Why does complex number multiplication cause rotation?

Why does complex number multiplication cause rotation?
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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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This video was created using generativepy, a free and open source Python library for creating mathematical art and animations, available from github.com/mar...

[@johnpolo7160]

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

[@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?

[@seth111yta1]

@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)

[@miguelcerna7406]

Quality content.

[@graphicmaths7677]

Thank you

[@pelasgeuspelasgeus4634]

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

[@graphicmaths7677]

It's not a silly question at all. Possibly the simplest hand-waving explanation is that real number and imaginary numbers are different things, so every complex number is a unique combination of a real and imaginary number, and vice versa. A standard way to represent that sort of idea is to use orthogonal axes. Also, viewing complex numbers that way turns out to be very useful, and it doesn't lead to any contradictions, so it is very convenient to treat it as being true. It is a good way to understand Euler's formula, and multi-valued complex roots. There are probably deeper reasons too.

[@pelasgeuspelasgeus4634]

​@@graphicmaths7677So, it's an assumption. And, why do you treat complex numbers, which are points, as vectors? I mean that multiplication is common in vector analysis.

[@user-ug2ty8rh9f]

@@pelasgeuspelasgeus4634 multiplying by i rotates by 90 degrees