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So, I've been doing a little bit of research lately and stumbled upon two neat explanations, cases in favor maybe, of complex numbers. One said that complex numbers tried to explain rotation on a plane. The other explained that imaginary numbers can be thought of as a way of scaling a circle backwards (shrinking it). But, why would I favor the use of complex numbers when I can do all that using polar and spherical coordinates?

I guess my question is, how can we physically think about complex numbers? To set the tone: I understand we can use natural numbers to talk about finite measurable quantities such as 3 apples. We can use integers to express debt. We can use rational numbers to express proportions. And we can use real numbers to explain approximations. But why do we have imaginary numbers other than to explain rotation and scaling?

Thanks in advance.

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In EE complex numbers are often used to describe waves. When many waves of same frequency but with varying phases and amplitudes interfere, it is simpler to use complex numbers to add them up. This is actually similar to prof. Israel's suggestion. – Jyrki Lahtonen Mar 14 '14 at 6:28

If you want something really physical: quantum mechanics requires complex numbers.

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