Complex numbers – further maths revision notes to download as PDF

Starter activity

A GPS app places your position at \(3+2i\) on a map in kilometres. Walk 4 km east and 1 km north: what is the new complex number? Now turn the whole map through \(90^\circ\) anticlockwise about the origin. Where does your original position go? Try multiplying \(3+2i\) by \(i\) and see what you get. Compare the result with the rotated point. What do you think multiplying by a complex number does to a position on the map?

Note: greatest and least modulus

On the circle \(|z-c|=r\), the greatest and least values of \(|z|\) are \(|c|+r\) and \(\big||c|-r\big|\), found along the line through the origin and the centre.

Note: why multiplication rotates

Write \(z=r(\cos\theta+i\sin\theta)\) and multiply by \(w=s(\cos\phi+i\sin\phi)\). The product has modulus \(rs\) and argument \(\theta+\phi\), so multiplying by \(w\) stretches the Argand diagram by the factor \(s\) and turns it through the angle \(\phi\). This is why multiplying by \(i\) (modulus 1, argument \(\tfrac\pi2\)) is a quarter turn anticlockwise. In engineering the same idea describes a phasor: an alternating voltage is a rotating arrow, and multiplying by a complex impedance both scales it and shifts its phase.

Reading the question: what examiners expect

Further maths questions on complex numbers usually mix several ideas, so it helps to recognise the typical patterns. If the question says "find the roots" and gives you one complex root of a polynomial with real coefficients, you are expected to write down the conjugate immediately and build a real quadratic factor. If it says "express in the form \(a+bi\)", you must remove every \(i\) from the denominator and leave no brackets. If it says "show that", you must display every step, because the answer is already given and the marks are for the reasoning. When the question uses the words "exact value", keep surds and fractions of \(\pi\) in your answer instead of rounding. Finally, if a diagram is asked for, label the axes \(\operatorname{Re}\) and \(\operatorname{Im}\), mark the key points with their coordinates, and shade the required region clearly.

Applications in science and engineering

Complex numbers are not just an exercise in algebra. In electrical engineering, resistors, coils and capacitors are combined by adding complex impedances, and the modulus and argument of the result give the size and timing of the current. In physics, the rotating arrow of a wave is the number \(\cos\omega t+i\sin\omega t\), so adding two waves of the same frequency means adding two complex numbers. In computer graphics, rotating an image about the origin through an angle \(\theta\) is done by multiplying every pixel position by the same complex number of modulus 1. Because multiplication combines a scaling and a rotation in a single operation, the algebra of complex numbers is a very economical way to describe plane geometry.

Understanding why the Argand diagram works

Every real number lives on a single number line, but a complex number needs two pieces of information, so it lives on a plane. The horizontal axis holds the real part and the vertical axis holds the imaginary part. Adding two complex numbers is exactly the same as adding two position vectors with the parallelogram rule, so the geometry you already know from vectors carries over unchanged. The new idea is multiplication. Because multiplying by \(i\) sends \(1\) to \(i\), \(i\) to \(-1\), \(-1\) to \(-i\) and \(-i\) back to \(1\), it behaves like a quarter turn. Applying it twice gives a half turn, which is multiplication by \(i^2=-1\). This single observation explains why the square of the imaginary unit is negative: two quarter turns make a half turn, and a half turn takes every number to its negative. Once you see multiplication as a turn combined with a stretch, the rules for moduli and arguments become natural rather than something to memorise.

Practical tips for the written paper

Work with exact values wherever you can, because exact answers are easier to check and are usually what the mark scheme asks for. When you solve a quadratic with a negative discriminant, write the square root of the negative number as an imaginary multiple straight away, for example \(\sqrt{-36}=6i\). Use the conjugate whenever you see a modulus squared. Keep your Argand sketches large and neat, label each axis, and draw dashed lines for strict inequalities. Lastly, if you are stuck on a locus question, write \(z=x+iy\) and convert to Cartesian form; the algebra is often longer than the geometric argument, but it always works.

Connecting the topics in this chapter

It is worth stepping back to see how the sections of this chapter fit together, because examination questions deliberately cross the boundaries. Cartesian form is best when you add, subtract or divide, because the real and imaginary parts are visible. Modulus-argument form is best when you multiply, divide or take powers, because moduli multiply and arguments add. Loci link the algebra to geometry: a modulus condition measures distance, and an argument condition measures direction. Polynomial roots link everything to algebra, since a quadratic with a negative discriminant produces a conjugate pair, and the pair is symmetrical about the real axis on the Argand diagram. When you meet an unfamiliar problem, ask yourself which form makes the operation easiest, convert if necessary, and convert back at the end. Students who switch comfortably between the two forms tend to score highly in this topic, because they can choose the quickest route instead of forcing every question through the same method.

Vocabulary to remember

The real part and the imaginary part of \(x+iy\) are the numbers \(x\) and \(y\); notice that the imaginary part does not include the letter \(i\). A number whose real part is zero is called purely imaginary, and a number whose imaginary part is zero is real. The conjugate is the reflection in the real axis. The modulus is a distance, so it is never negative, and the argument is an angle measured anticlockwise from the positive real axis. The principal argument lies in the interval from \(-\pi\) to \(\pi\), excluding \(-\pi\). A locus is a set of points, and a half-line is a locus with a starting point that is not included. Using these words precisely will help you write clear solutions and avoid ambiguity in explanations.

Common misconceptions and how to avoid them

Many students believe that \(i\) is just another variable, but it is a fixed number defined by \(i^2=-1\), so any power of \(i\) can be reduced to one of four values. Others think that the argument of a negative real number is zero, when in fact it is \(\pi\). Some try to order complex numbers using inequality signs, but there is no meaningful way to say that one complex number is greater than another; only moduli, which are real, can be compared. Another frequent error is to treat \(|z|^2\) as \(z^2\). They are different: \(z^2\) is a complex number, whereas \(|z|^2=zz^*\) is real and non-negative. Keep these distinctions clear and you will avoid the majority of the errors seen in examination scripts.

How this chapter prepares you for the next

The ideas here are the foundation for the following chapter on de Moivre's theorem and roots of unity, where powers and roots of complex numbers are found by working in modulus-argument form. They also prepare you for polynomial equations, differential equations with oscillating solutions, and polar coordinates, where the pair of numbers modulus and argument becomes a new way of describing every point of the plane. Time spent making the basic skills automatic now will make those later topics much easier.

Exam strategy and timing

On a typical further maths paper, a question on complex numbers carries between eight and fifteen marks and should take roughly one mark per minute. Spend the first minute reading and planning, because the most efficient method is not always the first one that comes to mind. Write each line of working clearly and underline or box the final answer. If you cannot complete one part, state the result you would expect and move on, since later parts often give you enough information to continue. Method marks are awarded for correct steps even when the final value is wrong, so never leave a blank. Leave a few minutes at the end to check signs, quadrant choices and the form in which the answer has been requested.

Did you know?

The number \(i\) was first used in the sixteenth century by Italian mathematicians such as Cardano and Bombelli, who found that solving certain cubic equations needed square roots of negative numbers, even when all three answers were real. Jean-Robert Argand published his diagram in 1806, and Gauss later used it widely. Today complex numbers are essential in electrical engineering, quantum physics, signal processing and the computer graphics used in films and games.