Multiplication of Complex Number by -1 is Equivalent to Rotation through Two Right Angles

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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.

$\blacksquare$


Sources