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Is infinity a number? Why or why not?

Some commentary:
I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school — but a difficult one to answer in an intelligent manner. I'm hoping to see a combination of strong citations and solid reasoning in the answers.

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For the purposes of IEEE arithmetic, yes (but it's certainly not an ordinary number in that respect). In analysis, however, no; it can however be given a sign (for real analysis) or direction (for complex analysis). Then you have the set-theoretic stuff... –  J. M. May 1 '11 at 22:44
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To summarize J.M. and Qiaochu's answer, "It depends on what the meanings of 'infinity' and of 'number' is." –  Arturo Magidin May 1 '11 at 23:05
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I have read your question and decided what to write as answer, but then I read your announcement that you will delete your question if you do not like some answers, so I decided that I will not spend my time on your question. –  Phira May 1 '11 at 23:10
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@user9325, for what it's worth, I believe the OP cannot unilaterally delete the question once answers have been posted. There may be extra stipulations like at least one answer must have at least one upvote, but I'm not entirely sure. –  yunone May 2 '11 at 1:43
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@user, I apologize if I came across as harsh. I've spent a lot of time on the SE network, but this is my first post on Math SE specifically. I was afraid of sparking a slew of answers that didn't follow the site/network quality policies and seeming like a discussion question-loving newbie. Considering the average quality of the answers here, I have no desire to delete anything. In any case, I'm going to delete the second half of the commentary now because it's no longer relevant. On-topicness is clearly not in question, and as others have said, auto-delete is not an option. –  Pops May 2 '11 at 14:09
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8 Answers

up vote 108 down vote accepted

It comes down to the definition of "number," as well as the definition of "infinity." Personally I don't think it's worth having an opinion on this subject; there are more precise words than "number" and "infinity" in mathematics. Historically the word "number" has come to mean an increasingly general list of things:

The word "infinity" has also come to mean an increasingly general list of things: it might refer to

Some of these meanings are compatible, as the above list demonstrates. But again, there are more precise words than "number" and "infinity" in mathematics, and if you want to get anywhere you should learn what those words are instead.

Here are some of those more precise words.

  • A set is a formalization of the intuitive notion of a bag of objects, and we can talk about finite or infinite sets. For example, $\{ 1, 2, 3 \}$ is a finite set, whereas the set of natural numbers is an infinite set. One can do arithmetic with sets in a way that leads to the arithmetic of the natural numbers: for example, taking the disjoint union corresponds to addition, and taking the Cartesian product corresponds to multiplication. These ideas lead to the arithmetic of the cardinal numbers, and similar ideas lead to the arithmetic of the ordinal numbers.
  • A ring is a formalization of the intuitive notion of a set of things you can add and multiply, so in some sense one can regard elements of rings as "generalized numbers" (but note that not every generalization I listed above can be interpreted in this way). When certain people say that "infinity is not a number," what they mean is that you can't adjoin an element called $\infty$ to the ring $\mathbb{R}$ of real numbers such that addition and multiplication do what you want them to do, the basic problem being that $\infty - \infty$ can't be consistently defined to satisfy the other rules of arithmetic if you also want it to be true that $n + \infty = \infty$ for any finite $n$.
  • A field is a commutative ring in which it's also possible to divide by nonzero elements. Some people would like to say that $\frac{1}{0} = \infty$, but by mathematical convention the element $0$ never has a multiplicative inverse in a field, the basic problem being that $0 \cdot \infty$ can't be consistently defined to satisfy the other rules of arithmetic. However, one can make sense of the expression $\frac{1}{0}$ in projective geometry; it describes the point at infinity on the projective line.
  • A topological space is an abstract setting for ideas like nearness and taking limits. Sometimes we don't want to view $\mathbb{R}$ as a ring, but as a space (the real number line), and we can talk about embedding this space into a larger space where more limits exist: this is known as compactification, and is an extremely useful tool in mathematics and physics. For example, we would like to say that the sequence $1, 2, 3, ... $ has limit $\infty$ in some sense, and we can do this compactifying $\mathbb{R}$.
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+1 For "Personally I don't think it's worth having an opinion on this subject..." I agree about the lack of precision in terminology. –  JavaMan May 2 '11 at 0:43
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Coming to think about it, this should really be the tag wiki for the infinity tag. –  Asaf Karagila May 10 '11 at 8:47
    
