Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, you can define the "extended real number line" (cf. the Wikipedia article of that title) which includes infinity and satisfies a list of axioms which you can write down.
This article is dangerously misleading. "Numbers" is not a well-defined set and there is no way to say that infinity does or does not belong to it. As is often the case in mathematics, you need to make the discussion more precise before you can reasonably answer questions.
No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of. You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently.
A lot of the comments here are getting hung up on trying to pin down the mathematics of what you can and can't do with infinity. That's fine, it's been keeping mathematicians busy for centuries, but this article is for the layman who doesn't know about rings, groups, algebras or any other mathematical structure which you might call 'numbers'. It's for someone who thinks the obvious when someone says 'number'.
In this context, infinite limits, I think the distinction is beginning to become important. I agree that colloquially it often doesn't matter, but I think it does here.
"Something which you can have that many of" is a terrible definition: You can have sets of infinite size. Do you mean physical things? What, then, is a "thing"? I can have infinity intervals of different length on my arm. So are intervals not a "thing"? You appear to think irrationals are numbers, so what physical thing can you have an irrational number of?
This is a dangerous path to walk down, but that doesn't make it an unimportant one. And that's why this article is ultimately flawed. The idea is important though. We need to understand how to operate formally on mathematical objects. If there's one thing I learned in my Philosophy of Mathematics class as a math undergrad, it is that we shouldn't struggle over defining what it means to be a number. We can just use them. I'll leave the philosophy to the philosophers.
I pretty much agree with you. I'm just worried that with all the comments about 'what is a number?', and 'sure, we can make infinity a well defined number', it will dilute the message of the article. Which is that infinity is not a 'number' (in the sense of real numbers). It's important to understand why it's not a number and what it means when we write it on one side of an equality.
> Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently.
Well, if your definition doesn't work for irrational numbers, then your argument isn't good enough: you exclude infinity on criteria that also exclude "good" numbers. In the context of the article, π is a number. And you cannot count things with an irrational number, by definition.
You have to know whether your definition of "numbers" makes them into a field or not, before you can go applying field operations to them and expecting usual results.
Including infinite ordinals in the set of numbers is legitimate, as parent points out that "number" is not strictly defined, but if you include infinity, numbers are no longer a field, and you have to cut someone off when they try to use field axioms in theorems, the uniqueness of multiplicative and additive inverses, which must be how people end up with nonsense like 1=0.
The article deals with the infinity symbol as it is used in calculus courses. With such a usage in mind the infinity symbol is not a number and does not represent a number. The usage of the infinity symbol in calculus is merely a shorthand notation for a more complicated statement.
Suppose that we have
Lim x->5 f(x) = infinity
What is meant by the use of the infinity symbol is that the limit is not bounded in the real number system. More specifically that given any large real number I can find a number d such that whenever |x - 5| < d then f(x) > M.
Here the use of infinity is not meant to be as a number though making such an association is helpful to beginners in terms of visualizing what is going on. Students have trouble with the precise definition of being unbounded and so it's convenient to say "it's infinity" and treat the symbol as a number.
The argument "Addition breaks" proves just as well that zero "isn't a number", since it breaks division rather badly. (Yes, division is a badly-behaved version of the more upstanding multiplication. The article uses subtraction in what is nominally a complaint about addition). It doesn't address ordinal numbers at all, since addition works just fine with infinite ordinals, exactly the way the article claims you'd expect (what have I missed?).
The immediately obvious uses of infinity in calculus (the topic of the wikibooks article) are, according to wikipedia, termed the "affinely extended real number system" (I learned to just refer to the "extended reals"), which heavily implies that the points within are considered extended real _numbers_.
http://en.wikipedia.org/wiki/Extended_reals
The terms "cardinal number" and "ordinal number" both definitively include infinite quantities -- infinitely many, even.
The IEEE standard for floating point defines two infinite numbers.
Essentially, I'm in full agreement with tokenadult; the only relevant question is "what do you mean by number?". But we can easily observe that varying infinities, including the calculus uses of infinity, are referred to as numbers all up and down the chain, including in the most unimpeachably correct sources, and that it walks and quacks like a duck, even if it may not quack in the precise manner of Anas Platyrhynchos.
> The argument "Addition breaks" proves just as well that zero "isn't a number", since it breaks division rather badly.
Mathematically, numbers (be it natural, rational, real, or complex) are defined as a field. Fields (or, more accurately, rings, which all fields are) are defined by addition and multiplication, not both. [1]
It turns out that treating infinity like it is a number is very convenient (e.g. in convex optimization).
I think these black and white statements are not a good way to explain things. Just start with the set of real numbers. Point out infinity is outside the set. Then introduce the extended reals (R + infinity), with special rules for arithmetic involving +/-infinity. This theoretical convenience extends to the computer when the rules for Inf are implemented correctly. Follow the rules and you might sensibly get an Inf result, break them and you should get a NaN. It's all implemented in IEEE 754. In fact Inf is in your computer but not all the reals are. So there.
