went the Peano axioms, right out the window. The failure of

the Peano axioms isn't some trivial, obscure theoretical issue. If the field axiom fails, then *every* proof about the natural numbers, *every* statement about how the natural numbers work, loses its validity. Nothing is safe. 1+1=2? Nope: field axioms say that if x=y, the

x+z=y+z. Let z=∞. Then ∞+1+1 = ∞ + 2

. ∞+1 = ∞, so then ∞+1 = ∞+2. Remove the ∞ from both sides,

and 1=2. But wait, you say, you can't remove the ∞ from both sides! In infinity is a number, if 1/0=∞, then *yes you can*.

The natural comeback to that is something like "Well, so ∞ isn't a *natural* number, but it's still a number."

Still no good. First, we normally define numbers using the Peano naturals as a starting point. But even if we don't, if we start with some other construction, most of the math we do with numbers ultimately relies on the fact that numbers form a *field*. Whether you're looking at

rational numbers, real numbers, complex numbers, or whatever, they form

a mathematical structure called a *field*. Fields are defined by

a set of fundamental axioms. If infinity is a number, then the field axioms

fail - and if the field axioms fail, then pretty much *everything* that we do with math - every proof about numbers, every numerical fact,

it's all rubbish.

The typical comeback to this is something like "So ∞ isn't a number, but it's still something, and 1/0 = ∞." Nope, still no good. The field axioms define division in terms of the multiplicative inverse,

and both multiplication and the multiplicative inverse are *closed* - meaning that you can't get a value outside of the real numbers from anything defined using multiplication or the multiplicative inverse in the real number field.

But wait, you might say, I distinctly remember talking about infinity in calculus class: lim_{x→∞}1/x=∞!

Limits aren't really talking about *numbers*, they're talking about *curves*. When we talk about infinity in limits, we're talking about *trend lines*, not necessarily numbers. *Some* limits

trend towards a number; some limits don't follow a trend *at all*;

and some follow a trend towards an unbounded increase. That last one

is what we mean when we say a limit trends towards infinity.

There's an easy illustration of what I mean when I say that a limit

talks about the trend of a curve. Think of one of the simplest curves: y=1/x.

What's the limit of 1/x as x approaches 0? That's not a meaningful question. You need to state from *which direction* you're following the curve. If x≥0, then lim_{x→0}1/x=+∞;

if x≤0, then lim_{x→0}1/x=-∞. We're not talking

about numbers there, but about the direction of an unbounded trend. When

we say "+∞" there, we're talking about the fact that the curve increases without bound in the positive direction. When we say that the

limit is -∞, we're not saying that the curve converges on a specific *number* called -∞; what we're saying is that the curve increases without bound in the negative direction.

∞ isn't a number. If it were, it would break the fundamental axioms that define numbers. Nullity is even worse; it breaks *even more* of the fundamental axioms of math. The guy who came up with

nullity is a true idiot: in the presentation shown by the BBC in which

he demonstrates the supposed properties of nullity, he uses several steps that only make sense under the field axioms, which are violated by nullity. So the results are fundamentally wrong in a *very* strong way: if you define nullity as he defines it, then you can use the existence

of nullity to prove statements like 1≠1.

It just doesn't work. There's no way of taking values representing true infinity, and turning them into numbers. You *can* do things like

Cantor's transfinite numbers, but they get very strange very quickly - and

they *don't* work the way that we expect numbers to work - for example, you need to distinguish between cardinal numbers and ordinal numbers, and you don't get fractions. And they don't even really behave like

infinity! For example, if we treat infinity as a number, there's no number

greater than infinity. But for Cantor's transfinite numbers, if N is a transfinite number, there's always another transfinite number larger than N.

And even with transfinites, you can't cause an expression like 1/0 to

have any meaning. 1/0 is fundamentally undefined. It's not ∞. It's not ω. It's not nullity. It's not anything. The moment you see something use the statement that 1/0=∞, you *know* that they're an idiot who doesn't really have the slightest clue of what they're talking about.

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Mark, I explain 1/0 and 0/0 this way, which is related to how the nuns at St. Teresa's taught me all those years ago in the 3rd and 4th grade:

If 12/3 means "how many $3 items can you buy with $12 in your pocket?", then 1/0 is akin to "how many free items can you buy with $1 in your pocket?" and 0/0 = "how many free items can you buy when you are broke?"

It doesn't take too much prodding on my part (can you take *none* if you don't like the item? Can you take 3? 4? a billion?) for the student to see that anything/0 is, interestingly, all numbers simultaneously and no one number in particular.

It's zen!

We have a word (I made up when I was in 2nd grade) to describe it: "gothagoo," which -- since only I and one other playmate know the real definition -- is a meaningless word to my students.

So, 0/0 = 1/0 = k/0 = GOTHAGOO.

yeh, like if 3%2= 1.3R

so .3 repeates endlessly

so 1.3r x 1.3r = 2.666666666666666666666666666r not 3

Interesting post. I never actually got confused with taking infinity as a number but this post has clarified lots of things for me, thanks!

BTW, make sure you correct the first occurrence of "limxââ1/x=â" should be the limit when x tends to 0 not infinity!

Could you say that introducing infinity makes arithmetic useless, so it's a practical decision to keep it out? We must choose between infinity and useful arithmetic.

Error reading drive C:. Abort, Retry... FAIL?

limxââ1/x=â!

Don't you mean

limxâ01/x=â!

It's worth noting that in the surreal numbers, the reciprocal of ω (the "earliest" infinity) is a well-defined number denoted by ε or ι. 1/0 is still undefined, 0 times ω is still 0, and moreover, the solution to 1/x = ω is x = ε.

The surreal numbers are an ordered field, just like the reals. I know it handles the transfinite ordinals just fine. Does it handle 1/0?

Re Don #4:

No, it's got nothing to do with practicality. It's got to do with fundamental semantics.

Infinity

isn'ta number. We don't "exclude" infinity from the numbers. That would imply that itisa number but that we just arbitrarily leave it out to make other things work better.That's not what's happening. Infinity is a mathematical concept, but it's not a number. "Number" has a specific meaning. And infinity is absolutely, totally incompatible with the meaning of "number".

I tend to like metaphors. Suppose I've got a value {a, b, a, c}. Is that a set? No. It's got two copies of "a". By the

meaningof set, {a,b,a,c} isn't a set. I'm notexcluding{a,b,a,c} from the sets for practicality. I'm doing it because it'snota set, and if I pretend that it is, then by including it, setsstop working.That's the situation with infinity. It's a concept related to numbers, but it fundamentally isn't a number. It's got a precise meaning, but that meaning isn't compatible with it being a number.

Re: #8:

No, the surreal numbers do not define 1/0. They're great at doing transfinites and infinitessimals. But 1/0 is fundamentally meaningless. You can't assign a meaning to it and end up with anything that makes sense.

The closest the surreals come is the relation between transfinites and infinitessimals: the first non-finite surreal is ω; 1/ω=ε. For any finite number N,

ε<1/N. So it's "infinitely small". But it's not zero. So you get 1/ε=ω, but never 1/0=∞.

Some remarks on infinity

There are two basic characterizations of infinity: potential infinity and actual infinity. Take a set with an infinity of objects in it, e.g., the set N of natural numbers {1, 2, 3,....}. One can characterize N as a potentially infinite set, that is, a set which is "open" in the sense of being incomplete: given any number of elements in it, it's always possible to add another one (we are not going to run out of natural numbers). However, one can think of N as an actually infinite set, that is, as a set which is actually completed, with all the natural numbers in it.

Historically, mathematicians and philosophers have preferred to restrict their considerations to potential infinity because some of the implications of actual infinity are at odds with seemingly obvious truths. To see why this is so, one has to understand the notion of biunivocal (or one-to-one) correspondence. [truncated]

I suggest reading the whole (short) web page.

For that matter, see the article and citations on Wikipedia's "Actual infinity"

Thank you, Mark. I now understand why I fail to get to sleep when counting sheep to infinity.

regards, grassie

P.S. Counting to infinity twice doesn't work either.

Isn't the 1/0 argument just a special case of the fallacy that the limit point of a sequence does not have to have the same properties as all the elements of the sequence?

E.g. all the tired and incorrect arguments about why 1.999999... doesn't equal 2.

First off, 1/0 is infinity because if you were to divide 1 into zero pieces, you would need to divide an infinite number of times. Same goes for -1/0 .No way, man! You would do it ONCE and then you'd be like "What is this shit" and walk away. One divide by zero is one the first time you do it, and one divided by zero is zero the second time you do it (because you know it is futile and don't even try).

"The moment you see something use the statement that 1/0=â, you know that they're an idiot."

I object to that. Using convenient tricks like that make a lot of engineering math more compact, particularly used in electronics. Of course, in that context 0 just means a very small positive number and â its inverse.

