Question

What are the arithmetic properties of infinity?

Answer 1

Some thoughts...

Explanation:

Infinity means different things in different contexts, but in the context of calculus we usually have two objects `+oo` (a.k.a. `oo`) and `-oo` which act a bit like very large positive and negative numbers.

Some operations have defined values. For example:

• If `x in RR` then `x/(+oo) = x/(-oo) = 0`

• If `x in RR` then `x + +oo = +oo` and `x + -oo = -oo`

• If `x > 0` then `x * +oo = +oo * x = +oo` and `x * -oo = -oo * x = -oo`

• `+oo + +oo = +oo` and `-oo + -oo = -oo`

• `+oo * +oo = +oo` and `-oo * -oo = +oo`

• `+oo * -oo = -oo * +oo = -oo`

That all looks good, but `+oo` and `-oo` are not really numbers, and all of the following expressions and similar ones are indeterminate:

`0 * oo`, `" "oo * 0`, `" "oo/oo`, `" "oo - oo`, `" "+oo + -oo`

Answer 2

Basically, there are two rules: `n/oo = 0`, for all numbers `n`, and `oo/oo` is undefined.

Explanation:

Technically, any number `n` divided by `oo` produces an infinitesimal value (i.e. an infinite amount of zeroes followed by a `1`)---however, this value, is for all intents and purposes, `0`.

`oo/oo` is undefined because there are different levels of infinity. The size of `RR`, for example is a larger infinity than `NN` (both the real numbers and natural numbers are infinite sets, but the natural numbers are a countable infinite set, whereas the real numbers are an uncountable infinite set).

Of course, `oo` isn't a number, which is why you run into inconsistencies (with the exception of `0`, a a number divided by itself equals `1`).