It's worse than that. "Infinity" isn't even a single concept.
There are at least two distinguishable uses of infinity (there may be more, but I haven't figured them out yet, not that my opinion counts for much). There's the adjective "infinite" that refers to a property of sets. This is the type that Cantor studied, and it turns out to have many different types, which are pretty strictly ordered into layers. Then there's the noun "infinity" which is either a point or a location that points can exist at, and while it's usually possible for there to be several such infinities in several different directions, they don't come in layers. I believe Rider of Giraffes refers to these as "set-theoretic" and "geometric" infinities, respectively.
For the first definition, it's easy to distinguish between it and traditional numbers: an infinite set is bijective to a proper subset of itself. However, it's also useful to consider it a generalization of numbers, so you can count forever.
The second definition is more problematic. Here, infinity is just a point, just like all your other points. You can choose to add it to your set, or not. It usually behaves a little funny (like it makes certain operators not invertible), but you may have to get subtle to define it, or a set that contains it. Sometimes, it's not different than normal points at all, and sometimes it depends on the context. For example, if you take the real line and add +/-infinity, in topology you just get a closed interval like [0,1], whereas in analysis based on metric spaces you get something outright broken (fails to satisfy the axioms of the objects being studied).
"Most people seem to struggle with this fact when first introduced to calculus..."
When I took calculus in college, I _did not_ struggle with this idea, despite at that time not having had any deeper background in advanced mathematics. Intuitively, the idea that you _approach_ some absolute as you edge the denominator ever larger made perfect sense to me.
Formally, my instructor made it clear that the _limit_ as you approach something was, in nature, different from any particular fixed value (of x). So, in a clearly defined manner, as you _apply the limit operator_ to the left side of the equation, the right side correspondingly behaves differently.
This concept never troubled me. As other comments here imply, this is an idiom specific to (differential) calculus. The only caveat might be in the use of strict equality, since limit operations by definition indicate asymptotic behavior. One could argue that a different type of relation is described (such as 'approximately equal': ≈). But then it's not infinity itself which is at issue.
Infinity can be relative and can be defined unlike a division by zero which in undefined (as in ... it can't be relative to anything else, and can't be used in a formula).
And hence infinity can be used in a formula and can cancel out with another relative infinity...
Example:
1) There are an infinite amount of real numbers between 1 and 2.
2) The amount of real numbers between 2 and 4 is twice the amount of real numbers between 1 and 2.
I would guess that if numbers are defined in terms of relativity/relationship, then infinity is a number.
But it seems that people wrongly define numbers in absolute terms, as if they exist outside the mind, and are separate from one another. Like the Universe cares about 1.24545434 and 7656.45433477.
Correct me if I'm wrong, but I believe the cardinality of the two sets you describe are equal, as there exists a bijective mapping between them, meaning there are an equal "number" of real numbers in both.
Not only does there exist a bijective mapping between [1,2] and [1,4], there exists infinite different bijective mappings between a subset of [1,2] and [1,4].
i.e.: One could map bijectively from [1,1.5] to [1,4] and map bijectively from [1.5,2] to [1,4] (1)
To talk about there being "twice as much" in one uncountable infinity than in another uncountable infinity is nonsense, since you can't apply words like "twice", since the infinities can't be counted.
The article only shows it is not an ordinal number.
Because I'm a Wikipedian, I've learned that whenever I visit an article on Wikipedia, Wikibooks, etc. I can visit the article talk page too. Many of the usual misconceptions about infinity can be found in the talk page of that article. Another article showing that infinity is not a number
by a young mathematician with some demonstrated chops in mathematics.
The last discussion of this issue on HN, which appears to have been from about two years ago, was interesting, so when I saw the article submitted here today (while looking up sources for the teaching I do), I thought I'd invite HN participants to discuss the issue again.
Two follow-up questions:
1) What do you mean by number?
2) Supposing the claim is that infinity is a number, how would that claim be verified by accepted principles of mathematics?
I think it is fair to say that a number should be an object of a ring. I'm an algebraist though. That is, there should be an addition operation and a multiplication operation that satisfy certain conditions.
1. Calculus. One sees in calculus things like
Lim x->5 f(x) = infinity
In this case infinity is not a number but rather a shorthand notation for a more complicated statement. What is meant by the use of the infinity symbol is that the limit is not bounded in the real number system. More specifically that given any large real number I can find a number d such that whenever |x - 5| < d then f(x) > M. Here the use of infinity is not meant to be as a number though making such an association is helpful to beginners in terms of visualizing what is going on.
