Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle

Theorem

Let $z \in \C$ be a complex number.

Let $z$ be interpreted as a vector in the complex plane.

Let $w \in \C$ be the complex number defined as $z$ multiplied by the imaginary unit $i$:

$w = i z$

Then $w$ can be interpreted as the vector $z$ after being rotated through a right angle in an anticlockwise direction.

Proof

Let $z$ be expressed in polar form as:

$z = r \left({\cos \theta + i \sin \theta}\right)$

From Polar Form of Complex Number: $i$:

$i = \cos \dfrac \pi 2 + i \sin \dfrac \pi 2$

and so:

the modulus of $i$ is $1$

the argument of $i$ is $\dfrac \pi 2$.

By Product of Complex Numbers in Polar Form:

the modulus of $i z$ is $r$

the argument of $i$ is $\theta + \dfrac \pi 2$.

That is, the result of multiplying $z$ by $i$ is the same as rotating $z$ through $\dfrac \pi 2$, which is a right angle.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations