# Infinity minus Infinity

Holocene
Does subtracting infinity from infinity leave you with zero?

Or could you subtract infinity from infinity and still have infinity?

## Answers and Replies

Werg22
Subtracting infinity from infinity has no mathematical meaning. Unless we are talking about limits and orders of magnitude.

Homework Helper
Gold Member
Subtracting infinity from infinity has no mathematical meaning. Unless we are talking about limits and orders of magnitude.

You can't subtract but you can add infinity from infinity.

Let N be the natural numbers and R be the real numbers. And card(X) denote the cardinality of X.

card(R) + card(N) = card(R)
card(N) + card(N) = card(N)

Staff Emeritus
Gold Member
Does subtracting infinity from infinity leave you with zero?

Or could you subtract infinity from infinity and still have infinity?
What "infinity" are you talking about? What notion of subtraction are you talking about? Your question really cannot be answered unless these are specified. (Though we can guess at what you meant, in order to give an answer)

Some examples:
The extended real line contains two points "at infinity": $+\infty$ and $-\infty$. The extended real line also comes with a notion of subtraction that's defined for most, but not all arguments. $(+\infty) - (-\infty) = +\infty$ and $(-\infty) - (+\infty) = -\infty$, but $(+\infty) - (+\infty)$ and $(-\infty) - (-\infty)$ are undefined. (And any combination involving at least one finite number is defined)

In the cardinal numbers, there are lots of infinite cardinals. (literally, too many to count) Subtraction makes little sense for them, because if $\alpha \leq \beta$ and $\beta$ is infinite, then $\alpha + \beta = \beta$.

The hyperreal line contains many infinite and infinitessimal numbers, and in a certain sense, the hyperreals behave exactly like the reals. (e.g. you can subtract any hyperreal from any other hyperreal)

Last edited: