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: limx→∞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 limx→01/x=+∞; if x≤0, then limx→01/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.
Physical Science
Comments
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.
Posted by: Wry Mouth | October 13, 2008 1:25 PM
yeh, like if 3%2= 1.3R
so .3 repeates endlessly
so 1.3r x 1.3r = 2.666666666666666666666666666r not 3
Posted by: tomg | October 13, 2008 1:31 PM
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 "limx→∞1/x=∞" should be the limit when x tends to 0 not infinity!
Posted by: stavros | October 13, 2008 1:31 PM
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.
Posted by: Don Henry | October 13, 2008 1:36 PM
Error reading drive C:. Abort, Retry... FAIL?
Posted by: Blake Stacey | October 13, 2008 1:40 PM
limx→∞1/x=∞!
Don't you mean
limx→01/x=∞!
Posted by: Charles | October 13, 2008 1:45 PM
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.
Posted by: Blake Stacey | October 13, 2008 1:46 PM
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?
Posted by: Blaisepascal | October 13, 2008 2:16 PM
Re Don #4:
No, it's got nothing to do with practicality. It's got to do with fundamental semantics.
Infinity isn't a number. We don't "exclude" infinity from the numbers. That would imply that it is a 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 meaning of set, {a,b,a,c} isn't a set. I'm not excluding {a,b,a,c} from the sets for practicality. I'm doing it because it's not a set, and if I pretend that it is, then by including it, sets stop 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.
Posted by: Mark C. Chu-Carroll | October 13, 2008 2:17 PM
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=∞.
Posted by: Mark C. Chu-Carroll | October 13, 2008 2:22 PM
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"
Posted by: Jonathan Vos Post | October 13, 2008 2:41 PM
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.
Posted by: grasshopper | October 13, 2008 2:42 PM
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.
Posted by: Charles Tye | October 13, 2008 2:49 PM
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).
Posted by: Greg Laden | October 13, 2008 2:55 PM
"The moment you see something use the statement that 1/0=∞, 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 ∞ its inverse.
Posted by: Flaky | October 13, 2008 3:05 PM
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.
Posted by: bruno | October 13, 2008 3:06 PM
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.
Posted by: mds | October 13, 2008 3:08 PM
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?
Posted by: Juraj Lörinc | October 13, 2008 3:11 PM
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.
Posted by: Alex Besogonov | October 13, 2008 3:13 PM
"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:-)
Posted by: Maya Incaand | October 13, 2008 3:18 PM
"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:-)
Posted by: Maya Incaand | October 13, 2008 3:20 PM
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 measure position.
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.
Posted by: Mark C. Chu-Carroll | October 13, 2008 3:25 PM
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.
Posted by: Lout | October 13, 2008 3:31 PM
@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.
Posted by: Beowulff | October 13, 2008 4:09 PM
If I remember correctly, in the theory of (complex) rational functions, concepts like 1/0 = ∞, ∞+1 = ∞ and -∞ = ∞ are used quite liberally... Statements like these are probably to be understood to be about limits in the context of the topology of ℂ ∪ ∞, but mathematicians' own usage does not help the confusion.
Posted by: lukas | October 13, 2008 4:21 PM
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.
Posted by: neop | October 13, 2008 4:57 PM
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.
Posted by: micah | October 13, 2008 5:01 PM
You mean that infinity is not a natural number. Infinity (which one?) is a cardinal number.
http://en.wikipedia.org/wiki/Cardinal_number
Posted by: JS | October 13, 2008 5:03 PM
What about the projective line?
Posted by: Sam Iam | October 13, 2008 5:06 PM
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.
Posted by: Alex Smith | October 13, 2008 5:10 PM
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.
Posted by: Dan | October 13, 2008 5:25 PM
I would argue that the situation you are describing is division by one.
Posted by: duus | October 13, 2008 5:27 PM
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.
Posted by: Ross Duncan | October 13, 2008 5:38 PM
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.
Posted by: Ross Duncan | October 13, 2008 5:41 PM
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?
Posted by: x5315 | October 13, 2008 5:42 PM
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.
Posted by: Ross Duncan | October 13, 2008 5:49 PM
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?
Posted by: Nomen Nescio | October 13, 2008 5:54 PM
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.
Posted by: Joshua Zelinsky | October 13, 2008 5:55 PM
There is one method for defining and working with ∞ 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 ∞ 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 Möbius transformations. Lines are just circles that pass through the point at infinity.
Posted by: Mark Reid | October 13, 2008 5:56 PM
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.
Posted by: Sam Iam | October 13, 2008 6:01 PM
@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
Posted by: George Pollard | October 13, 2008 6:04 PM
#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.
Posted by: Egbert B. Gebstadter | October 13, 2008 6:12 PM
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.
Posted by: JimFiore | October 13, 2008 6:14 PM
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.
Posted by: Sam Iam | October 13, 2008 6:29 PM
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.
Posted by: AA | October 13, 2008 6:33 PM
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.
Posted by: AA | October 13, 2008 6:41 PM
Mark -
Your counterpart Zeno has very recently made this concept graphical and topical. Coincidence? I think not. There are no accidents, man.
Posted by: jre | October 13, 2008 7:08 PM
@15, if 1/0=∞ is used to actually mean 1/(a very small number) = (a very large number), why not use 1/ε=ω instead?
Posted by: Anonymous | October 13, 2008 7:11 PM
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!
Posted by: paul | October 13, 2008 7:27 PM
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!
Posted by: paul | October 13, 2008 7:29 PM
#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!
Posted by: Nomen Nescio | October 13, 2008 7:47 PM
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.
Posted by: sigfpe | October 13, 2008 7:47 PM
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.
Posted by: sigfpe | October 13, 2008 7:56 PM
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.
Posted by: sigfpe | October 13, 2008 7:58 PM
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.
Posted by: AA | October 13, 2008 8:45 PM
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.
Posted by: Jonathan | October 13, 2008 8:58 PM
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.
Posted by: Paul Murray | October 13, 2008 9:04 PM
I've always heard infinity described as a direction rather than a vector or a scalar.
Posted by: Anon | October 13, 2008 11:55 PM
@Anonymous #48
"@15, if 1/0=∞ is used to actually mean 1/(a very small number) = (a very large number), why not use 1/ε=ω instead?" They're engineers? I suppose 1/0=∞ 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.
Posted by: Flaky | October 14, 2008 1:32 AM
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.
Posted by: Mark | October 14, 2008 2:05 AM
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.
Posted by: Carsten | October 14, 2008 3:00 AM
There is another way to calculate on R with ∞ as a number: http://en.wikipedia.org/wiki/Max-plus_algebra
The problem with ∞ is: what is ∞-∞? 0? ∞? 42?
Posted by: andreas | October 14, 2008 3:13 AM
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
Posted by: MarkW | October 14, 2008 6:44 AM
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)à number groups increases by a-unit ß (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'!
Posted by: Raghuthaman | October 14, 2008 7:11 AM
@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
Posted by: Matt Heath | October 14, 2008 8:09 AM
@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.
Posted by: Freak | October 14, 2008 8:10 AM
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.
Posted by: ObsessiveMathsFreak | October 14, 2008 8:10 AM
Why must I fail at every attempt to close blockquotes?
Posted by: Matt Heath | October 14, 2008 8:11 AM