Mark, at the risk of asking you to repeat yourself (and i apologise in advance for that), i have to take the opportunity to ask: is there any axiomatic system in which the concept of infinity is actually expressed as a number?

it does sound hard to come up with a system in which infinity is a number and still you can proove basic algebraic truths --- and of course one such system would be highly impractical and, deep down, meaningless in what concerns algebra (it would just be borrowing the names for its entities)... but still. i have a hard time questioning mathematicians' capacity to come up with this sort of stuff.

There is an algebraic structure that handles division by 0, named a wheel. You can take any semigroup and expand it into a wheel, and it will start allowing division by 0. I can't remember all the details, but for the natural numbers/reals/etc., k/0 = 1/0 for all k but 0, and 0/0 is its own value. They act like little Hotel Californias of the wheel - you can add with them and multiply with them, but most of the time you'll just get the same number back or a more restrictive one (e.g. 3 * 1/0 = 1/0, but 0 * 1/0 = 0/0. 0/0 plus or times anything is still 0/0).

More details can be found here.

What I miss both in the post and in comments, it is the definition of number. Mark, you do talk a lot abouf natural numbers, Peano axioms, rational numbers, real numbers, complex numbers, and even state that infinity is not a number. What is a number then - without any adjective - if you state your point so strongly?

IIRC my course of mathematical logic, it IS possible to construct a consistent set of axioms based on Peano arithmetic but with positive and negative infinities.

Of course, you can't compare infinities, divide one infinity by another infinity, and so on.

But such system is certainly possible and it might be useful to demonstrate different ways to treat infinities.

"Of course, you can't compare infinities, divide one infinity by another infinity, and so on"

Why not, physicists do it all the time.

ZFC is NOT the only game in town:-)

"Of course, you can't compare infinities, divide one infinity by another infinity, and so on"

Why not, physicists do it all the time.

ZFC is NOT the only game in town:-)

Re Juraj #18:

Really defining numbers is beyond the scope of a comment in a post.

Basically, everything starts with the natural numbers. The Peano axioms define the natural numbers operationally. Semantically, they can be defined in a lot of different ways, but my favorite is from set theory, where there are fundamentally two different kinds of numbers: cardinals and ordinals. Cardinals are things that fit the Peano axioms, and measure

size, and ordinals are things that measureposition.Integers are a generalization of the naturals produced by creating a simple notion of forwards/backwards direction. Rationals are defined by parts of things - the positions between the numbers. Reals come by generalizing rationals. Complex come by generalizing reals. And so on.

But it all comes back to cardinals and ordinals: size and position. "Infinity" doesn't measure the size of anything: it's *bigger than* the size of anything. Infinite doesn't measure a specific position within a group: there's no set which has an "infinite" position.

To divide a thing by a number is to part the thing into that number of equal pieces. The idea of dividing a thing by zero would be to not divide the thing at all: no division takes place, nothing happens, so the idea is a meaningless notion.

@Alex Besogonov: yes, the IEEE 754 standard for floating point numbers does something like that. See http://en.wikipedia.org/wiki/IEEE_754-1985. It defines +Infinity, -Infinity and NaN (Not a Number) and defines arithmetic rules for them. They aren't considered real numbers though. See also http://en.wikipedia.org/wiki/Affinely_extended_real_number_system.

If I remember correctly, in the theory of (complex) rational functions, concepts like 1/0 = â, â+1 = â and -â = â are used quite liberally... Statements like these are probably to be understood to be about limits in the context of the topology of â âª â, but mathematicians' own usage does not help the confusion.

Saying that infinity is not a number is not quite right. Numbers are the elements of the number system you want to use. If you are using the extended reals, then infinity (and minus infinity) is indeed a number in the sense that it belongs to the number system you are working with.

What I'm saying is that everything you mention is true except for the sentence "infinity is not a number", since you can define a number system that contains infinity, it's just no longer a field, and has some odd operations which do not treat all elements equally. But it's still a number systems, and in fact a very useful one.

I was about to post something about complex analysis. Certainly you can think about these statements as limits, but you can also (and equivalently) just think about them as statements about a complex manifold, at which point no more limit-taking is required to make the statement than for any other point on the manifold.

You mean that infinity is not a natural number. Infinity (which one?) is a cardinal number.

http://en.wikipedia.org/wiki/Cardinal_number

What about the projective line?

It may make the situation worse or better, but people may want to read up on the extended real number line:

http://en.wikipedia.org/wiki/Extended_real_number_line

The summary is something like: We take the real numbers and add positive and negative infinity. This makes some things nicer and some things worse. For example,

lim_{x \to 0} 1/(x^2) is now an element of our space. On the other hand, lim_{x \to 0} 1/x is still undefined, as is 1/0 (for both, the reason is -- more or less -- you don't know if the zero is a positive zero or a negative zero).

Read the article for more info.

I detect an excuse to post my favorite math riddle:

Proof that 2 = 1

Start with: X = Y

multiply both sides by X: X*X = X*Y

subtract Y^2 from both sides: X^2 - Y^2 = X*Y - Y^2

do some factoring: (X+Y)*(X-Y) = Y*(X-Y)

divide both sides by (X-Y): (X+Y) = Y

substitute Y for X: (Y+Y) = Y

divide by Y and simplify: 2 = 1

There you have it, a mathematical proof that two is equal to one!

I used to get a lot of mileage out of that as a substitute math teacher. To the kids credit, most classes figured out on their own the fallacy of dividing by zero.

I would argue that the situation you are describing is division by one.

On the subject of field axioms: As Mark rightly said above, allowing division contradicts the field axioms, but we can actually say something stronger by looking at how it goes wrong.

The axioms of a field say that x/x = 1 for all choices of x. So if 1/0 was an element of the field we would have 0/0 = 1, which implies that 1 = 0. The field axioms require 1 and 0 to be different, so this thing is not a field.

Suppose we can live with that. Let's carry on looking at this weird structure. Since 1=0, for any choice of x we have x.1 = x.0, which in turn gives x = 0. Hence the "field" contains only one element.

So you see, division by zero is not forbidden for reasons of "practicality": it's that if you want to have an algebraic structure which looks somewhat like normal arithmetic, and allow division by zero, then you're stuck with the one-point set. If you want to call that a "number system" then you're welcome to, but you'll be on your own.

On the subject of field axioms: As Mark rightly said above, allowing division contradicts the field axioms, but we can actually say something stronger by looking at how it goes wrong.

The axioms of a field say that x/x = 1 for all choices of x. So if 1/0 was an element of the field we would have 0/0 = 1, which implies that 1 = 0. The field axioms require 1 and 0 to be different, so this thing is not a field.

Suppose we can live with that. Let's carry on looking at this weird structure. Since 1=0, for any choice of x we have x.1 = x.0, which in turn gives x = 0. Hence the "field" contains only one element.

So you see, division by zero is not forbidden for reasons of "practicality": it's that if you want to have an algebraic structure which looks somewhat like normal arithmetic, and allow division by zero, then you're stuck with the one-point set. If you want to call that a "number system" then you're welcome to, but you'll be on your own.

Ah, i think i get it.

What you're saying is that if i had an infinite number of donuts, i'd be infinitely happy?

On the subject of field axioms: As Mark rightly said above, allowing division contradicts the field axioms, but we can actually say something stronger by looking at how it goes wrong.

The axioms of a field say that x/x = 1 for all choices of x. So if 1/0 was an element of the field we would have 0/0 = 1, which implies that 1 = 0. The field axioms require 1 and 0 to be different, so this thing is not a field.

Suppose we can live with that. Let's carry on looking at this weird structure. Since 1=0, for any choice of x we have x.1 = x.0, which in turn gives x = 0. Hence the "field" contains only one element.

So you see, division by zero is not forbidden for reasons of "practicality": it's that if you want to have an algebraic structure which looks somewhat like normal arithmetic, and allow division by zero, then you're stuck with the one-point set. If you want to call that a "number system" then you're welcome to, but you'll be on your own.

Lout's line of thought in #23 is close to how i like to think about it. division by a natural number is intuitive (i have to fall back on intuition, because i'm not a mathematician and don't have the brains to be one); everybody's seen a cake being cut. division by rationals less than one is only a slight extension of that; it's asking "if this piece was such-and-such a fraction of a greater whole, how big would the whole be". division by rationals greater than one, and by reals in general, come by slight extension of that in turn.

but asking what the result of splitting this whole into zero pieces would be just makes no sense at all. what's zero parts of anything? the most sensible intuitive answer would be zero, in every case, but that would mean disregarding --- considering as meaningless --- the thing you're supposedly "dividing". that's no more reasonable a way to define division by zero than just defining the whole operation as being equally meaningless.

what i (the non-mathematician, mind) can't quite wrap my head around is how this doesn't prove division as being not the inverse of multiplication after all. as far as i know, there is no number such that multiplying by it becomes meaningless; yet with division, there is zero. shouldn't that mean they're not really such closely related operations after all?

I disagree. Numbers are whatever we choose to consider as numbers. I could make a number system that consisted of some set of objects with hideous operations defined so that they never associated or commuted or did anything else nice. That's a number system. It is a thoroughly useless and uninteresting one. The real real why adding infinity in as a "number" is that we don't then have a field and fields are nice objects. But that doesn't mean "infinity" is not a number anymore than for some purposes we rule out complex numbers when want to have nice order properties or rule out everything but Z when we want to have a well-ordered set. There are no known where we would want to ever add an element that behaved as 1/0 (for the reasoning outlined by Ross above). But saying that makes it not a number is an almost theological claim that isn't necessary to get the real point across: If we want to do interesting math we can't include this.

There is one method for defining and working with â that I've seen that makes it behave like a number (at least algebraically): the one point compactification of the complex plane. Otherwise known as the Riemann Sphere.

Here, you still cannot deal with â arithmetically (the extended complex numbers are not a field) but geometrically and algebraically it stands in as a multiplicative inverse of 0. Specifically, it's the point opposite 0 on the sphere sitting on the complex plane.

Some really beautiful mathematics appears when you consider transformations of the complex plane in this way. In particular, lines and circles can be seen as the same type of object: those that are preserved under MÃ¶bius transformations. Lines are just circles that pass through the point at infinity.

That's the problem with adding two infinities: division by zero is still undefined. Things depend on which direction you are approaching the zero from in your limits.

Just add one infinity. That way you can divide by zero and refer to that result as infinity. Unfortunately, 0/0 will still be undefined. I don't think 0 times infinity is defined either.

@Alex Smith

On the other hand if you restrict yourself to one infinity, you get http://en.wikipedia.org/wiki/Projectively_extended_real_numbers , where you can do x/0.

Again, there are lots of things you can't do in this system :P

#37: The reason that this doesn't prove that "division isn't the inverse of multiplication" is that 0 is special as far as multiplication is concerned: there is no other number a with the property that ax = ay for all x, y. In other words, 0 is the only number such that, when you multiply by it, you completely obliterate all information about what number you originally had. In this sense, it makes sense that "undoing" multiplication by 0 should be impossible.

If I had a dollar for every time I've told my students "Infinity is not a number, it's a concept", I'd have a lot more money than I do now...

Here's a fun bit I do to show them the idea: I ask, "Is .9999...carried out to infinity less than, equal to, or greater than 1, no rounding." After the usual arguments about it being just a teensy amount less than 1, I ask "Well, isn't .9999... just 3 times .3333..."? Everyone agrees. "And isn't .3333... the same as 1/3?" Everyone agrees. "And isn't 3 times 1/3 equal to 1?"

Sometimes I have to get a mop.

Anyway, the response to the original commenter in the post reminds me of something one of my profs used to say in a graduate math class I took. If someone gave an answer that was completely "out there" he'd look at them and say "You might as well have said 'a can of green beans'". That is, a response that was so wild that it would take considerable work in order to elevate it to the right kind of wrong thing. I stole that from him, so now when a student says something goofy (like specifying a voltage in watts) I say "asparagus". It's a code word that they all understand now.

If appears that the problem is that everyone approaches the question from different points of view. Some look at it from the point of view of analysis (calculus), others - set theory. Some from the point of view of algebra, others - geometry.

Certainly, if what we want is a field such that division by zero is defined, then we are out of luck.

On the other hand, if we look at the set theoretic approach (the "infinity + 2" sort of thing), and we work with cardinal numbers, then we can define addition and multiplication (as well as powers) using unions and maps, but as Mark points out, subtraction and division do not extend to infinities too well. Notice, however, that mathematicians to refer to these constructs as infinite cardinal numbers.

One can also look at ordinal numbers. In that case, subtraction can be defined, since for these numbers things are not reduced modulo bijections. In essence, the difference between two ordinals is how many steps (possibly infinite number) it takes to get from one to the other.

If one goes into algebraic geometry, then one can start adding infinities left and right. For example, if you add just one infinity to your favorite field, then you get the projective line over that field. The projective line over the complex numbers is referred to as the Riemann Sphere. Once again, not all of the operations are well-defined, but one can still get a kick out of those fractional linear (Mobius) transforms.

One can also add two infinities to the real line, and make some bits of calculus a bit easier to stomach. Unfortunately, division by zero will still be undefined, unless one specifies from which direction one is approaching zero.

The problem with Mark's post is that he wants infinity to be everything to everyone. As he correctly points out, that's impossible. That does not mean that particular versions of infinity with well-specified properties for whatever situation you find yourself in are unacceptable.

Mark,

I'm not fundamentally disagreeing with your point (or the silliness of arguing around 1/0, nullity and all that crap). But some of the statements you make may be inaccurate or at least easily misinterpreted, which could cause some more unnecessary confusion.

You highlighted the role of the Peano axioms. Well, there's nothing in Peano's axioms to rule out infinitely large numbers: the axioms are totally consistent with the existence of such numbers. In fact, so-called non-standard models of PA are widely discussed and researched.

To see why this is so, just add a new symbol to your language (call it I), and add infinitely many axioms to PA saying I>0, I>1, I>2, I>3, etc. The new system is consistent because obviously every finite sub-system of it is (compactness argument). Thus, PA has a model with an element corresponding to I, and that element has a perfectly legitimate unique successor I+1, and it does indeed satisfy I>n for every ordinary natural number n.

(If you don't like systems with infinitely many axioms, remember that PA itself is such a system).

To probe beyond a shadow of doubt that I'm right, here's a link to Wikipedia :-)

http://en.wikipedia.org/wiki/Non-standard_arithmetic

At a higher level, I disagree that PA "defines" the natural numbers or that it "constructs" them. PA describes various properties of what we perceive to be natural numbers, and it allows us to prove various things about them if we agree that its axioms are sound, but it doesn't "create" them. Without access to some rudimentary form of natural numbers it's hard to see how a complex construct as an "axiom system", with its language, strings of terms as formulas, axioms etc. can even get started.

All of this, of course, doesn't support any sort of tiresome discussion around 1/0 being whatever.

Mark,

I'm not fundamentally disagreeing with your point (or the silliness of arguing around 1/0, nullity and all that crap). But some of the statements you make may be inaccurate or at least easily misinterpreted, which could cause some more unnecessary confusion.

You highlighted the role of the Peano axioms. Well, there's nothing in Peano's axioms to rule out infinitely large numbers: the axioms are totally consistent with the existence of such numbers. In fact, so-called non-standard models of PA are widely discussed and researched.

To see why this is so, just add a new symbol to your language (call it Z), and add infinitely many axioms to PA saying Z>0, Z>1, Z>2, Z>3, etc. The new system is consistent because obviously every finite sub-system of it is (compactness argument). Thus, PA has a model with an element corresponding to Z, and that element has a perfectly legitimate unique successor Z+1, and it does indeed satisfy Z>n for every ordinary natural number n. Namely, it's true that Z>0, and that Z>0', and Z>0'' etc where ' is the "successor" function. Unfortunately, the statement "Z>n for every ordinary integer n" is not provable in PA since PA has no idea what an "ordinary integer" is. In other words, you can't "fix" the problem by somehow positing "a natural number must be ordinary, and not this stupid infinite artifact" in PA.

(If you don't like systems with infinitely many axioms, remember that PA itself is such a system).

To prove beyond a shadow of doubt that this is true, here's a link to Wikipedia :-)

http://en.wikipedia.org/wiki/Non-standard_arithmetic

At a higher level, I disagree that PA "defines" the natural numbers or that it "constructs" them. PA describes various properties of what we perceive to be natural numbers, and it allows us to prove various things about them if we agree that its axioms are sound, but it doesn't "create" them. Without access to some rudimentary form of natural numbers it's hard to see how a complex construct as an "axiom system", with its language, strings of terms as formulas, axioms etc. can even get started.

All of this, of course, doesn't support any sort of tiresome discussion around 1/0 being whatever.

Mark -

Your counterpart Zeno has very recently made this concept graphical and topical. Coincidence? I think not.

There are no accidents, man.@15, if 1/0=â is used to actually mean 1/(a very small number) = (a very large number), why not use 1/Îµ=Ï instead?

Two thoughts:

(1) cardinality

The idea of "number" can be tricky to grasp. On the one hand, a number could be identified with a cardinality. Infinity is a cardinality, so there is a very naturally constructed set out there containing "numbers" and infinity: the set of all possible cardinalities.

The confusion arises when folks cannot discriminate as "numbers" as cardinalities and "numbers" as objects in a group, ring, field, or other structure with an operation. To those not initiated, all numbers are automatically tied to the mathematical operations they learned as children. The operations cannot be decoupled with symbols, even when the symbols take on other meanings.

That is, the field (and other) operations associate with "numbers" are mentally coupled with cardinal numbers, even though they shouldn't be.

(2) limits

You said:

Awful. This exposition is lacking rigor and misleading. It sounds like something a naive freshmen would blurt out trying to explain limits to a classmate who was struggling. You aren't living up to your normal standards of quality!

Consider the definition of a limit (of a function): either the delta-epsilon version, or something more abstract for a general topological setting. Limits are not about curves. Nor are they about "trends" as nothing is moving. Limits are about the behavior of a function in a neighborhood.

