The Argand Diagram, Modulus and Argument

A2 · P3 · 3 min

A complex number $x + iy$ is a point $(x, y)$ in the plane, with the real part along the horizontal axis and the imaginary part along the vertical axis. This picture is the Argand diagram, and it turns algebra into geometry: the modulus is a distance and the argument is an angle.

Plotting

Key result

The point representing $z = x + iy$ is $(x, y)$ . The conjugate $z^*$ is the reflection in the real axis, and $-z$ is the rotation through $180^\circ$ about the origin.

(3, 2) (3, -2) (-3, -2) (0, 0) -> (3, 2)

Modulus and argument

Key result

For $z = x + iy$ :

|z| = \sqrt{x^2 + y^2}, \qquad \arg z = \theta \text{ where } \tan\theta = \frac{y}{x},

with the quadrant chosen from the signs of $x$ and $y$ . The principal argument satisfies $-\pi < \arg z \leq \pi$ : positive angles are anticlockwise from the positive real axis, negative angles clockwise.

Method

To find $\arg z$ :

Compute the acute angle $\alpha = \tan^{-1}\left|\dfrac{y}{x}\right|$ .
Place the point in its quadrant: first $\theta = \alpha$ ; second $\theta = \pi - \alpha$ ; third $\theta = -(\pi - \alpha)$ ; fourth $\theta = -\alpha$ .
Points on the axes: positive real $0$ , positive imaginary $\tfrac{\pi}{2}$ , negative real $\pi$ , negative imaginary $-\tfrac{\pi}{2}$ .

Modulus and argument in each quadrant

Find the modulus and argument of $1 + i$ , $-2 + 2i\sqrt{3}$ , $-3 - 3i$ and $4 - 3i$ .

Solution

$1 + i$ : $|z| = \sqrt{2}$ , first quadrant, $\arg z = \tfrac{\pi}{4}$ .

$-2 + 2i\sqrt{3}$ : $|z| = \sqrt{4 + 12} = 4$ ; $\alpha = \tan^{-1}\sqrt{3} = \tfrac{\pi}{3}$ , second quadrant, $\arg z = \pi - \tfrac{\pi}{3} = \tfrac{2\pi}{3}$ .

$-3 - 3i$ : $|z| = 3\sqrt{2}$ ; $\alpha = \tfrac{\pi}{4}$ , third quadrant, $\arg z = -\tfrac{3\pi}{4}$ .

$4 - 3i$ : $|z| = 5$ ; $\alpha = \tan^{-1}\tfrac{3}{4} = 0.6435$ , fourth quadrant, $\arg z = -0.644$ .

From modulus and argument to Cartesian form

A complex number has modulus $6$ and argument $-\tfrac{5\pi}{6}$ . Write it in the form $x + iy$ .

Solution

$x = 6\cos\left(-\tfrac{5\pi}{6}\right) = -3\sqrt{3}$ , $y = 6\sin\left(-\tfrac{5\pi}{6}\right) = -3$ . So $z = -3\sqrt{3} - 3i$ .

Modulus of a product and a quotient

$z = 3 + 4i$ and $w = 1 - i$ . Find $|zw|$ and $\left|\dfrac{z}{w}\right|$ without computing $zw$ or $\dfrac{z}{w}$ .

Solution

$|z| = 5$ , $|w| = \sqrt{2}$ . $|zw| = 5\sqrt{2}$ and $\left|\dfrac{z}{w}\right| = \dfrac{5}{\sqrt{2}}$ .

Distance between points

$|z_1 - z_2|$ is the distance between the points $z_1$ and $z_2$ . This is the idea behind every locus in Loci in the Argand Diagram.

Distance and midpoint

Points $A$ and $B$ represent $z_A = 2 + 5i$ and $z_B = -4 - 3i$ . Find $|AB|$ and the complex number represented by the midpoint of $AB$ .

Solution

$z_A - z_B = 6 + 8i$ , so $|AB| = 10$ . Midpoint: $\tfrac{1}{2}(z_A + z_B) = -1 + i$ .

Watch out

$\tan^{-1}\dfrac{y}{x}$ on a calculator always gives an angle between $-\tfrac{\pi}{2}$ and $\tfrac{\pi}{2}$ , which is wrong for the second and third quadrants. Sketch the point first, every time.

Exam tip

Unless told otherwise, give arguments in radians in the range $-\pi < \theta \leq \pi$ , to 3 significant figures or as an exact multiple of $\pi$ . If the question uses $0 \leq \theta < 2\pi$ , follow it.

Practice

Question

Find the modulus and argument of $-1 + i$ , $2 - 2i\sqrt{3}$ and $-5$ .
Write the number with modulus $2$ and argument $\tfrac{3\pi}{4}$ in Cartesian form.
Show on an Argand diagram the points $z$ , $z^*$ , $-z$ and $iz$ for $z = 2 + i$ , and describe the effect of multiplying by $i$ .
Find the distance between the points representing $3 - 2i$ and $-1 + i$ .

Answers

$\sqrt{2}$ , $\tfrac{3\pi}{4}$ ; $4$ , $-\tfrac{\pi}{3}$ ; $5$ , $\pi$ .
$-\sqrt{2} + i\sqrt{2}$ .
$iz = -1 + 2i$ ; multiplying by $i$ rotates the point $90^\circ$ anticlockwise about the origin.
$|4 - 3i| = 5$ .

Plotting

Modulus and argument

Distance between points

Practice

Console