Why $\infty - \infty \neq 0 ?$

## marked as duplicate by user26486, leonbloy, Matthew Towers, Claude Leibovici, user26857Feb 28 '15 at 12:39

• Sorry i did'nt knew – AAkash Feb 28 '15 at 12:26
• See The Chaz 2.0's comment here: "If I have an infinite number of marbles, each either red or blue, I can give you all of them. Then $\infty−\infty = 0$. Or say I give you all of the blue marbles. Then $\infty−\infty = \infty$. Or say I give you all but $11$ marbles. Then $\infty−\infty = 11$. So $\infty−\infty$ could be anything. Ok... these symbols aren't right." – user26486 Feb 28 '15 at 12:30
• Your title also tells us nothing about what you want to know. – PJTraill Oct 4 '17 at 20:47

Suppose you lived in a world where $\infty - \infty = 0$. Then you might have this.

Since $$5 = x + 5 - x,$$ we then have $$5 = \lim_{x\to\infty} (x + 5) - \lim_{x\to\infty} x = \infty - \infty = 0.$$

The consruct $\infty - \infty = 0$ is just not meaningful.

The statement $\infty - \infty \neq 0$ is not even well-defined.

You must specify by what you mean with "$\infty$".

Do you think of it as "number" that is larger than all other numbers? In that case, minus is not defined, because that would lead to inconsistencies. Namely, let $\infty - \infty = a$. Then also $2 \infty - 2\infty = a$ since $2 \infty = \infty$. But then, dividing by $\infty-\infty$ on both sides, you get $a=1$ and $a=2$. Which doesn't make sense.

In case you think of "$\infty$" as "limits", then consider the example $\lim_{x \to \infty} x^2 - \lim_{x \to \infty} x$. This is certainly equal to $\infty$, even though this has the form $\infty-\infty$.