How do I find the argument of a complex number?

Draw it on the Argand diagram and measure the angle from the positive real axis, anticlockwise positive. tan⁻¹(b/a) only gives the right answer in the first and fourth quadrants: in the second add π, in the third subtract π, so the principal argument lands in (−π, π].

Why does multiplying complex numbers add the arguments?

Write z = r(cos α + i sin α) and w = s(cos β + i sin β). Multiplying out and using the addition formulas gives zw = rs(cos(α + β) + i sin(α + β)). The lengths multiply and the angles add.

What are the roots of unity?

The n solutions of zⁿ = 1. They are e^(2πik/n) for k = 0, 1, ..., n − 1, equally spaced round the unit circle as the corners of a regular n-sided polygon, and for n ≥ 2 they add up to 0.

What is the principal argument of −1, and of 0?

−1 lies on the negative real axis, so its principal argument is π (not −π, because the interval is (−π, π]). 0 has no argument at all, since it has no direction.

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Complex numbers on the Argand diagram

Every complex number is a point and an arrow. Adding slides arrows end to end; multiplying multiplies the lengths and adds the angles.

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A complex number z = a + bi is the point (a, b) on the Argand diagram, or equally an arrow of length |z| = √(a² + b²) at angle arg z from the positive real axis. Adding complex numbers adds the arrows, so z + w is the far corner of a parallelogram. Multiplying is a rotation and a scaling: |zw| = |z||w| and arg zw = arg z + arg w, which is why the triangle 0, 1, z and the triangle 0, w, zw are similar, and why multiplying by i (length 1, angle 90°) is a quarter turn. Dividing undoes it. Repeating the multiplication gives de Moivre's theorem: zⁿ has modulus |z|ⁿ and argument n arg z, so the powers spiral out (|z| > 1) or in (|z| < 1). Going backwards, the n roots of z share the modulus |z|^1/n and sit 2π/n apart, the corners of a regular polygon; for z = 1 they are the roots of unity, and they always add up to 0. The principal argument is the one in the interval (−π, π].

z = a + bi = r(cos θ + i sin θ) = re^iθ|zw| = |z||w|, arg zw = arg z + arg wzⁿ = rⁿ(cos nθ + i sin nθ)z^1/n = r^1/n e^iθ/n e^2πik/n

A-Level Further Maths (AQA, Edexcel, OCR, OCR MEI): Argand diagrams, modulus-argument form, loci, de Moivre's theorem and nth roots. First-year university complex analysis.

Work through the numbers with Unit Circle and Scientific Calculator.

Challenge

Predict first: z = 1 + i. What are the modulus and argument of z⁸? Use de Moivre, then switch to zⁿ, set n = 8 and check.

FAQ

How do I find the argument of a complex number?: Draw it on the Argand diagram and measure the angle from the positive real axis, anticlockwise positive. tan⁻¹(b/a) only gives the right answer in the first and fourth quadrants: in the second add π, in the third subtract π, so the principal argument lands in (−π, π].
Why does multiplying complex numbers add the arguments?: Write z = r(cos α + i sin α) and w = s(cos β + i sin β). Multiplying out and using the addition formulas gives zw = rs(cos(α + β) + i sin(α + β)). The lengths multiply and the angles add.
What are the roots of unity?: The n solutions of zⁿ = 1. They are e^2πik/n for k = 0, 1, ..., n − 1, equally spaced round the unit circle as the corners of a regular n-sided polygon, and for n ≥ 2 they add up to 0.
What is the principal argument of −1, and of 0?: −1 lies on the negative real axis, so its principal argument is π (not −π, because the interval is (−π, π]). 0 has no argument at all, since it has no direction.

Complex numbers on the Argand diagram

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