Geometry of Complex Operations

A2 · P3 · 3 min

Every operation on complex numbers moves points in the Argand diagram in a predictable way. Addition is a translation, multiplication is a rotation with a stretch, conjugation is a reflection. Knowing these lets you sketch results without calculating and explains why loci take the shapes they do.

The effects

Key result

Operation	Geometric effect on the point $z$
$z^*$	reflect in the real axis
$-z$	rotate $180^\circ$ about the origin
$z + w$	translate by the vector representing $w$ (parallelogram rule)
$z - w$	translate by $-w$ ; $\lvert z - w \rvert$ is the distance from $w$ to $z$
$kz$ ( $k > 0$ real)	enlarge from the origin by scale factor $k$
$iz$	rotate $90^\circ$ anticlockwise about the origin
$zw$	rotate by $\arg w$ and enlarge by $\lvert w \rvert$
$z / w$	rotate by $-\arg w$ and shrink by $\lvert w \rvert$

The multiplication rule is the key one: $|zw| = |z||w|$ and $\arg(zw) = \arg z + \arg w$ . It is proved in Polar Form.

Multiplying by a unit complex number

$z = 4 + 2i$ and $w = \tfrac{1}{2} + \tfrac{\sqrt{3}}{2}i$ . Describe geometrically the transformation from $z$ to $zw$ , and find $zw$ .

Solution

$|w| = 1$ and $\arg w = \tfrac{\pi}{3}$ , so multiplying by $w$ rotates $z$ by $60^\circ$ anticlockwise about the origin with no change in distance.

$zw = 2 + 2\sqrt{3}i + i + \sqrt{3}i^2 = (2 - \sqrt{3}) + (2\sqrt{3} + 1)i$ .

Vector addition

$O$ is the origin and $A$ , $B$ represent $z_1 = 3 + i$ and $z_2 = 1 + 4i$ . Show that $OACB$ is a parallelogram where $C$ represents $z_1 + z_2$ , and find the complex number represented by the diagonal $\overrightarrow{AB}$ .

Solution

$C = 4 + 5i$ . $\overrightarrow{OA} = \overrightarrow{BC}$ since $C - B = 3 + i = A$ , and $\overrightarrow{OB} = \overrightarrow{AC}$ since $C - A = 1 + 4i = B$ . Opposite sides are equal and parallel, so $OACB$ is a parallelogram.

$\overrightarrow{AB}$ is $z_2 - z_1 = -2 + 3i$ .

A right angle from multiplication by i

Show that for any non-zero $z$ , the points $0$ , $z$ and $iz$ form a right angle at the origin, and hence that the triangle with vertices $0$ , $z$ , $z + iz$ is isosceles right-angled.

Solution

$\arg(iz) = \arg z + \tfrac{\pi}{2}$ , so $iz$ is $z$ rotated through $90^\circ$ ; $|iz| = |z|$ . The angle between $\overrightarrow{0z}$ and $\overrightarrow{0(iz)}$ is $90^\circ$ and the two sides are equal, so the triangle $0$ , $z$ , $z + iz$ (with $z + iz$ completing the square) has a right angle at $0$ and two equal sides.

Reading a diagram

Questions may show points and ask which represents $z^*$ , $2z$ , $z - w$ , or $iz$ . Use the table: the conjugate is the mirror image below or above the real axis; $2z$ is on the same line from the origin, twice as far; $iz$ is a quarter turn round.

Identifying a point

$z = -2 + 3i$ . Find the numbers represented by (a) the reflection of $z$ in the imaginary axis; (b) the point half-way between $z$ and $z^*$ ; (c) $z$ rotated $90^\circ$ clockwise about the origin.

Solution

(a) $-z^* = 2 + 3i$ . (b) $\tfrac{1}{2}(z + z^*) = \operatorname{Re} z = -2$ . (c) $-iz = -i(-2 + 3i) = 3 + 2i$ .

Watch out

Rotation is about the origin, not about the point itself. To rotate a point about a different centre $c$ , use $c + i(z - c)$ : translate to the origin, rotate, translate back.

Exam tip

"Describe geometrically" wants the transformation named with its parameters: "rotation through $\tfrac{\pi}{3}$ anticlockwise about $O$ and enlargement scale factor $2$ , centre $O$ ". A calculation alone does not answer the question.

Practice

Question

$z = 1 + 2i$ . Find and plot $z^*$ , $-z$ , $iz$ and $2z$ .
Describe the transformation from $z$ to $(1 + i)z$ .
$A$ represents $2 + i$ and $B$ represents $-1 + 3i$ . Find the number represented by the point $C$ such that $OACB$ is a parallelogram, and the length of $OC$ .
Find the image of $3 + i$ under a rotation of $90^\circ$ anticlockwise about the point $1 + i$ .

Answers

$1 - 2i$ ; $-1 - 2i$ ; $-2 + i$ ; $2 + 4i$ .
$|1 + i| = \sqrt{2}$ , $\arg(1 + i) = \tfrac{\pi}{4}$ : rotation through $\tfrac{\pi}{4}$ anticlockwise and enlargement by $\sqrt{2}$ , both about the origin.
$C = 1 + 4i$ ; $|OC| = \sqrt{17}$ .
$1 + i + i\big((3 + i) - (1 + i)\big) = 1 + i + 2i = 1 + 3i$ .

The effects

Reading a diagram

Practice

Console