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zkmon

This is just scratch on the surface.

* Enter quaternions; things get more profound.

* Investigate why multiplicative inverse of i is same as its additive inverse.

* Experiment with (1+i)/(1-i).

* Explore why i^i is real number.

* Ask why multiplication should become an addition for angles.

* Inquire the significance of the unit circle in the complex plane.

* Think bout Riemann's sphere.

* Understand how all this adds helps wave functions and quantum amplitudes.

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Any hints towards the answers? I've spent a lot of time with complex numbers, and my answers would be

Quaternions: not profound, C is complete, quirky but useful representation of SO(3)

Inverses: fun fact coincidence

1+i/1-i: not sure what to experiment with here

i^i: gateway to riemann surfaces.

Adding angles: comes out like this, that's the point of exp(i phi)

Unit circle: roots of unity?

Riemann sphere: cool stuff!

Quantum stuff: mathematical physicist here, no need to sell this one!

i^i isn't anything. Please don't write this. Of the two inputs to the function (w, z) -> w^z = exp(z ln(w)), only z is a complex number, so that bit is OK. The problem is that w is NOT a complex number but a point on a particular Riemann surface, namely: The natural domain of the function ln. That particular Riemann surface looks like an endless spiral staircase. The more grown-up term might be "a helix". When you write informally "w=i", that could mean any of ln(w) = i pi/2, i (2pi + pi/2), i(4pi + pi/2), etc. Incidentally, w^z is then always a real number. However, there's an infinite sequence of those numbers that it could equal.

I suppose that by pure convention, "w=e" is understood as denoting a single unique point on the helix. But extending that convention to w=i starts to look like a recipe for confusion.

Is your argument that complex powers isn't anything?

Their argument is that ln(z) where z is a complex number is a multi-valued function, so the statement "Explore why i^i is real number" could be misinterpreted as i^i = a single well-defined real value.

Yes, but it seems strange to claim that i^i isn't anything. That just completely ignores what's interesting, namely that i(π/2 + 2πk) is real for all k ∈ Z.

In maths, an expression only ever equals a single number. You can't say i^i = e^(-pi/2) and then also say that i^i = e^(3 pi / 2), because if i^i equals two things, then those two things are also equal to each other, and then we get that e^(-pi/2) = e^(3 pi / 2), which is wrong.

Riemann surfaces are the only way to fix this. And they're not even that hard to understand, but I'm not sure if you do.

Stop making people confused.

Apologies if this is pedantic but "multi-valued function" is not a thing. The function doesn't have multiple values here. Saying "multi-valued function" is not just a way of misleading people about what's really happening, but it's almost the perfect way to stop people from finding out. Do people who say "multi-valued function" know what's really happening? Do you know what a Riemann surface is?

Do you know what a Riemann surface is? Because if you don't, then you don't know what you're talking about - and you should stop getting people confused.

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