2. There are ordinal numbers that are infinite. That is, that represent the order type of an infinite well ordered set. Ordinal numbers do not for a ring but they do have an arithmetic defined. They do form a semi-ring though. If one wants to say that objects of a semi-ring are numbers then there are infinite numbers. This also applies to cardinal numbers.
3. There is the extended real number system which has the symbols -infinity and positive infinity attached to the real number system to form a compact set. Think of the compact closure of the reals. Again not a ring though.
4. That said, there are infinite sets. An infinite set is one that is not finite. Or, in more precise terms, and infinite set is one that can be put into 1-1 correspondence with a proper subset of itself. Cardinal numbers and cardinal number arithmetic deal with comparing sizes of sets. Mostly useful for dealing with infinite sets. Some infinite sets are bigger, in a meaningful way, than other infinite sets. The set of reals is much bigger than the set of integers.
A while back we had a discussion about the halting problem, and someone had difficulty with the existence of the paradox in the canonical proof: http://news.ycombinator.com/item?id=1323407
The difficulty was, I think, in realizing that if we make a bunch of claims, and those claims lead to a paradox, the paradox exists, and that is not allowed. We can't "work around" the paradox, because paradoxes are not allowed to exist. Nor can we conveniently try to redefine some of our constructs such that they don't encounter the paradox if those constructs live in a reality with constructs that do encounter the paradox. I think it's a similar mental hurdle that some other people have with understanding that infinity is not a number.
The discussion that has begun in this thread suggests that the Wikibooks chapter submitted here could use some more work. (Its last revision was a 9 October 2011 reversion of I.P. edits to restore a version from 22 May 2011.) Evidently, not every reader of Hacker News is convinced that infinity is not a number, despite several websites by mathematically learned people who say exactly that,
so equally evidently, some readers here are not convinced that there is a rationale for drawing a distinction between infinity and numbers. Does it help to take a look at a discussion of "not a number" concepts
as they are implemented in computer science? What I see here, from my view as an educator in primary mathematics (in a program in which I can define "primary" to include topics like Hilbert's Hotel), is that some readers here have had educational experiences in which they "remember" seeing infinity treated as a number. The classic case, which prompts the Wikibooks chapter, is taking the limit of a rational quantity as the denominator approaches zero. This appears (based on previous HN discussion
from more than two years ago) to suggest that physical quantities can be divided by zero with a quotient that becomes infinity. Perhaps this is an example of how an engineering calculus course isn't always interpreted by learners quite the way it was presented by teachers. (I presume all but the tiniest number of teachers of engineering calculus would agree that infinity is not a number, and that no one can divide any number by zero.)
What would be a good way to clarify the point so that people are communicating with one another well as they speak about infinity and about what numbers are?
AFTER EDIT: impendia's kind top-level comment here
and there is a discussion of the affinely extended real numbers. Does the limit example in the submitted Wikibooks chapter fit the characteristics of that number system fully?
> Evidently, not every reader of Hacker News is convinced that infinity is not a number, despite several websites by mathematically learned people who say exactly that
Like impendia said, it depends on the context.
To give a simpler example: at elementary school I was told that I couldn't write "2 - 3". Essentially my teacher was saying that numbers less than zero did not exist! She wasn't lying to me. She was telling me that we were working with natural numbers only, and that our operation "-" was supposed to be closed on the natural numbers.
In the context of calculus, infinity is not a number. lim(n->c) foo = ∞ is just another way of saying that "foo does not converge as n goes to c".
But in realm of surreal numbers (or combinatorial game theory), infinities are first-class numbers. You can add them and subtract them alright! Besides, in that realm ω + 1 is not the same as ω. It is greater than ω, as expected. I can even show you a children's game -- that is, we can teach kids how to play it without ever uttering the words sets and cardinals -- that has positions with such values.
We have to define what is meant by "infinity". Do you mean the infinity symbol one encounters in calculus with regard to limits? Do you mean infinite cardinal number? Infinite ordinal numbers? Do you mean the symbols added to the reals in order to compactify the set (extended real numbers)? Some other concept?
Once that gets settled then we have to define what is meant by a number. I suggest that the most broad, reasonable definition of this concept would be elements of a ring. There should be some algebraic structure that is broad but not too broad. If we use such a definition then the various concepts of infinity mentioned above are not numbers since they are not elements of a ring.
The limit example in the Wikibooks article is merely a shorthand notation and isn't a number. It's mostly to be suggestive and make things a bit easier to comprehend. It's not intended that a person think of it as a number.
Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, you can define the "extended real number line" (cf. the Wikipedia article of that title) which includes infinity and satisfies a list of axioms which you can write down.
This article is dangerously misleading. "Numbers" is not a well-defined set and there is no way to say that infinity does or does not belong to it. As is often the case in mathematics, you need to make the discussion more precise before you can reasonably answer questions.