Complex numbers: why one multiplication is a quarter-turn

Here is a fact that makes a lot of complex number questions shorter: multiplying by \(i\) rotates a point a quarter-turn anticlockwise about the origin.

Three lines of algebra

Let \(z = a + bi\). Then

\[ iz = ai + bi^2 = -b + ai. \]

So the point \((a, b)\) moves to \((-b, a)\). That new point is the same distance from the origin, because \(\sqrt{(-b)^2 + a^2} = \sqrt{a^2 + b^2}\), and it is at right angles to the original, because the dot product of the two position vectors is \(-ab + ab = 0\).

Where this helps

• Rotating a point \(90^\circ\) clockwise? Multiply by \(-i\).
• Finding the fourth vertex of a square with sides from \(O\) to \(z\) and \(iz\)? It is \(z + iz\).
• In polar form, \(i = \operatorname{cis}\frac{\pi}{2}\), so this is a special case of the rule: multiply the moduli and add the arguments.

The free notes below go further, with six questions and worked solutions.