why $\infty$ +$\infty$, $\infty$ - $\infty$ and 0⋅$\infty$ are left undefined.

Question

I'm reading http://en.wikipedia.org/wiki/Riemann_sphere, and having the following question.

1. What's the mean of symbol $\infty$?Is it a surreal number?

2. they write

note that ∞ + ∞, ∞ - ∞ and 0 ⋅ ∞ are left undefined.

Why we can't define ∞ + ∞?

why not define ∞ + ∞=∞

Answer 1

"Infinity" here refers to a single point which you simply label as $\infty$.

Imagine taking a real line, and "wrapping" it around into a circle, then count both $-\infty$ and $+\infty$ as a single point of intersection. This is the point which is $\infty$ in the Riemann sphere.

Since both $-\infty$ and $+\infty$ are in fact represented by this point, $\infty + \infty$ is undefined.

The nice thing about this sphere is that it can be generalized to structures like $\mathbb{R}^2$ and $\mathbb{C}$. For instance, all of the points on the complex plane with infinite modulus are referred to as $\infty$.

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In particular it's important to distinguish the Riemann sphere with Euclidean space. Neither "$\infty$" nor "$-\infty$" are actually points on $\mathbb{R}$ for instance, but $\infty$ (with no sign) IS a point on the Riemann sphere.

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So finally, to answer your last question regarding $\infty + \infty$, it won't be well-defined since, for instance, taking $\lim_{x\rightarrow\infty} x+x$ one obtains $\infty + \infty = \infty$ but taking $\lim_{x\rightarrow\infty} x + (-x)$ one obtains $\infty + \infty = 0$.

Answer 2

The "formal" answer is that $\infty$ is simply an "object" in this new "number system," and that the rules of the system do not permit those "quantities" to be well-defined.

If you are referring to $\infty$ as we usually think of it, the best way to understand these facts is to think in terms of limits. The expressions $\infty - \infty$ and $0\times \infty$ (and several others) are called indeterminate forms because as limits, they can equal anything at all.

Specifically, say we have $f(x)\to \infty$ and $g(x)\to\infty$ as $x\to a$. What is $$\lim_{x\to a}(f(x)-g(x))$$ The answer is that it can be anything at all, depending on what the functions are. And similarly for other indeterminate cases. Having a "rule" that, for example, $\infty-\infty=0$ would lead to erroneous limit calculuations.

You are correct that the "limit law" $\infty+\infty=\infty$ is perfectly valid. But keep in mind that "$\infty$" being discussed here is not the number so much as it is an abstract object, defined to be a part of a number system with certain properties. You could just as well write that "thing" as $x$ or $\text{Turkey}$, it is denoted "$\infty$" because it has certain properties reminiscent of that idea, but it is not the same as infinities seen in limits.