"When the input is close to a point X, the output is close to a point Y," one might say, although a lot of details are buried in the word "close", especially when talking about infinities or a general topological setting.

Limits are not about things moving!

Two thoughts:

(1) cardinality

The idea of "number" can be tricky to grasp. On the one hand, a number could be identified with a cardinality. Infinity is a cardinality, so there is a very naturally constructed set out there containing "numbers" and infinity: the set of all possible cardinalities.

The confusion arises when folks cannot discriminate as "numbers" as cardinalities and "numbers" as objects in a group, ring, field, or other structure with an operation. To those not initiated, all numbers are automatically tied to the mathematical operations they learned as children. The operations cannot be decoupled with symbols, even when the symbols take on other meanings.

That is, the field (and other) operations associate with "numbers" are mentally coupled with cardinal numbers, even though they shouldn't be.

(2) limits

You said:

Awful. This exposition is lacking rigor and misleading. It sounds like something a naive freshmen would blurt out trying to explain limits to a classmate who was struggling. You aren't living up to your normal standards of quality!

Consider the definition of a limit (of a function): either the delta-epsilon version, or something more abstract for a general topological setting. Limits are not about curves. Nor are they about "trends" as nothing is moving. Limits are about the behavior of a function in a neighborhood.

"When the input is close to a point X, the output is close to a point Y," one might say, although a lot of details are buried in the word "close", especially when talking about infinities or a general topological setting.

Limits are not about things moving!

#42: ah, thank you, that approach makes perfect sense. don't know how come i couldn't think of it myself, but i'll certainly remember it!

Speaking as someone whose blog is named after the antipode of zero in the Riemann sphere I really have to differ with this article. There are perfectly sensible rules for working with the infinity on the Riemann sphere. We can add, subtract, multiply and divide by infinity, as well as define functions that are continuous and differentiable at infinity. In fact, moving from the complex plane to the Riemann sphere can make many functions much better behaved, and that's why people do it.

I can see arguments for not calling this infinity a number. But the argument that it's not a number because it breaks some rules is very unconvincing, after all the negative numbers and complex numbers also break many rules that some people would say were essential properties of numbers.

Ultimately you draw the line between numbers and non-numbers using taste and utility.

Speaking as someone whose blog is named after the antipode of zero in the Riemann sphere I really have to differ with this article. There are perfectly sensible rules for working with the infinity on the Riemann sphere. We can add, subtract, multiply and divide by infinity, as well as define functions that are continuous and differentiable at infinity. In fact, moving from the complex plane to the Riemann sphere can make many functions much better behaved, and that's why people do it.

I can see arguments for not calling this infinity a number. But the argument that it's not a number because it breaks some rules is very unconvincing, after all the negative numbers and complex numbers also break many rules that some people would say were essential properties of numbers.

Ultimately you draw the line between numbers and non-numbers using taste and utility.

Speaking as someone whose blog is named after the antipode of zero in the Riemann sphere I really have to differ with this article. There are perfectly sensible rules for working with the infinity on the Riemann sphere. We can add, subtract, multiply and divide by infinity, as well as define functions that are continuous and differentiable at infinity. In fact, moving from the complex plane to the Riemann sphere can make many functions much better behaved, and that's why people do it.

I can see arguments for not calling this infinity a number. But the argument that it's not a number because it breaks some rules is very unconvincing, after all the negative numbers and complex numbers also break many rules that some people would say were essential properties of numbers.

Ultimately you draw the line between numbers and non-numbers using taste and utility.

Mark,

I'm not fundamentally disagreeing with your point (or the silliness of arguing around 1/0, nullity and all that crap). But some of the statements you make may be inaccurate or at least easily misinterpreted, which could cause some more unnecessary confusion.

You highlighted the role of the Peano axioms. Well, there's nothing in Peano's axioms to rule out infinitely large numbers: the axioms are totally consistent with the existence of such numbers. In fact, so-called non-standard models of PA are widely discussed and researched.

To see why this is so, just add a new symbol to your language (call it I), and add infinitely many axioms to PA saying I>0, I>1, I>2, I>3, etc. The new system is consistent because obviously every finite sub-system of it is (compactness argument). Thus, PA has a model with an element corresponding to I, and that element has a perfectly legitimate unique successor I+1, and it does indeed satisfy I>n for every ordinary natural number n.

(If you don't like systems with infinitely many axioms, remember that PA itself is such a system).

To probe beyond a shadow of doubt that I'm right, here's a link to Wikipedia :-)

http://en.wikipedia.org/wiki/Non-standard_arithmetic

At a higher level, I disagree that PA "defines" the natural numbers or that it "constructs" them. PA describes various properties of what we perceive to be natural numbers, and it allows us to prove various things about them if we agree that its axioms are sound, but it doesn't "create" them. Without access to some rudimentary form of natural numbers it's hard to see how a complex construct as an "axiom system", with its language, strings of terms as formulas, axioms etc. can even get started.

All of this, of course, doesn't support any sort of tiresome discussion around 1/0 being whatever.

Accch. I thought I had it. I tell kids infinity is not a number (good), it is the size of a set, a cardinality (apparently bad?).

Help. I need an accurate version for 13-14 year olds that doesn't leave them thinking I'm snowing them.

I don't know that this argument:

"In fact, it's not just not a number, it's nothing. It's a meaningless expression."

helps the cause a great deal. You could say the same thing of the square root of -1.

I've always heard infinity described as a direction rather than a vector or a scalar.

@Anonymous #48

"@15, if 1/0=â is used to actually mean 1/(a very small number) = (a very large number), why not use 1/Îµ=Ï instead?" They're engineers? I suppose 1/0=â could also be interpreted as an implicit limit. Typically one is talking about positive values, or the sign is otherwise known, e.g. resistance/conductance, so no need to consider negative infinity.

Excellent article.

Infinity and division by zero call into question the nature of the axiomatic systems of numbers that we have developed, and an interesting line of thought about that all comes from Kurt Godel.

Kurt Godel proved in 1936 that any formal axiomatic system (like numbers) must be either incomplete or inconsistent. Incomplete means that there are true statements which have no proofs within the system. Inconsistent means that there are proofs for two statements which form a direct contradiction. Inconsistency or Incompleteness, there must be one!

It is called Godel's Incompleteness Theorem because most people agree that inconsistency renders truth (mathematical equality) meaningless, and therefore can not be the case. Douglas Hofstadter has written some excellent books on the subject.

Graham Priest of the University of Melbourne is one logician who makes a case for inconsistency, and his ideas are interesting to consider.

I think the problem with 1/0 is that people learn to divide without understanding what's that is all about.

Ok they learn that "/" is the inverse operation to multiplication or learn nice statements like "x/y = how many items costing y$ can you buy if you have x$.

BUT they never understand the statement with the "inverse operation" - and that is because they learn that multiplication is a form of iteratet additions - and that is just not true!

They never get the catch that 0 is a special number (just like 1).

It's really so simple: if you could divide by zero that is if there is a mult. inverse to 0 - let's call it o then 0*o=1 and of course 0=0+0 so 1 = (0+0)*o = 0*o + 0*o = 1+1

ok so 1 = 1+1 - now substract 1 on both sides and you've got 1=0 ... now this is interessting because whenever you have 1=0 you just grin and shake your head ... because an object with this property is very uninteressting because it's only member is 0

Bye the way lim_{x->0} 1/x = \infty is just a fancy way of saying the limes does not exists! Its divergent! So the limes does NOT exists at all but we mathematicans are lazy and so we just write it anyway.

There is another way to calculate on R with â as a number: http://en.wikipedia.org/wiki/Max-plus_algebra

The problem with â is: what is â-â? 0? â? 42?

The way I was taught it:

0/0 = x

: implies 0 = 0x

: which is true for all x

: so there's no (single) number 0/0

1/0 = x

: implies 1 = 0x

: which is true for no x

: so there's no number 1/0

The case 1/0 can be generalised to n/0

Basics of 'infinity'

1) Your explanations are very good.

2) "1/0 = infinity" is wrong

3) "1/(infinitesimal) = infinity" too is an 'indefinite relation'

4) 'Infinity' and 'infinitesimal' are mutually related concepts!

5) Whole numbers (say a-unit increases) are 1, 2, 3...infinity

5) A unit less than '1,2,3...infinity' is '0,1,2...(infinity-1)'

6) (Infinity-1) is " endless all 9 digits" number.

7) Infinity is {(endless 'all 9') +1}. It is (1& endless 'all 0')

8) Difference in between infinity and (infinity-1) is one.

9) Infinity cannot be used as a "definite/fixed" number

10) A theorem that relates 'infinity' confuses a user!

11) 2D square matrices are visual matrix positions

12) 2D has equal-digit 'row' and 'column' numbers

13) 2D matrix positions are merged row-column numbers

14) Matrix 0...9 has 10 0 positions 00...99

15) Matrix 00...99 has 100 00 positions 0000...9999

16) Matrix 000...999 has 1000 000 positions 000000...999999

17) Matrix n 0s...n 9s has 1&(n 0s) (n0s) positions.

