The Argand diagram

An Argand diagram is a graph for complex numbers. The horizontal axis is the real axis and the vertical axis is the imaginary axis, so a + bi sits at the point with coordinates (a, b). The distance from the origin to the point is the modulus, |z|, and the angle from the positive real axis is the argument, arg z.

Plotting complex numbers on an Argand diagram

Read the real part across and the imaginary part up, just like x and y coordinates. So 3 + 2i goes 3 right and 2 up, and −1 − 3i goes 1 left and 3 down. A real number such as 4 sits on the real axis, and a purely imaginary number such as 2i sits on the imaginary axis.

The complex conjugate of z = a + bi is z* = a − bi. On the diagram it is the reflection of z in the real axis, as with 3 + 2i and 3 − 2i above. In AQA's A-level Further Maths specification, using and interpreting Argand diagrams is a named skill, alongside the fact that non-real roots of polynomials with real coefficients come in conjugate pairs.

Modulus and argument

For z = a + bi, the modulus is the length of the line from the origin to z. By Pythagoras' theorem:

The argument θ is the angle that line makes with the positive real axis, measured anticlockwise (counterclockwise in the US) in radians. The principal argument is chosen between −π and π, so angles below the real axis are negative. For z = 3 + 4i, |z| = √(9 + 16) = 5 and arg z = tan⁻¹(4/3) = 0.927 radians, which is 53.1°.

Arguments in the other quadrants

tan⁻¹(b/a) on a calculator only gives angles between −π/2 and π/2, so for points on the left of the imaginary axis you must adjust. Always sketch the point first.

Modulus-argument form

Knowing r = |z| and θ = arg z, any complex number can be written as:

For example, −1 + √3i has r = 2 and θ = 2π/3, so −1 + √3i = 2(cos 2π/3 + i sin 2π/3). Converting between this and a + bi is in AQA's specification. The complex number calculator converts both ways and plots the result on an Argand diagram.

Adding and subtracting on an Argand diagram

Complex numbers add like vectors. Draw a line from the origin to each number; the sum is the fourth corner of the parallelogram they make. Subtraction works the same way: z₁ − z₂ is the line from z₂ to z₁, so |z₁ − z₂| is the distance between the two points. That distance idea is the key to loci.

Multiplying on an Argand diagram

In modulus-argument form, multiplication has a neat picture: multiply the moduli and add the arguments. Dividing divides the moduli and subtracts the arguments. So multiplying by i, which has modulus 1 and argument π/2, rotates a point a quarter turn anticlockwise about the origin: i(2 + i) = −1 + 2i.

Loci in the Argand diagram

A locus is the set of points that obey a rule. AQA lists constructing and interpreting simple loci, such as |z − a| > r and arg(z − a) = θ. Two shapes cover most questions:

• |z − a| = r is a circle with centre a and radius r, because every point is distance r from a. Replace = with < for the inside, or > for the outside.
• arg(z − a) = θ is a half-line starting at a, at angle θ to the positive real direction. The point a itself isn't included, because its argument is undefined.
• |z − a| = |z − b| is the perpendicular bisector of the line from a to b: every point the same distance from both.

Roots of a complex number

The n different nth roots of a complex number are spread evenly round a circle centred on the origin, so AQA notes they form the vertices of a regular polygon with n sides. The cube roots of 1 are 1 and −½ ± (√3/2)i, at angles 0, 2π/3 and 4π/3, making an equilateral triangle.

Common mistakes

• Swapping the axes. Real part across, imaginary part up.
• Using the calculator's tan⁻¹ answer in every quadrant. Sketch the point and adjust for the top-left and bottom-left quadrants.
• Leaving i in the modulus. |3 + 4i| = √(3² + 4²), not √(3² + (4i)²).
• Degrees instead of radians. Arguments are normally given in radians at A-level.
• Wrong sign in a locus. |z − 2 + i| = 3 is centred on 2 − i, because z − 2 + i = z − (2 − i).

To practise complex numbers under time pressure, try today's A-level Further Maths short timed paper. Each paper mixes topics. For the basics of imaginary numbers, the square root guide covers the square roots of negative numbers.

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