I have found Georg Cantor's two papers from the 1895 and 1897 particularly interesting and informative on infinity and the transfinite numbers. See "Contributions to the founding of the theory of Transfinite Numbers"- Dover Publications. –  Sean Jun 1 '12 at 12:27
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The word "number" carries some baggage with it. If $a$, $b$, and $c$ are numbers, and $a+c=b+c$, one expects $a=b$. For infinity, that doesn't work; under any reasonable interpretation, $1+\infty=2+\infty$, but $1\ne2$. So while for some purposes it is useful to treat infinity as if it were a number, it is important to remember that it won't always act the way you've become accustomed to expect a number to act.

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"under any reasonable interpretation, $1+\infty=2+\infty$": This would rule out nonstandard analysis, in which infinite numbers exist, and $1+x=2+x$ is false for all $x$, including infinite $x$. It would also rule out the surreals. –  Ben Crowell Aug 8 '12 at 15:46
    
For the sake of precision, I want to point out that @Ben is interpreting "$\infty$" in non-standard analysis to denote an infinite (nonstandard) number, rather than to denote to the transfer of a standard use of the symbol $\infty$. –  Hurkyl Feb 11 '13 at 7:07
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@Hurkyl: IIRC, we have an unresolved disagreement about the use of the transfer principle for the symbol $\infty$ from the extended reals. I claim that the transfer principle cannot be applied to it, and you claim that it can. –  Ben Crowell Feb 12 '13 at 16:24
    
@Ben: Which is why I took care to phrase my comment as one that should be acceptable whether or not you believe the domain of the transfer map includes the entire superstructure over $\mathbb{R}$. –  Hurkyl Feb 12 '13 at 17:04
    
I would strongly disagree that one 'expects' $a + c = b + c \implies a = b$ for numbers a, b, and c. This might be true in general, but I don't think it's part of the intuitive understanding of what a number is. –  jwg Jul 29 '13 at 12:08
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To answer your question directly: no, infinity is not a number. That is, for most people in most situations most of the time, infinity is not included in the set of numbers.

Qiaochu gave a very good answer and I agree with his opinion parts. As Qiaochu pointed out, it depends largely on which definitions you choose to accept, or use, at any given time. Sometimes it's useful for infinity to be a number, other times it isn't. When Qiaochu talks about rings, sets, fields and spaces, and when I say "sometimes yes, sometimes no", what we're doing is referencing a group of axiom sets, or rule sets, which has different definitions and rules as to what numbers and infinity are.

If you're interested, this topic bleeds into a discussion about the history of math, mostly the last 200 years. Basically, the idea that any given definition or rule is "absolutely right" or "absolutely wrong" was discarded in favor of a more axiomatic approach. That is, things may be one way one day, and another way another day; and we, as mathematicians, are ok with that.

For a quick concrete example, ask yourself, "Is 2.5 a number?" If you're counting people, it is often absurd to consider 2.5 a number (thus the joke in the T.V. show). On the other hand, if you're measuring mass, it may be absurd to ignore 2.5 as a number. So the question, "Is 2.5 a number?" has no one answer all the time. In the same manner, the question "Is infinity a number?" has no one answer all the time.

Some interesting highlights on the history of math and infinity: -Alice's Adventures in Wonderland, by Lewis Carroll, is supposed to be a sarcastic take on the emergence of modern math, notably Non-Euclidean Geometry and Abstract Algebra. Carroll insists that math has was fine for the last 2000 years, and it doesn't need to be adjusted. He portrays emerging math concepts as absurd nonsense in wonderland. -Georg Cantor rocked the boat with his work on infinity.

As far as describing it in an intelligent way, just say that infinity is not a number because infinity is a meta word not in the set but used to describe the set. Just as the words "unbounded" and "non-empty" are (usually) not considered as numbers, infinity is (often) not considered as a number.

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What's that about Lewis Carroll and "one true math"? –  ShreevatsaR May 2 '11 at 11:20
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I think Carroll held out for Euclidean geometry as the one true way, against the claims of non-Euclidean geometry. Maybe this is what Rodney is referring to. –  Gerry Myerson May 2 '11 at 13:05
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To me infinity is just a way to represent something which cannot be written or expressed with common mathematics writings.

