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

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.

• 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
• 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
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.