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.

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    $\begingroup$ I don't know why this got a -1, it's a fine question. $\endgroup$ – Noah Schweber Mar 26 '20 at 21:29
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    $\begingroup$ Have you thought about the polar form of complex numbers? $\endgroup$ – Eric Towers Mar 26 '20 at 21:30
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    $\begingroup$ 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 $re^{i\theta}$ ...). $\endgroup$ – Noah Schweber Mar 26 '20 at 21:30
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    $\begingroup$ 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. $\endgroup$ – Servaes Mar 26 '20 at 21:31
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    $\begingroup$ Multiplying by a real number corresponds to a scaling; multiplying by $i$ corresponds to a $90^\circ$ rotation $\endgroup$ – 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.


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