# B Is infinity an imaginary number?

#### PeroK

Homework Helper
Gold Member
2018 Award
I am an engineer, and not a mathematician, so I understand I do not speak in the same precise terms as one. But I struggle to think of an infinity that can't be inverted, and, when it is, would not equal zero. Is there one?
An example where inverting infinities goes wrong is. Let $f(x) = 2x$ and $g(x) = x$, then $f(\infty) = \infty$ and $g(\infty) = \infty$, so:

$\frac{f(\infty)}{g(\infty)} = \frac{\infty}{\infty} = 1$

But

$h(x) = \frac{f(x)}{g(x)} = 2$

So $h(\infty) = 2$

But $h(\infty) = \frac{f(\infty)}{g(\infty)} = \frac{\infty}{\infty} = 1$

So, which is correct? Is $\frac{\infty}{\infty} = 1$ or is $\frac{\infty}{\infty} = 2$?

One of the reasons you need to develop rigorous, well-defined mathematics is to resolve this sort of issue. And, one of the things you have to do is to stop using $\infty$ as a number.

#### DrClaude

Mentor
Invoking the hyperreals in a "B" level thread is the maths equivalent of invoking the stress-energy tensor to explain the SHM of a pendulum!

The hyperreals are at an advanced undergraduate level and depend on a solid grasp of real analysis. They are not suitable for a "B" level thread, IMHO.
I think that this thread has now long been flying over the OP's head. Time to close.

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