Complex Multiplication as Geometrical Transformation/Corollary

Corollary to Complex Multiplication as Geometrical Transformation

Let $z = \polar {r, \theta}$ be a complex number expressed in polar form.

Let $z$ be represented on the complex plane $\C$ in vector form.

The effect of multiplying $z$ by $e^{i \alpha}$ is to rotate it about the origin of $\C$ by $\alpha$ in the positive direction

Proof

From Complex Multiplication as Geometrical Transformation, the effect of multiplying a complex number by $r e^{i \alpha}$ is:

to rotate it about the origin of $\C$ by $\alpha$ in the positive direction

to multiply its modulus by $r$.

In this instance $r = 1$.

Hence the result.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $24$