Arithmetic of complex numbers, the Argand diagram, modulus-argument form, de Moivre's theorem, nth roots, complex roots of polynomials and loci.

What this dot point is asking

Edexcel wants you to work fluently with complex numbers in Cartesian, modulus-argument and exponential form, plot them on an Argand diagram, use de Moivre's theorem to find powers and roots, solve polynomial equations with complex roots, and sketch loci defined by modulus and argument conditions. Complex numbers appear in every Core Pure paper, both as standalone questions and as a tool inside differential-equation and matrix problems.

Arithmetic and the conjugate

For $z = x + iy$ the real part is $\operatorname{Re}(z) = x$ and the imaginary part is $\operatorname{Im}(z) = y$. Addition is component-wise: $(a + bi) + (c + di) = (a + c) + (b + d)i$. Multiplication uses $i^2 = -1$:

The complex conjugate of $z = x + iy$ is $z^* = x - iy$ (Edexcel also writes $\bar{z}$). Multiplying a number by its conjugate clears the imaginary part:

which is always real and non-negative. This is the key to division: to divide, multiply numerator and denominator by the conjugate of the denominator so the denominator becomes real.

The Argand diagram and modulus-argument form

On an Argand diagram the horizontal axis is the real part and the vertical axis is the imaginary part, so $z = x + iy$ is the point $(x, y)$. The modulus is the distance from the origin,

and the argument $\theta = \arg z$ is the angle measured anticlockwise from the positive real axis, taken in the principal range $-\pi < \theta \le \pi$. Always sketch the point first so you choose the correct quadrant for $\theta$, because $\arctan\frac{y}{x}$ alone cannot tell the second quadrant from the fourth.

De Moivre's theorem and nth roots

For any integer $n$,

Equivalently, in exponential form, $(re^{i\theta})^n = r^n e^{in\theta}$. This single result is the engine for three things: raising a complex number to a power, finding the $n$ distinct $n$th roots of a complex number, and deriving multiple-angle identities such as $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ by expanding $(\cos\theta + i\sin\theta)^3$ and comparing real parts.

To find $n$th roots, write the number with a general argument by adding $2\pi k$, then divide the argument by $n$ and let $k$ run over $n$ consecutive integers.

Roots of polynomials

For a polynomial with real coefficients, complex roots occur in conjugate pairs. If $2 + 3i$ is a root then so is $2 - 3i$. The sum and product of a conjugate pair are both real, so they generate a real quadratic factor.

Loci on the Argand diagram

Conditions on $z$ describe curves in the plane. The three standard loci are:

• $|z - a| = r$ is a circle of radius $r$ centred at the point $a$.
• $|z - a| = |z - b|$ is the perpendicular bisector of the segment joining $a$ and $b$.
• $\arg(z - a) = \theta$ is a half-line (ray) starting at $a$ and pointing at angle $\theta$, with the start point itself excluded.

For exam work, translate the algebra into geometry, sketch it, and then read off distances. For a maximum or minimum of $|z|$ on a circle, use the centre-to-origin distance plus or minus the radius.

Examples in context

Complex numbers thread through the whole Core Pure course. De Moivre's theorem is the standard route to summing series such as $\sum \cos k\theta$ by treating them as the real part of a geometric series in $e^{i\theta}$. The exponential form $z = re^{i\theta}$ reappears when you solve second-order differential equations with complex auxiliary roots, where the solution $e^{(\alpha \pm i\beta)x}$ becomes $e^{\alpha x}(A\cos\beta x + B\sin\beta x)$ via Euler's formula. In matrices, the eigenvalues of a rotation matrix are complex conjugates $\cos\theta \pm i\sin\theta$, tying the topic back to modulus-argument form. Loci questions, meanwhile, blend complex algebra with coordinate geometry, and frequently ask for the greatest or least value of $|z|$ or $\arg z$ subject to a constraint.

Try this

Q1. Express $z = 1 + i$ in modulus-argument form. [3 marks]

• Cue. $r = \sqrt{2}$, $\theta = \frac{\pi}{4}$, so $z = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)$.

Q2. Given that $3 - 2i$ is a root of $x^2 + bx + c = 0$ with real $b$ and $c$, find $b$ and $c$. [3 marks]

• Cue. The other root is $3 + 2i$; sum of roots $= 6 = -b$ so $b = -6$, product $= 13 = c$.

Q3. Find the four fourth roots of $-16$, giving them in the form $re^{i\theta}$. [5 marks]

• Cue. $-16 = 16e^{i(\pi + 2\pi k)}$, so $z = 2e^{i(\pi + 2\pi k)/4}$ for $k = 0, 1, 2, 3$, giving arguments $\frac{\pi}{4}, \frac{3\pi}{4}, -\frac{3\pi}{4}, -\frac{\pi}{4}$.