Infinity is not a number , thus we cannot perform the usual arithmetic operations that we do with real numbers
This is the usual reason given when asked why we can't perform the usual arithmetic operations with infinities. However the terminology used in these types of arguments are very loose and aren't well defined mathematically. For example, there are multiple types of infinities.
I will attempt to formalize this question in the context of Real Analysis below.
Let's assume for the purposes of this question that we are only dealing with the Real Field, $\mathbb{R}$, the set of Real Numbers (Scalars), satisfying the Axioms of Addition and Multiplication.
Correct me if I'm wrong, but infinities of all types are not elements of the Real Field ($\infty \not\in \mathbb{R}$), therefore they do not obey the Axioms of Addition and Multiplication (hence they do not obey the usual 'arithmetic operations', that Real Numbers do).
Some of the results of these operations (not limited to infinities) are not well defined, and leads us to the 7 Indeterminate Forms in Real Analysis:
- $\frac{0}{0}$
- $\frac{\infty}{\infty}$
- $0 \cdot \infty$
- $\infty – \infty$
- $0^{0}$
- $\infty^{0}$
- $1^{\infty}$
But why then are the following results true :
$$\text{Given} \ a \in \mathbb{R}$$
$$(+\infty)\cdot( -a) = -\infty $$
$$(-\infty)\cdot( -a) = +\infty$$
$$(+\infty)\cdot( +\infty) = +\infty$$
$$(-\infty)\cdot( +\infty) = -\infty$$
$$(-\infty)^{2k} = +\infty, \ \text{where} \ k\in \mathbb{Z} \geq 1$$
In these cases infinities behave the way we expect elements of the Real Field to, with regards to whether the product of the two multiplicands is either positive or negative. Why is that so?
I have framed this question within the context of Real Analysis, however if there are interesting observations from other areas such as Complex Analysis or Abstract Algebra regarding operations with infinity, I would be highly interested in reading about it.
Best Answer
Strictly speaking we can define the symbols to work however we want. We choose to define the symbols to work that way because it matches up with the behavior of limits. For example, if $a>0$ and $f(x) \to \infty$ as $x \to c$ then $af(x) \to \infty$ as $x \to c$ while $-af(x) \to -\infty$ as $x \to c$. (Note that $f(x) \to \infty$ and similar expressions are written using the symbol $\infty$ but their definitions can be stated entirely in terms of real numbers.) We generally leave the "indeterminate forms" undefined because there is no way to define them that is consistent with all the ways that they would appear in limits. (One exception: some people define $0^0=1$ and $0 \log 0 = 0$, even though $f(x,y)=x^y$ and $g(x,y)=x \log y$ cannot be continuous at $(0,0)$.)
Also, you should make sure to understand that here we are talking about infinity in the sense of the extended real numbers. There are other notions: infinite cardinal numbers, infinite ordinal numbers, and complex infinity come to mind.