B Is infinity an imaginary number?

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PeroK

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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

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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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