18) Just ahead of last matrix position (all 9) lay a related infinity!

19) 100(matrix 0...9), 100000 (matrix 00...99) etc are 'infinity'

20) Either 'a-zero' or 'a related-infinity' can be used to compute.

21) Vedic 2D matrix virtues have been compiled as Vedic sutras.

22) Vedic sutras help all users "to compute" in a best possible manner

23) Vedic sutras are few but each one of it works in many ways!

24) Related Number application is too simple and straightforward!

25) Least matrix 0...9 and any huge (n 0s... n 9s) has 'similar' virtues!

1/9^2= 0.0 1 2 3 4 5 6 7 9 & (second last 8 missing)

1/99^2=0.00 01 02 03 04...95 96 97 99 & (second last 98 missing)

1/999^2= 0.000 001 002 003...995 996 997 999 & (998 is missing)

1/(thousand times 9)^2= (1000 times 0)(999 times 0 & 1)(999 times 0 & 2)(999 times 0 & 3)Ã number groups increases by a-unit Ã (999 times 9 & 6)(999 times 9 and 7) & (1000 times 9). Each bracketed 'number group' is part of 'a huge digits number'. It has {(1000 times 9)^2 -1000) digits in it and each digit can be remembered accurately!

This example shows that ancient Indians had analyzed/used infinity far more sensibly. Vedic 'matrix by matrix' number application reveals it!

Ancient Indians have used relations like these for mental computing! Vedic sutras that have been disclosed by Shri Jagadguru Sankaracharya (1884-1960) are proof to "infinite digits related mental computing" by ancient Indians! It is simplest of logical computing!

We can apply practical matrix-by-matrix number application even today. Computers will help us to do it! Vedic Mathematics is a classic example of using endlessly recurring number groups. They perfectly knew 'sense infinity', which differ from modern mathematics knowledge (infinity).

I have already explained it!

Related basics I have also published on a site (please web search Vedic matrix). Concerned number applications are given on 'Orkut' forum NUMBER ZERO, link kk Raghuthaman, and community/forum.

Vedic Mathematicians have used whole number-unit to arrive at infinity and infinitesimal -unit to arrive at one.

Infinite a-units (ones) make 'infinity'

Infinite least-units (infinitesimal units) make 'one'

26) To understand a-infinity we have to grasp number of 'a-units' in it

27) An increase from "0 to 1" is like an increase from "1 to infinity"! Both are digital a-unit increases

28) 'One' has infinite 'infinitesimal-units' within it and 'infinity' has infinite 'a-units' within it. When both said infinites are technically same both 'infinitesimal' and 'infinity' becomes an easier to grasp knowledge!

29) Please don't ignore one (which is made up of infinite*infinitesimal units). Your conclusion, "infinity is not a number" will pose a new question "is one a number?"

30) Zero, infinitesimal, (1- infinitesimal), which is 0.999...endless digits...999 and 'one' is a 'zero to 1' decimal number range.

31) Zero, one, (infinity-1), which is '999...endless digits...999' and 'infinity' is a 'zero to infinity' whole number range! There is no need to use 'numbers less than zero', which is not-liked number application!

32 Number system includes all numbers in a range 0 to infinity. Mathematics binds all numbers, (which includes zero and infinity).

33) 0 to (infinity-1) which is 'all 9' numbers is good enough for practical number applications. 'Vedic mathematics' makes use of virtues of all 2D square matrix positions (excluding infinity). Vedic sutras carry 'sense of doing it'!

@Mark:

I have to quibble a bit on the basis that "number" isn't a cleanly defined technical term in mathematics.

The argument that "Infinity doesn't obey the rules of arithmetic we expect from numbers and therefore isn't a number" cannot stand unless you insist on saying that the "ordinal numbers" are not after all numbers (when considered with the "usual" ordinal addition):

omega +1+omega=omega+omega.

But common usage amongst mathematicians ensures these are numbers. Where else will we find the meaning of "number" except in usage (since there is no single technical definition as there is of "natural number" or "real number")?

"Number" was one of Wittgenstein's main examples of a "family resemblance" term. We call different thing (ordinals, cardinals, complex numbers...) "number" because they have enough in common with each other that we can recognise as "numberness".

But there doesn't seem to be a specific set of properties they all share which all "non-numbers" lack that can be taken as "fundamental axioms that define numbers" (e.g. the ordinals aren't nice algebraically, the complexes have no well-behaved order).

Of course there is no complex number, or (single) ordinal called "infinity", but the Riemann sphere (as mentioned above) is one of several sets where you would reasonably call the elements "numbers" and one of the elements is called "infinity". (Also it makes a lot of sense to extend the definition of reciprocal here to allow 1/0=infinity and 1/infinity = 0)

The problem with the cranks you get is that they are applying their intuition about real numbers to their intuition about infinity without grounding it in any formal basis that can produce meaning.

@Carsten

@60: That's not quite true; Godel showed that any system strong enough to model PA (integers with addition and multiplication) was incomplete or inconsistent; Presburger Arithmetic (integers with addition) or the reals with addition and multiplication are complete and consistent.

I will admit that when working with mathematics, and indeed when teaching it, I do sometimes write down 1/0 = infinity or 1/inf = 0, but always, _always_ with a caveat. Essentially, it's a shorthand used to skirt around some rather tedious limiting arguments, and it _will_ blow up in your face if any stress whatsoever is placed on it. It's important to mention this, especially if you use it when teaching. So, I always try to mention that I am using it as a shorthand, that it is not rigorous, and that it is dangerous.

Infinity is a pain. One of the reason that fourier analysis remained on shaky mathematical grounds for over a hundred years is because the theory involves infinities all over the place. Nevertheless, the utility of taking all these infinite integrals and sums was so great that people can be forgiven for being slightly less than rigourous. However, they paid the price for this with a capricious and cantankerous "results" of fourier analysis, which caused much consternation for many years. To this day, taking a fourier transform is still a tricky enterprise.

While not exactly analogous to the 1/0=inf matter, the history of fourier analysis does show that sometimes, even in mathematics, one needs to be a little less than rigorous to move ahead. However, there is no excuse for abandoning rigor entirely. And there is no justification for trying to elevate 1/0=inf to be anything other than the mathematical equivalent of a hack.

Mark, I don't know if you've done a post on this before, but if you want to do a follow up to this discussion, and a far more interesting and subtle one at that, you should investigate the debate surrounding the definition of 0^0 (google, 'zero to the power of zero'). Far more elusive than infinity or nullity, this one continues to provoke debate. It will be interesting to see what names have "recently" weighed in on this one.

Why must I fail at every attempt to close blockquotes?

Paul Murray @ 57:

The difference between â(-1) and 1/0 is that the square root of minus one still obeys all the usual laws of mathematics. You can pretend that there is this number,

jsuch thatj*j= -1, and it behaves exactly as though it were a real number. You just can't count outjof anything.However, division by zero

does notobey the usual laws of mathematics. Since 0 *x= 0 for allx, the operation of multiplying by zero is irreversible: there is no way to recover whatever it was that you multiplied by zero.You could argue that since -1 * -1 = 1, then the operation of squaring is also not reversible. But at least there are only ever two possible values for

xsuch thatx*x=y, and we can deal with that.>The problem with the cranks you get is that they are applying their intuition about real numbers to their intuition about infinity without grounding it in any formal basis that can produce meaning

To add to the total confusion let me mention that in ZF in fact 1/0 = 0 .

Here is why: this is the application of a binary operation "/" to a pair (1,0) that is outside of the domain. The value of a function on a point that is not in the domain is the empty set - see theorem apply_0 in Isabelle/ZF func.thy . I was unable to find a similar theorem in Metamath, but see the definition of function application there . The empty set in ZF is the same as zero of natural numbers. Of course this is typically different than the zero on the other side of the uequality in 1/0 = 0.

Awesome post, mark.

Just to echo what I think I saw elsewhere in the comments, the surreal numbers first defined by John H. Conway do treat infinite numbers, but they do not behave exactly the way many people would expect and even there 1/0 != \omega.

The square root of -1 does not obey all the usual laws of mathematics. It doesn't obey the laws of comparison, so they must be given up. But in the process, algebraic completeness / fundamental theorem of algebra is gained. That's a worthwhile tradeoff.

Similarly, in going from the nonnegative integers to all integers or the rationals, well-ordering / induction is lost in return for additive / multiplicative inverses.

It is a theorem of the axioms of the real / complex numbers that, for all x, x*0 = 0. In order to have a system for which 1/0 is well defined, it is necessary to give up or weaken one or more of the axioms.

So, in adding nullity to a system, what's being given up and what's being gained?

Very interesting post. Thanks.