I don't really understand infinity as a precise number, I just understand it as some kind of domain for huge numbers, some place beyond our imagination.

The Knuth arrow notation http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation is a very good way to represent big numbers and compare them, but it won't help you understand why some fractions with the exp and ln functions will behave: infinity is there as some kind of numerical evaluation, it has no precise use.

A pretty good way to understand what infinity really means in term of reality, so that you can understand infinity is just an abstract and flawed, unprecise way to represent a value, try to trace the sin(1/x) function with some tool or other, try to zoom and try to understand what is happenning.

http://www.math.tamu.edu/~sottile/teaching/03S/sin1x-movie.html

Try also to understand convergent and divergent sequences.

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Infinity is no more beyond our imagination than the precise construction of the real numbers or any of a number of abstract constructions in mathematics. With the proper training all of these things can be understood as well as any human can understand anything. –  Qiaochu Yuan May 3 '11 at 3:06
    
That's finitist talk. Personally I can say that I am able to imagine infinite orders much much better than I can imagine things in two or three dimensions. –  Asaf Karagila May 3 '11 at 4:33
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well I certainly may lack mathematical skill and knowledge then... –  jokoon May 3 '11 at 15:56
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@jokoon Laugh out loud man! cheer out! Of course you could understand maths and you would have all the skills! But if you don't make effort and emerge in maths, you will never learn anything. It's the same thing like someone says I never win lottery, with the fact that he never buys a ticket. –  Ginger Head May 24 '12 at 6:27
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To simplify (with respect to Qiaochu's answer), most people think of: - 'number' as an integer or a real (a count or length), with their usual properties of addition and multiplication. - 'infinity' as some vague unreachable bigger than any integer or real, so that it doesn't follow all the same rules as integers or reals.

So in common English, 'infinity' is not a 'number'.

But of course in math, there are technical methods for dealing with a particularly defined 'infinity' so that it can be manipulated in the same sentence as an integer or real, so that one notices that though it doesn't share all properties with things called 'numbers', it shares some.

Even then, one normally does not call 'infinity' a 'number', but it does share some properties with them and can (sometimes be manipulated with them). So one might say that 'infinity' can be treated as a number sometimes.

That is, 'X is a Y' is not such a straightforward thing to answer.

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Agree with some earlier comments that it depends on the definition, but assume that a number can be used in algebraic expressions. then the following follows: Assume the following numbers exist.

In case x holds a number (x=∞) then there would exist another number y=∞+1. Then x≠y e.g. x+1=y But infinity +1 is still infinity e.g. ∞+1=∞ so if we insert that we get x=x+1 : x-x+1=0. meaning that 0=1 which is a contradiction. So infinity is not a real number.

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This doesn't seem very different from my answer from yesterday. I'll just note (at the risk of confusing people who don't know about ordinal arithmetic) that as ordinal numbers infinity and infinity plus one are different (but infinity and one plus infinity are equal). –  Gerry Myerson May 3 '11 at 6:35
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If you want to take time to listen to something about the history and use of "infinity" then the BBC Radio 4 programme "In Our Time" ran a programme on this subject in 2003 with Ian Stewart, Professor of Mathematics at the University of Warwick; Robert Kaplan, co-founder of The Math Circle at Harvard University and author of The Art of the Infinite: Our Lost Language of Numbers; and Sarah Rees, Reader in Pure Mathematics at the University of Newcastle all contributing. This may not quite go into the depth and rigour that you are after, bearing in mind the "general" audience it is aimed at, but it may well be of interest still.

I'm not sure of the availability of this to people that reside outside the UK however, so this may not be of use to the OP.

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Infinity is not a number, but some things that can reasonably be called numbers are infinite. This includes cardinal and ordinal numbers of set theory and infinite non-standard real numbers, and various other things.

There are various different things called infinity. There's the infinity of complex analysis, and there are the $\pm\infty$ encountered in calculus. There is the "infinity" that is the value of Dirac's delta function, which changes when multiplied by a real number. And there are many others.

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