Complex addition and multiplication (generic)

Geometric Interpretation of the Addition and Multiplication of Complex Numbers

There is a direct geometric interpretation for both the addition and the multiplication of complex numbers. While the addition of a constant summand results in a translation, complex multiplication with a constant factor can be interpreted as a scaling rotation.

Complex Addition

Basically, a complex addition is nothing but an addition of 2-dimensional vectors. The real part and the imaginary part are added independently. Geometrically this sum can be found using a parallelogram construction.

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Complex Multiplication

When multiplying two complex numbers, their lengths are multiplied while their angles with respect to the real axis are added. This is best seen in the polar form of a complex number. If

$a=r_a\cdot e^{i\psi_a} \;\;\;\mbox{and} \quad b=r_b\cdot e^{i\psi_b},$

then the product becomes

$a\cdot b=r_a r_b\cdot e^{i(\psi_a+\psi_b)}.$

One can also interpret the multiplication by a complex number $r_a\cdot e^{i\psi_a}$ as a spiral similarity transformation. In this case, the object is rotated by an angle $\psi_a$ and expanded (resp. contracted) by a factor of

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Complex addition and multiplication (integral) $\hookleftarrow$ Contents $\hookrightarrow$ Complex conjugation