# Is $\infty$ undefined? [duplicate]

This question already has an answer here:

I am confused at $$\infty$$ in a lot of ways. First, we sometimes say that $${1\over 0}=\infty$$. That gives us these confusing calculations.$${1\over0}=\infty$$ $$1=0*\infty$$ $$2=(0*2)*\infty$$ $$2=0*\infty$$ $$1=2$$ But certainly, this is not true. $$1\ne 2$$. What is happening?
Another confusion: let's say $$x=x+1$$. This is never true. But with $$x=\infty$$, it works.
Third confusion: let's say $$2x=x$$. Subtracting both sides by $$x$$ gives $$x=0$$. But what about $$\infty$$?
Fourth confusion: $$x-x=0$$ is always true, but not for $$\infty$$. $$\infty-\infty$$ is undefined.
What is actually happening??! Any help will be very appreciated.

## marked as duplicate by GNUSupporter 8964民主女神 地下教會, Dietrich Burde, Lord Shark the Unknown, Kemono Chen, Delta-uFeb 7 at 10:17

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• $\infty$ is not a number. You have shown that calculating with it leads to contradictions. Division by zero is not allowed. Surely, some users will come with the "extended real line", but this does not make division by zero meaningful. Better avoid it ... – Peter Feb 6 at 18:14
• Yes, that's all very confusing. In fact, you produced a reductio ad absurdum against the very claim that $\frac{1}{0}=\infty$. – Bram28 Feb 6 at 18:14
• However, to determine limits, setting $1/0=\infty$ can be useful, but never consider this equation to actually hold. – Peter Feb 6 at 18:16
• Related to 1/0 = infinity – GNUSupporter 8964民主女神 地下教會 Feb 6 at 18:18
• Infinity isn't something you can operate with. For starters, you can't define $\frac10$ to be a number because the operation itself is unperformable. Diving by zero has no meaning, physical or abstract. Infinity isn't a number – Christopher Marley Feb 6 at 18:39

## 2 Answers

Infinity does not exist in the typical real number line: it is a construct that is contained in the extended real number line, which is used in certain fields of mathematics, especially limit calculus. However, the extended real number line does not share all of the same properties of the normal real number line, which means that the apparent contradiction of 1=2 that you have produced does not hold, as certain operations which you performed to reach that point are not valid in the extended real number line. Whenever we manipulate infinity, we must enter the extended real number line, with different properties.

You may calculate with $$\infty$$, in a certain sense. For that purpose let $$\infty$$ be an object which is not a real number, e.g., $$\infty:=\{\mathbb R_+\}$$. (You may define $$-\infty$$ as $$\{\mathbb R_-\}$$.) Declare for any real number $$r$$ the set $$N_r(\infty\}:=\{x\in\mathbb R\mid x>r\}$$ as a neighbourhood of $$\infty$$.

Apply the usual definition of a limit to define if as sequence has $$\infty$$ as limit: in each neighbourhood of $$\infty$$ there are almost all elements of the sequence.

Now prove that for the limit of sequences some of the usual rules are valid, for example $$\infty\cdot\infty=\infty$$, which means the product of two sequences which limit is $$\infty$$ has limit $$\infty$$ as well. Also $$\infty+\infty=\infty$$ as well as $$\infty\pm a=\infty$$ for any sequence with limit $$a\in\mathbb R$$ and $$|a|\cdot\infty=\infty$$ for $$a\neq0$$. $$a/\infty=0$$ is another one for non-negative $$a$$.

On the other hand there are so called "indetermined expressions" as $$0\cdot\infty$$ or $$\infty-\infty$$, which can be any number (including $$\infty$$).