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

 * Rotation-by-minus-1.png

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.