User:Qerez/sandbox

I'm looking for a better source for the geometric interpretation of multiplication, preferably a textbook. This interpretation seems to be well e.g. heavily featured in 3blue1brown video on complex numbers: https://www.youtube.com/watch?v=5PcpBw5Hbwo&t=1480. Grant refers to https://www.geogebra.org/m/mbhbdvkr Ben Sparks but I did not find a text book that features it. The only source beside these two I found was deemed not reliable enough. Happy for any suggestion. A bit (together with two pictures) I'd like to add in the section Complex_number#Multiplication_and_square (please note that Grant Sanderson and Ben Sparks distribute b rather a, but due to commutativity this does really matter):

Multiplication of complex numbers ab can be found geometrically as the point that has the same coordinates as a in a scaled and rotated Cartesian coordinate system generated by b and ib.

The commutativity property of multiplication can be shown geometrically by showing that the point ba which has the same coordinates as b in a scaled and rotated Cartesian coordinate system generated by a andi ia coincides with ab.

The distributive property of multiplication over addition can be used to visualize multiplication geometrically. In particular, distributing b to $a=x+yi$:

$$a\,b=(x+yi)\,b=xb+yib.$$

and using the visualization of complex numbers in the complex plane, multiplication has the following geometric interpretation: the product of two complex numbers a and b is the point that has the same coordinates as the point a in the complex plane but in a new scaled and rotated Cartesian coordinate system that has the x-axis and y-axis going through b and ib and a unit of measure $|b|$.

Qerez (talk) 11:06, 20 February 2022 (UTC)