Multiplication of Complex Number by -1 is Equivalent to Rotation through Two Right Angles
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 $-1$:
- $w = \left({-1}\right) z$
Then $w$ can be interpreted as the vector $z$ after being rotated through two right angles.
The direction of rotation is usually interpreted as being anticlockwise, but a rotated through two right angles is the same whichever direction the rotation is performed.
Proof
By definition of the imaginary unit:
- $-1 = i^2$
and so:
- $-1 \times z = i \paren {i z}$
From Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle, multiplication by $i$ is equivalent to rotation through a right angle, in an anticlockwise direction.
So multiplying by $i^2$ is equivalent to rotation through two right angles in an anticlockwise direction.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations