# Does infinite equal infinite?

I have a question.
Let $x$ be infinite.
$$2x=\infty\times2, \quad 2x=\infty$$

So actually, does $2x=x$?

-
infinity is not a number – sigmatau Jan 25 '14 at 9:12
It should first be clear to you what does $\;2x\;,\;x=\infty\;$ mean...is it? – DonAntonio Jan 25 '14 at 9:14
Yes, that's what I mean. – Jamie Jan 25 '14 at 9:17
You have numbers and multiplication on them means that every pair of numbers is connected with a number: $(2,5)$ with $10$. You can add 'infinity' to this set of numbers, but after that conventions must be made to get an extending of this multiplication. This in such a way that the rules of multiplication remain valid as far as possible. – drhab Jan 25 '14 at 9:20

Start with a set, let's take $\mathbb Z$. Its elements are integers and on it there is a multiplication. That means that every pair of integers is connected with a integer wich we call the product. E.g $(2,5)$ has product $10$ . You can add 'infinity' (whatever it is) to this set and denote the result by $\mathbb{Z}\cup\left\{ \infty\right\}$. If you want to extend the multiplication then it must be 'decided' what the product is for pairs like $(2,\infty)$ and $(\infty,\infty)$. These decisions/conventions must be taken in such a way that the rules of multiplication (e.g. $x\times y=y\times x$) remain valid as much as possible. Quite a job! Your intuition says that for $(2,\infty)$ it is a good thing to choose $\infty$ as product. That confirms to me that your intuition is to be respected. And remember: intuition is very important in mathematics!

-
Well, clear answer. – Jamie Jan 26 '14 at 0:40

As drhab stated in his answer, your intuition tells you that $\infty \times 2$ should be $\infty$. But that intuition depends on what you understood $\infty$ to mean. A very 'layman' definition could go something like "a quantity with larger magnitude than any finite number", where "finite" = "has a smaller magnitude than some positive integer". Clearly then $\infty \times 2$ also has larger magnitude than any finite number, and so according to this definition it is also $\infty$. But this definition also shows us why, given that $2x=x$ and that $x$ is non-zero but may be $\infty$, we cannot divide both sides by $x$. It is akin to asking, if John runs twice as fast as Jack and both run off away from me, can I divide John's final position by Jack's final position, which are both further away from me than I can ever go, and get $2=1$? (Of course neither John nor Jack themselves can reach their "final position", but the process by which they 'approach' it explains the situation quite well.)

-
Note that you are also assuming "there is only one quantity with larger magnitude than any finite number" in order to be able to use $\infty$ as a proper noun. – Hurkyl Jan 25 '14 at 12:48
Not really.. I left it to the reader whether he wants to assume that, or simply consider the definition to be a description of what is infinite. Indeed there can be many things that "have larger magnitude than any finite number", and double any such thing also satisfies the same property. For instance, if you replace "$\infty$" in my answer with "something infinite", you get something that makes sense! Which of course means that you assumed "$\infty$" to be a proper noun, not me! =P – user21820 Jan 25 '14 at 13:06

In every commonly used number system that has a number called $\infty$, it is indeed true that $2 \infty = \infty$ -- e.g. the Riemann sphere, the extended real line, or the cardinal numbers.

However, there are various number systems -- e.g. the hyperreal numbers or the ordinal numbers -- that have infinite numbers that do not satisfy this property. Note that we usually never use the symbol $\infty$ when referring to an infinite number in these number systems.

(the principal exception I know of is the extended hyperreal line, which has many infinite numbers obeying the 'usual' laws of arithmetic, and a pair of additional numbers we call $+\infty$ and $-\infty$ that have the largest magnitude of all infinite numbers, and do not obey the 'usual' laws of arithmetic)

The answer to your question of whether $2x = x$ when $x$ is infinite, thus, depends very much on what number system you're using.

