# 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

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.

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. – user14972 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

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 $0 \neq 0 \cdot 2$. – user14972 Jan 25 '14 at 12:51
• ... 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. – user14972 Jan 25 '14 at 13:29
• 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. – user14972 Jan 25 '14 at 13:47
• But you still have the point that is being approached. Would you ever eschew "$x$ approaches $0$" in favor of saying "$x$ is a quantity whose magnitude is deceasing so as to eventually be smaller then any positive real number"? Probably not. But that's exactly what you're doing when you tell people that $\infty$ needs to be thought of as a concept rather than a point that can be approached.... – user14972 Jan 25 '14 at 14:31
• ... and really, the point of limits is not that the variable never reaches something, but that we can assign a single quantity at a single point that describes the behavior of a function near that point in a way that is often useful and relatively easy to compute and compute with. When I compute $\lim_{x \to 0^+} 2 \log(x) = -\infty$, I simply plug in $0$ since I know the function is continuous; I don't take a mental detour through an $\epsilon-N$ argument to think "Okay, $\log(x)$ decreases without bound as $x$ gets small, and twice that value will also decrease without bound". – user14972 Jan 25 '14 at 14:33

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...

You can't do this, as I have learned. Take all numbers between 0,1 and compare it to all numbers between 0,2. We assume there are double the amount of numbers between 0,2 compared to 0,1, but its not that simple.

Imagine all numbers between 0,1 that only go out one decimal place.$$0.0,0.1,0.2,0.3,\cdots1.0$$Now double them.$$0.0,0.2,0.4,0.6,\cdots2.0$$We can clearly see this is only half of the possible one digit values between 0,2.

However, let us add the next digit.

Then we can see that I can indeed reach all of your values between 0 and 2, but you will argue that now I am missing some of the values in the 2nd decimal place.

So we go out another decimal, and another, in fact, we go infinitely.

There will always be a number $n$ that when doubled, equals a number $m$, where $n$ is between 0,1 and $m$ is between 0,2.

However, there is never a number $m$ where there is not some $n$ I can double to reach your $m$.

Therefore, I have argued that there is, indeed, the same amount of numbers between $0,1$ and $0,2$. Measuring infinity like this is called the Cardinality of a set.

However, I will say that there is both an equivalent amount of numbers between 0,1 and 0,2 and double the amount of numbers between them, at the same time.

There is a solution to $2x=x$. It's $x=0$. That being said, where does infinity come from? If it's the result of adding one to itself forever, $1+1+1+...$, then if we say $f = 1+1+...$, it seems intuitive that if we define $g$ as $2+2+...$, then if we divide $g$ by $f$, then the result will be 2. To be more precise, if $f(x) = x$ and $g(x)=2x$, then it seems obvious that for all values of x, even $∞$, that $g(x)/f(x) = 2$, and $f(x)/g(x) = 1/2$. In calculus, with limits one might say that $1/2$ and $2$ are the limits of f(x)/g(x) and g(x)/f(x), respectively, as $x → ∞$ or 'as x increases without bound'. Some say this means there are different sizes of infinities, or that some infinities or functions grow at different rates. If you think a system which handles infinity well has certain properties, then find one which has those properties. If you think systems should say infinity doesn't exist, and questions concerning it are nonsense, impossible, or undefined, then find one. Many types of systems exist (however much the adherents of a given system may deny it).

If we say that 'infinite' is the property that a number is greater than any finite number, then since doubling a number does not decrease its value, if the function $i(x)$ takes x and returns $1$ if x is infinite, and $0$ otherwise, then for any number $n$ such that $i(n) = 1$, $i(2n)$ and $i(n+1)$ is also $1$.