Geometric interpretation of the multiplication of complex numbers?I've always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the x component and the imaginary part is the y component.In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.However, is there a geometric interpretation for the multiplication of two complex numbers?I tried out two test ones, 3+i and −2+3i, which multiply to −9+7i. But no geometrical significance seems to be found.Is there a geometric significance for the multiplication of complex numbers?

Question

Geometric interpretation of the multiplication of complex numbers?I&#039;ve always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the x component and the imaginary part is the y component.In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.However, is there a geometric interpretation for the multiplication of two complex numbers?I tried out two test ones, 3+i and −2+3i, which multiply to −9+7i. But no geometrical significance seems to be found.Is there a geometric significance for the multiplication of complex numbers?

marktje28 · Accepted Answer

Suppose we multiply the complex numbers       z    1   and       z    2  . If these numbers are written in the polar form as       r    1        e          i              θ        1             and       r    2        e          i              θ        2            , the product will be       r    1        r    2        e          i      (              θ        1            +              θ        2            )      . Equivalently, we are stretching the first complex number       z    1   by a factor equal to the magnitude of the second complex number       z    2   and then rotating the stretched z1 counter-clockwise by an angle       θ    2   to arrive at the product. There are several websites that expand upon this intuition with graphics and more explanation.

skylsn · Accepted Answer

Add the angles and multiply the lengths.

Geometric interpretation of the multiplication of complex numbers? I've always been taught that one

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