Examples include: if you're studying calculus of real variables, you're probably using the extended real line; if you're quantifying the number of elements in a collection, you're probably using the cardinal numbers.

-

It depends on how you use the term "infinite". If you speak in terms of cardinal numbers (for counting of objects), then yes, they're the same infinity. This is because any countable set containing an infinite number of objects can be counted in a way to have exactly the same number of objects. For instance, the set of all integers is clearly twice as big as the set of all even integers... and yet, if you just multiply the set of all integers by 2, you get the set of all even integers, thus showing that there's just as many even integers as integers.

If, on the other hand, you're using it as a number, you cannot evaluate it, as infinity is not a number.

On a third hand, if you think in terms of limits, then they are not equal. That is,

$$\lim_{x\to\infty} \frac{x}{2x} = \frac12$$

and thus the limits are not the same.

-
While there is no member of the real line named $\infty$, that does not imply there is not a number named $\infty$. Incidentally, note that your final argument would suggest we should also have $0 \neq 0 \cdot 2$. – Hurkyl Jan 25 '14 at 12:51
@Hurkyl - the limit to infinity is different from the limit to zero, so the final argument doesn't apply in the limit to zero. And my choice of word was key - there is a difference between "cardinal number" and "number" - when people use "number" in English, they are referring to real numbers or natural numbers. If someone is asking whether $2\infty = \infty$, then they're nowhere near the point where the nuances involved in set theory are easily understood. – Glen O Jan 25 '14 at 13:20
... but (IMO) notions like the extended real line or projective infinity are not beyond the realm of understanding. They even become indispensable by the time you get to introductory calculus, although (unfortunately, IMO) they are introduced in a very ad-hoc fashion. – Hurkyl Jan 25 '14 at 13:29
I disagree - the only thing required at the point of introductory calculus is limits. Indeed, most students learning about calculus struggle with the idea that you should use limits to infinity, precisely because infinity is too often treated like a regular number when taught before that point. I usually explain it as "infinity is a concept, not a number" - indeed, it's why the concept of the aleph numbers was invented - because "infinity" doesn't work as a number. – Glen O Jan 25 '14 at 13:46
Limits (and limit forms and the suite of related concepts) are precisely what I was referring to the extended reals being introduced in an ad-hoc fashion. The alephs are a completely unrelated notion: cardinal numbers (and ordinal numbers, for that matter) have nothing to do with that topic. – Hurkyl Jan 25 '14 at 13:47

Yes. If you work with cardinals, you can show that if two cardinals are finite (i.e. integers), then the product rule is as usual ; if you have two positive cardinals that are infinite, say $\kappa, \lambda$, then $$\kappa + \lambda = \kappa \lambda = \max \{ \kappa, \lambda\}.$$ In particular, if $\kappa$ is infinite, then $\kappa + \kappa = \max \{\kappa,\kappa\} = \kappa$.

See http://en.wikipedia.org/wiki/Cardinal_number#Cardinal_addition to understand them a bit better.

Hope that helps,

-
Cardinals! Your formula sum=max is false for ordinal sum. – Martín-Blas Pérez Pinilla Jan 25 '14 at 11:12
I always switch between them... damn it. Thanks – Patrick Da Silva Jan 25 '14 at 21:06
I was wondering why I couldn't see my formula on that page actually, it was quite strange. Thanks @Martín-BlasPérezPinilla – Patrick Da Silva Jan 25 '14 at 21:08

From the question itself, arises the fact that you are looking at it at an intuitive way. The answer to your intuitive question "is 2x = x?" then is YES. But notice that x is an "infinite number", and so, saying that two times infinite is infinite is not a big deal. Anyway, as you pointed out, in maths there are indeed frames in which 2x = x...

From my point of view, the infinite question has not been solved. We still don't know WHAT infinite actually is, although we have described many of its properties pretty well. Maybe, if we start researching outside the boundaries of the stablished axioms we may get to some conclutions...

-