Why does complex number multiplication cause rotation?

Why does complex number multiplication cause rotation?

Complex multiplication of a and b has the effect of rotating b through angle that depends on a.

In this video, we will investigate an intuitive explanation for this, based on the similarities between multiplying complex numbers and matrix multiplication of vectors.

Suitable for UK A Level, IB Diploma Programme, US grades 11+. Typically taught to students aged 16 and above.

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[johnpolo]

I like the way to study some important ideas by comparison. Not so many people can explain it this way. You sir did it in a very nice way which is very important mainly for newcamers studying this subject

[sethyta]

@5:45 so you're saying the *true reason* complex number multiplication produces a rotation is because, at the end of the day, it is "arithmetically equivalent" to a matrix transformation? (great vid btw you should have more views)

[theclipreaper]

So did I get this straight: complex numbers do a specific matrix multiplication "in the background" and in a way that's more intuitive for us humans to understand?

[pelasgeuspelasgeus]

I have a silly question: how do you know that imaginary axis is vertical to the real axis?