Why does Infinity x Zero not Equal One?

Question

Why does Zero Times Infinity not equal One ($0 \times \infty \neq 1$)?

If Infinity = $\infty$ and Zero = $\frac{1}{\infty}$

Then Zero Times Infinity = $0 \times \infty = \frac{1}{\infty} \times \infty$ which is equal to '$1$'?

What Am I doing wrong?

Answer 1

$0 \neq \dfrac{1}{ \infty}$

It is true that we have $\lim_{x\to \infty} \dfrac 1{x}= 0$, but that is not to say that $\dfrac 1{\infty} = 0$.

And if you have evaluated a limit to get the indeterminate form $0 \cdot \infty$, that is simply an indeterminate form of a limit (not a value) that tells us more work needs to be done to find the limit.

Answer 2

$\infty$ is not a real number. If it were then it would lead to all kinds of contradictions. The fact that it does lead to contradictions is precisely why we can't treat it the way you do here. Instead, we are much more careful and treat it differently in different parts of mathematics.

Answer 3

Infinity isn't explicitly a number so it doesn't follow the rules of arithmetic, you can pseudo-prove all sorts of silly things like 1=2 if you assume that infinity is a regular number, which is clearly wrong.