While I would in no way argue with MarkCC's treatment of the total idiot whose email provoke his ire, I am somewhat surprised that apparently neither Mark nor any of his commentators have ever heard of Cantorian Transfinite Arithmetic which is a whole consistent mathematical system of whole hierarchies of both ordinal and cardinal infinite numbers. Of course they do not fulfil the Peano axioms but have their own, at first sight, somewhat bizarre calculation rules e.g. 1+omega is not equal to omega+1 where omega is the ordinal number representing the infinity of the natural numbers. A good introduction can be found in Randy Rucker's Infinity and the Mind.

Um, say what? Read the penultimate paragraph of the original post.

I would think non-standard analysis would be the more relevant topic than transfinite ordinals and cardinals. IIRC, Robinson obtained the usual rules for a field.

"What's the square root of a nice juicy plum?"

1) Begin with an arbitrary conceptualization X.

2) Replace X by the average between X and plum/X, (X + plum/X)/2. An approximate value (e.g., if you are a diversity admission) of the average is sufficient to ensure convergence.

3) Repeat (2) until X and plum/X are adequately close.

I'll just go away quietly and throw my self off an infinitely high cliff :(

Although I will say that I totally disagree with Mark's characterisation of the behaviour of infinity and what he seems to be assuming is the definition of number. There are many different number systems with many different definitions and from a mathematical or logical point of view none of them is more "correct" or "valid" than the others. The Peano axiom system does no deliver a clear definition of infinity, Cantor does and just because it then turns out that infinity does not behave the way our very limited intuition expects it to behave does not disqualify it as "a" or in Cantor's case infinitely many numbers.

A last comment for those who dragged Goedel into this discussion, Goedel showed that there is no finite proof of the consistency of arithmetic but Gerhard Gentzen proved the consistency of arithmetic using transfinite mathematical induction up to epsilon zero (that's omega tetrated omega!).

"Infinity is NOT a number"

It's a state of mind, right?

I for one find the concept of infinity very confusing, but how can you understand calculus, set theory, logic, or virtually any other branch of mathematics without some kind of understanding of infinity.

Just try imagining the first real number that is the successor to the number 0 or the successor to the number pi. It makes about as much sense as dividing 1 by 0. Nevertheless, when I imagine a number line between 0 and 1 filled with points that correspond to real numbers then I just assume that there is a first real number following 0.

It seems to me that mathematics is missing a fundemental explaination in regards to infinity. The Banach-Tarski paradox, Cantors diagonal arguement, the Axiom of Choice, and a lot more all make my head hurt.

Geez! You do get some weirdos! Your heckler apparently doesn't know that zero is considered an "even number."

I would just ask him/her to plug the relation into a computer program, sit back, and say, "Some "infinity!" when the program terminates. ;-)

I wish I knew 1/10 of what you know.

Well, you'll never hit the bottom, and maybe that's worth something. :-)

No I'll suffer infinite torment waiting for the crunch to come!

@71

If I recall (and I might not) the "square root of -1" is undefined, but "i" is a number such that "i-squared = -1".

"I'll just go away quietly and throw my self off an infinitely high cliff..."

"Well, you'll never hit the bottom, and maybe that's worth something."

It depends. If you accelerated sufficiently, your velocity could become infinite, and you could reach "infinity" in finite time.

See, for example:

"The Existence of Noncollision Singularities in Newtonian Systems", Zhihong Xia, Ann. Math. 135(1992)411-468.

Dan:

"The square root of -1" is not exactly undefined, but rather multivalued. Both i and -i do the job perfectly well.

I like to think of this in terms of the complex plane. Just as real numbers form a "number line", complex numbers form a "number plane", traditionally drawn such that the pure real numbers are along the horizontal axis and the pure imaginaries are along the vertical axis.

Multiplying complex numbers involves rotations and scaling. If we represent a familiar, ordinary real number by an arrow from the zero point, then addition just means sticking arrows head-to-tail, and multiplication involves stretching or shrinking an arrow. "2 + 3 = 5" means that if we take three more steps from the end of the "2" arrow, we land on the tip of the "5" arrow. Saying 2×3 = 6 means that if we take the "2" arrow and stretch it to 3 times its length, we get the "6" arrow.

Negation, going from positive numbers to negative, means flipping an arrow so that it points in the opposite direction from the zero point. If you negate twice, you get back where you started. Because multiplying by a negative number involves both a scaling and a flip, squaring a negative number will always give a positive result. For example, what's -π × -π? Well, start with the "1" arrow, which points in the positive direction, and flip it so that it points in the negative direction, and then stretch it out so that it has length π. Then take this "-π" arrow, flip it again, and stretch it by another factor π. The final arrow has length π2 and points in the positive direction.

If you work in a plane of numbers, you can do something else to your arrows: you can twist them, in addition to stretching or shrinking. The flip from +1 to -1 is just two successive twists by one quarter-circle each, so one quarter-circle twist — taking an arrow from horizontal to vertical — is "the square root of minus one". It's what you have to multiply by itself to reach the -1 arrow. However, you can twist clockwise or counterclockwise, equally well. This means you'll have two arrows, each of which can serve as the square root of -1, which are the flipped versions of each other.

Blake: Thank you for responding at such length to my comment, especially given that I was somewhat off-topic to start with. The point I hoped (and seemingly failed) to make was using an incorrect definition of i might get you into trouble just as must as an incorrect definition of 1/0, as I demonstrated in post #31.

To my understanding, "The square root of -1" as used in post #71 is an incorrect definition of i. Rather i is a number such that "i*i = -1" (and as you say, could be multivalued). I rarely stray from the real numbers myself these days, so I am not equipped to explain why one definition is correct and the other is not - but that is my recollection.

"The square root of -1" is multivalued in the sense that "the square root of 4" is multivalued.

"an incorrect definition of i might get you into trouble"

An incorrect definition of anything might get you into trouble.

The issue here is why 1/0 is incorrect.

An additional point about Presburger arithmetic (integers, addition, no multiplication) is not only that it is complete, but that we have a bound on the number of steps in a proof of any proposition within it. I think that it is (off the top of my head) 2^2^n where n is the length of the proposition.

That makes Godel incompleteness so surprising.

And note that we do not know with certainty that integer arithmetic is consistent.

re 71:

I don't know what this means. The "laws of comparison" don't have to be given up, they just need to be redifined to operate on vectors, or numbers as ordered pairs. Generally by defining the magnitude and only allowing comparison of magnitudes. Thus "5i %lt; 6" is illegal, but "|5i| %lt; |6|" is okay.

Regardless of that, I think it is still meaningful to say 2i %gt; i. "Laws of comparison" require one to be using a common unit, you cannot ask, "which is greater, 2 miles or 5 pounds". Likewise "imaginary" numbers are a kind of unit so you cannot compare 5i to 5, but you can compare 6i to 7i.

Mark: I completely agree with you about how frustrating it can be discussing mathematics with someone who hasn't yet grasped the key point that mathematical objects are not just syntax - that a piece of syntax like "1/0" doesn't mean anything at all until someone defines it.

However... I think you should be aware that there is one place in mathematics where 1/0 does make perfect sense, and that's in the context of the 'projective line' P^1 that one meets in algebraic geometry. Given any three arbitrary points x, y and z on the projective line, one can define arithmetical operations (regular maps P^1 x P^1 -> P^1) such that x plays the part of 0, y plays the part of 1 and z plays the part of infinity, and (if I recall correctly) satisfies most of the properties that one naively expects infinity to have. However, 0/0 remains undefined.

Axioms of comparison:

There exists a relation "<" with the properties of:

- Total order (trichotomy & transitivity)

- if a < b, then a + c < b + c

- if 0 < a and 0 < b, then 0 < a*b

A consequence of these axioms and the other axioms of the reals is that 0 = a*a or 0 < a*a.

Axioms of comparison:

There exists a relation "<" with the properties of:

- Total order (trichotomy & transitivity)

- if a < b, then a + c < b + c

- if 0 < a and 0 < b, then 0 < a*b

A consequence of these axioms and the other axioms of the reals is that 0 = a*a or 0 < a*a.

Freak @ 88:

Actually, there's something I didn't spot before. You can't tell

japart from -j(which also gives -1 when squared) in the way that you can tell 2 apart from -2. Because you can count out 2 physical objects, measure out 2 metres of string, or wait for time to move forward by 2 second; but sincejis imaginary, you can't* visualise it.*Well, unless you build an electronic circuit with capacitors or inductors, and you have a dual trace oscilloscope. And even then, you're really only observing an established convention.

Which infinity do you mean? +â and -â are certainly not real numbers.

But â is used quite often in the context of complex numbers---the regular complex plane augmented with the single number â even has a name, the extended complex plane. It is the one-point compactification of the complex plane. It is isomorphic to the Riemann sphere.

One can also create a one-point compactification of the real numbers. In that context, one augments the reals with a single quantity sometimes called Â±â and sometimes called just â.

With the extended complex plane, it is entirely reasonable to say that 1/0 = â. With the one-point compactification of the reals, it is also reasonable to say that 1/0 = Â±â.

It is not, however, correct to say that 1/0 = +â, or that 1/0 = -â. These uses of the infinity symbol indicate limiting processes.

How far does this get you? Not very, perhaps. We have not succeeded in making division always work to give a unique answer, because 0/0 is still indeterminate. We now have two numbers, 0 and â, that are neither positive nor negative. Some things work better, while other things break.

I talk about infinity to my classes, who like to write â but are usually thinking something more like +â. I tell them that there are number systems limited to: nothing, 1, 2, 3, many. We do a little calculating with this system, finding out that (many - 1) might be many or 3, but it can't be 2. (many - 1) is somewhat---but not completely---indeterminate in this system. (many - many), however, is completely indeterminate.

Infinity as they are used to using it behaves a lot like "many" in the "nothing, 1, 2, 3, many" system.

I then introduce them to what I describe to them as a unit infinity, Î©, and its reciprocal, Î´, the unit infinitesimal. These quantities are scalable, so that 2Î© is twice as much as Î©, and is not equal to Î©. Î© and Î´ behave like algebraic indeterminates, and can be manipulated using ordinary algebra just as if they were an x and a 1/x. Powers of Î© or Î´ have an obvious size order, and so do series expansions in powers of Î© or Î´. I can start them calculating jets (e.g. the jet of f at a, which is f(a+Î´) up to truncation), and therefore Taylor series, before they even know what a derivative is...

I don't use

iand-iorjand-j. I arrange the real axis vertically, with the positive direction upward. That way the complex plane is bilaterally symmetric in the way that people are. Instead ofjand-j, I then talk about Right and Left.jand-jcan be told apart---and not told apart---in about the same way as right and left: its obvious that there are two of them and that they're different, but it's not always easy to tell which is which.Before this, I will have introduced vectors in the plane, using North, East, South and West as unit vectors. (My students can be a little shaky on negative numbers before I get them.) I will have elicited how -1 Ã North is South and vice versa, so that they can see why -1 Ã -1 has to be 1. I will also have introduced Right as a quantity which, when you multiply North by it, gives East, or when you multiply East by it, gives South, and so forth.

Re: #94

"You can't tell j apart from -j (which also gives -1 when squared) in the way that you can tell 2 apart from -2."

Yes you can. It is merely arbitrary which one you pick first to name that way. Then you have no choice for the other. You can tell the difference, it's merely the nomenclature which has an arbitrary choice.

Please, everyone knows that sqrt(Plum) = hamburger.

Hey Mark,

Maybe you could swing a job at G for Randal Munroe. It may save his relationship...

Re Blake Stacey #5;

I've always been puzzled by that. Yes, the C: drive has not been found by DOS but what's the difference between Abort and Fail? Does DOS return you to your previous prompt, but this time with a pout?

PS - Comment #100 - yay me!

OK, so if infinity is not a number, what exactly is it?

Another situation where it's useful to consider infinity as a number is when you're dealing with continued fractions; to make your definitions consistent, you want to define the empty continued fraction [] to be infinity. (Since continued fraction expansions only really make consistent sense if you're dealing with positive reals, this isn't too problematic.)

re #101: "so if infinity is not a number, what exactly is it?"

Infinity, most often denoted as [insert figure 8 on its side symbol], is an unbounded quantity that is greater than every real number. The symbol [insert figure 8 on its side symbol] had been used as an alternative to M (1000) in Roman numerals until 1655, when John Wallis suggested it be used instead for infinity.

Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor's treatment of infinite sets.

Informally, 1/[insert figure 8 on its side symbol]=0, a statement that can be made rigorous using the limit concept,

lim_(x->[insert figure 8 on its side symbol])1/x=0.

Similarly,

lim_(x->0^+)1/x=[insert figure 8 on its side symbol],

where the notation 0^+ indicates that the limit is taken from the positive side of the real line.

In Mathematica, [insert figure 8 on its side symbol] is represented using the symbol Infinity.

SEE ALSO: Aleph, Aleph-0, Aleph-1, Cardinal Number, Complex Infinity, Continuum, Continuum Hypothesis, Directed Infinity, Division by Zero, Hilbert Hotel, Infinite, Infinite Set, Infinitesimal, Limit, Line at Infinity, L'Hospital's Rule, Point at Infinity, Transfinite Number, Uncountably Infinite, Zero

REFERENCES:

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 19, 1996.

Courant, R. and Robbins, H. "The Mathematical Analysis of Infinity." Â§2.4 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 77-88, 1996.

Hardy, G. H. Orders of Infinity: The 'infinitarcalcul' of Paul Du Bois-Reymond, 2nd ed. Cambridge, England: Cambridge University Press, 1924.

Lavine, S. Understanding the Infinite. Cambridge, MA: Harvard University Press, 1994.

Maor, E. To Infinity and Beyond: A Cultural History of the Infinite. Boston, MA: BirkhÃ¤user, 1987.

Moore, A. W. The Infinite. New York: Routledge, 1991.

Morris, R. Achilles in the Quantum Universe: The Definitive History of Infinity. New York: Henry Holt, 1997.

Owen, H. P. "Infinity in Theology and Metaphysics." In The Encyclopedia of Philosophy, Vol. 4. New York: Crowell Collier, pp. 190-193, 1967.

PÃ©ter, R. Playing with Infinity. New York: Dover, 1976.

Rucker, R. Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1995.

Smail, L. L. Elements of the Theory of Infinite Processes. New York: McGraw-Hill, 1923.

Thomson, J. "Infinity in Mathematics and Logic." In The Encyclopedia of Philosophy, Vol. 4. New York: Crowell Collier, pp. 183-190, 1967.

Vilenskin, N. Ya. In Search of Infinity. Boston, MA: BirkhÃ¤user, 1995.

Weisstein, E. W. "Books about Infinity." http://www.ericweisstein.com/encyclopedias/books/Infinity.html.

Wilson, A. M. The Infinite in the Finite. New York: Oxford University Press, 1996.

Zippin, L. Uses of Infinity. New York: Random House, 1962.

CITE THIS AS:

Weisstein, Eric W. "Infinity." From MathWorld--A Wolfram Web Resource.

Also, see this interesting stuff from the blog "A Neighborhood of Infinity" about how an "infinity" (specifically a transfinite ordinal) can describes a 1 Player Game that anyone can play and which is guaranteed to terminate after a finite number of moves.

What's the use of a transfinite ordinal?

This connects what Mark Chu-Carroll has been explaining about Game Theory and about Infinity, in a way that he may or may not endorse, but which I think is really fun.

Mr. Starvos, you are also wrong, for if x tends to zero then the function 1/x does not tend to anything particular. In this case the limit does not exist. Actually

it should be as follows........

lim 1/x = 0

(x tends to infinity )

Gribit @ #98 wrote:

It does?

{searches frantically for the Plum button on his calculator}

So what if this is a really late post!

When I did math(s) at collage I always thought about it like this:

In linguistic terms, numbers in mathematics are nouns (things).

Infinity is also a noun but it is a noun used to identify a process, so that it is really identifying a verb (action) which means something like 'going on forever.' So infinity means 'the process of going on forever'.

So when you see (3+1) you are seeing one thing plus another thing. When you see (infinity+1) you are seeing the process of going on forever plus 1.

I think mathematics shot itself in the foot when is started using the phrase 'to infinity.' It makes it sound like infinity is a point you can get to. If we replaced it with 'going on forever' things would be much easier.

I know that this is not the whole story and is a bit clunky but it sure beats treating infinity as big number.

This post is misguided. To begin with, you say: "For example, the most basic definition of numbers that I know of is Peano arithmetic." There are more basic definitions that I will turn to shortly, but even if there were not the line of reasoning you pursue is bad. You go on to argue that because extending the naturals to include infinity yields a system that violates the Peano Axioms, infinity is not a number. If that argument works then the real numbers are not numbers either, since it is not the case that every real number has a unique successor.

Furthermore, the Peano Axioms do not "define" the natural numbers. They define a certain kind of sequential structure, of which the natural numbers with the successor function are but one instance. But there are many sequences that satisfy the Peano Axioms. For example, in set theory the ZF numbers generated from the empty set by the successor function S(x)={x, {x}} satisfy the Peano Axioms. So do the VN numbers generated from the empty set by the successor function S(x)={x}. The ZF numbers are not the same as the VN numbers, clearly. In addition to the ZF and VN numbers there are countless sequences generated by the respective successor functions of the two systems beginning with various different objects other than the empty set. At best, the Peano Axioms define a structure shared by all of the systems that satisfy them, including perhaps the natural numbers, but they do not in any way pick out the natural numbers uniquely.

So, the Peano Axioms do not define the natural numbers and even if they did your argument using the Peano Axioms to show that infinity is not a number would also show that the reals are not numbers. Your fixation on the matter of adding one to infinity indicates to me that you are perhaps running together the conception of numbers as cardinals and numbers as ordinals. I will try to clarify a bit, beginning with the Frege-Russell conception of numbers as cardinals as applied by contemporary abstractionist philosophers of mathematics.

The modern abstractionist program begins with a principle employed by Frege that has come to be known as Hume's principle (Hume apparently stated it somewhat vaguely). I'll use the following notation: #F for "the number of Fs", where "F" stands for some concept. For example, #(books on my desk) means "the number of books on my desk". Terms of the form #F are logically singular referring terms that stand for numbers. Hume's Principle, without further ado:

HP: (#F = #G) iff (there is a 1-1 function from the Fs onto the Gs)

The left-hand side (lhs) of HP states an identity condition for the singular terms formed by the operator "#" and does refer to numbers. The right-hand side of HP states an equivalence condition for concepts and does not refer to numbers. Abstractionists believe that HP defines the numbers, that HP succeeds in giving a proper definition where the PA fail. If they are correct (and there is considerable debate) then infinity most certainly is a number. For example, take the concept F="finite number". Then (countable) infinity is #F. Notice that after using HP to define the numbers (as cardinals) it seems promising that one may be able to prove that the system of finite cardinals satisfies the Peano Axioms and it is certainly desirable to do so, but there would be no reason to suppose that numbers defined by HP that are not finite are thereby not numbers!

On the cardinality-based approach we may define the numbers as cardinals then define a successor function for the finite cardinals in an intuitive way so that they (viz., the finite cardinal numbers) satisfy the Peano Axioms. On this approach (countable) infinity is a number sure enough, since it is the number of the concept "finite number" but there is ambiguity as to how we might define its successor because we have only defined that function for the finite cardinals. How to proceed? We might define the successor of countable infinity to be the first non-countable cardinality, for example. But then it's not the case that infinity = infinity + 1 (i.e., infinity is not equal to its successor). If that's so, then the reductio you use to show that if infinity were a number then 1=2 fails. There seems to be some confusion brought about by the realization that adding one item to a countably infinite set yields a countably infinite set, but there is no reason to identify the successor function with that operation. That is, there is no reason to hold that (outside of the finite numbers for which this is a derivable theorem) #F + 1 = #(F and one more thing).

Another approach that has been taken which is closer historically to the methods of Cantor and Dedekind has been to think of numbers as ordinals. I won't dwell on this too much except to say that when thought of as ordinals the successor of infinity (i.e.: infinity plus one) is not equal to infinity. True, adding one more book to an infinite collection of books yields an infinite collection of books, but this is thinking is terms of cardinals not in terms of ordinals. Adding an item that is after every item in an infinite sequence of items does yield a sequence with a new and different order structure. For example, if the original sequence is the natural numbers (an infinite collection that does not have a greatest member) then adding an item that is after them all yields a sequence with a completely different order structure (an infinite collection that does have a greatest member). So, again (leaving out some details because I've already gone on too long) it will not be the case that infinity = infinity + 1 and your reductio to show that infinity is not a number fails.

To be sure, your correspondent was incorrect in many, many things including the statement that "what is interesting about infinity is that when you add or subtract a number from it, infinity remains." That is neither true nor interesting about the number infinity (though something like it is both true and interesting about infinite classes). You, however, were just as wrong to take this statement as an hypothesis for a reductio to show that infinity is not a number. The statement is false not because infinity is not a number but because on both the cardinal and the ordinal conceptions of infinity it is not equal to its own successor (i.e., nothing forces us to define it as such).

Re: Chris #107 -- "In linguistic terms, numbers in mathematics are nouns (things). Infinity is also a noun but it is a noun used to identify a process, so that it is really identifying a verb (action) which means something like 'going on forever.' So infinity means 'the process of going on forever'."

Mathematics is LIKE a language, but it is not as close to English as you hope. I think.

I have been entranced the past few months by the meta-language of Jeffrey Morton. I've kicked some implications back and forth with him by email, discussed it with some science PhD friends of mine, and even used it in teaching high school students.

Categorification and Matter

Posted by Jeffrey Morton under category theory, philosophical, physics, tqft

http://theoreticalatlas.wordpress.com/

... Right now I want to say something a bit broader than I have been doing - somewhere between "intuitive justification" and "philosophy". The motivation is that whenever I talk about ETQFT's {JVP: Extended Topological Quantum Field Theories} and how to see them as introducing

matter into quantum gravity, there's always some puzzlement about this "categorification" business. To people who think a lot about category theory, it may seem natural, but many of those interested in physical questions don't fall in this category, and the whole idea of "categorifying" a theory seems like a weird, arbitrary imposition.

So talking to these different audiences has forced me to think about how to give an intuitive account of why this might be a good idea. Ideally this will not be so precise as to be incomprehensible, or so vague as to be useless. In reality, this will be at best a rough sketch of such a justification.

Stuff, Structure, and Properties

One aspect of the relationship which I wanted to comment on, one that almost seems like a pun, is the trichotomy which John Baez and Jim Dolan like to use in describing mathematical, um, widgets (I would use the more standard term "objects", or maybe "structures", but both of these words have technical meanings in the following) in categorical terms. This is the distinction between "stuff", "structure", and "properties". (More details here and via subsequent links - some of which shows up in my first paper).

http://arxiv.org/abs/math.QA/0601458

Almost any usual mathematical widget can be broken down this way:

(1) they consist of some "stuff", often in the form of some sets;

(2) the stuff is equipped with "structure", often described by some functions;

(3) the structure satisfies some "properties", often expressed as equations.

For example: a group is:

(1) a set G of elements, equipped with

(2) a group operation (expressed as a function m : G x G --> G), and a special identity element (picked out by a function from the one-element set, 1 : * --> G), and an inverse for each element (given by an inverse function

inv : G --> G.

These satisfy

(3) the group axioms, which are some equations involving expressing some properties - associativity, the properties of 1 and inverses.

In this case, the structure live inside the category of sets and functions - but similar things could be said in any other category. For instance, in the category of topological spaces and continuous functions, the same setup gives the definition of a topological group, likewise divided into "stuff" (objects, in this case topological

spaces), "structure" (some morphisms), and "properties" (equations between morphisms).

Re: Chris #107 -- [resubmitted with one URL omitted to prevent filtering] "In linguistic terms, numbers in mathematics are nouns (things). Infinity is also a noun but it is a noun used to identify a process, so that it is really identifying a verb (action) which means something like 'going on forever.' So infinity means 'the process of going on forever'."

Mathematics is LIKE a language, but it is not as close to English as you hope. I think.

I have been entranced the past few months by the meta-language of Jeffrey Morton. I've kicked some implications back and forth with him by email, discussed it with some science PhD friends of mine, and even used it in teaching high school students.

Categorification and Matter

Posted by Jeffrey Morton under category theory, philosophical, physics, tqft

... Right now I want to say something a bit broader than I have been doing - somewhere between "intuitive justification" and "philosophy". The motivation is that whenever I talk about ETQFT's {JVP: Extended Topological Quantum Field Theories} and how to see them as introducing

matter into quantum gravity, there's always some puzzlement about this "categorification" business. To people who think a lot about category theory, it may seem natural, but many of those interested in physical questions don't fall in this category, and the whole idea of "categorifying" a theory seems like a weird, arbitrary imposition.

So talking to these different audiences has forced me to think about how to give an intuitive account of why this might be a good idea. Ideally this will not be so precise as to be incomprehensible, or so vague as to be useless. In reality, this will be at best a rough sketch of such a justification.

Stuff, Structure, and Properties

One aspect of the relationship which I wanted to comment on, one that almost seems like a pun, is the trichotomy which John Baez and Jim Dolan like to use in describing mathematical, um, widgets (I would use the more standard term "objects", or maybe "structures", but both of these words have technical meanings in the following) in categorical terms. This is the distinction between "stuff", "structure", and "properties". (More details here and via subsequent links - some of which shows up in my first paper).

http://arxiv.org/abs/math.QA/0601458

Almost any usual mathematical widget can be broken down this way:

(1) they consist of some "stuff", often in the form of some sets;

(2) the stuff is equipped with "structure", often described by some functions;

(3) the structure satisfies some "properties", often expressed as equations.

For example: a group is:

(1) a set G of elements, equipped with

(2) a group operation (expressed as a function m : G x G --> G), and a special identity element (picked out by a function from the one-element set, 1 : * --> G), and an inverse for each element (given by an inverse function

inv : G --> G.

These satisfy

(3) the group axioms, which are some equations involving expressing some properties - associativity, the properties of 1 and inverses.

In this case, the structure live inside the category of sets and functions - but similar things could be said in any other category. For instance, in the category of topological spaces and continuous functions, the same setup gives the definition of a topological group, likewise divided into "stuff" (objects, in this case topological

spaces), "structure" (some morphisms), and "properties" (equations between morphisms).

what is the square root of a plum?

I am a lesson plan editor at LessonPlansPage.com. We received a lesson on infinity, but I am not sure what math category to put it in. I want to put it into number sense, but would this be wrong since infinity is not a number? Hate to put it in "other." What would you suggest?

find out the square root of a plum and bobs your uncle

no infinaty is not a number b/c we do not know how much it is.