What would be geometrically analogous to adding or multplying points on the plane $\mathbb R^2$? ( On complex numbers).

Question

Why not a complete duplicate ( though a partial one) : This question deals both with multiplication of complex numbers,and with addition; hence , with the general idea of performing a binary operation on ordered pairs of reals. So, it is a bit more general as another post ( linked below) , and , as such, may be useful for complex numbers beginners, like me.

Geometric interpretation of the multiplication of complex numbers?

Complex numbers are defined as elements of $\mathbb R^2$, that is, ordered pairs of real numbers.
So, in a way, binary operations on complex numbers - such as addition or multiplication - are similar to adding or multiplicating points.
Can these operations be represented as movements in the real plane, in the same way as addition of integers is represented, at the basic level, as a movement on a line , or rather, on a series of aligned dots.
Maybe adding two complex numbers is analogous to moving from one point to another?
But I can't imagin to what movement could correspond multiplying two complex numbers.

Note : in comments, a link to a very helpful video by 3Blue1Brown.

That said, it's a duplicate. Briefly, adding corresponds to summing the position vectors, and multiplication corresponds to multiplying magnitudes and addiing rotational angles (think about polar form $re^{i\theta}$ ...). — Noah Schweber, Mar 26 '20 at 21:30
This video may provide some insight. The part from about 6:00 is relevant to your question, though the preceding part may be useful to watch as well. — Servaes, Mar 26 '20 at 21:31
Multiplying by a real number corresponds to a scaling; multiplying by $i$ corresponds to a $90^\circ$ rotation — J. W. Tanner, Mar 26 '20 at 21:31
Thanks for all these answers. ( New to complex numbers, I'll search in all these directions). — , Mar 26 '20 at 21:32

score 0 · Accepted Answer · answered Mar 26 '20 at 21:32

The multiplication by a non-zero complex number can be seen as the composition of a rotation (around $0$) with a homothety (with respect to $0$). That can be seen in the polar representation of complex numbers: if $z=\rho\bigl(\cos(\theta)+\sin(\theta)i\bigr)$, then multiplication of $w$ by $z$ is the same thing as rotating $w$ clockwise by an angle of $\theta$ radians, followed by a homothety with ratio $\rho$. Or you can do the homothety first and the rotation after. The result will be the same.

What would be geometrically analogous to adding or multplying points on the plane $\mathbb R^2$? ( On complex numbers).

1 Answers1