Chapter 4 · Introduction

Chapter 4 — Complex Numbers and Quadratic Equations

The equation $x^2+1=0$ has no real solution — so we invent a number $i$ with $i^2=-1$, and discover a whole new number system, $\mathbb{C}$, with its own arithmetic and its own plane.

Ask a calculator for $\sqrt{-1}$ and it answers "Error". That is fair: the square of every real number is positive or zero, so no real number squares to $-1$, and the simple-looking equation $x^2+1=0$ has no real solution. For centuries mathematicians simply stopped there. Then, in the 1500s, Italian algebraists found a formula for solving cubic equations, and something strange happened. Applied to $x^3=15x+4$, whose solution $x=4$ you can check in your head, the formula produced the expression $\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$. Rafael Bombelli decided to calculate with $\sqrt{-1}$ as if it were an ordinary number obeying one extra rule, $(\sqrt{-1})^2=-1$ — and the "impossible" square roots cancelled out to leave exactly $4$. A detour through numbers that did not exist had led to a perfectly real answer.

Today the number $\sqrt{-1}$ is written $i$, and numbers of the form $a+ib$ — complex numbers — are everywhere. Electrical engineers describe alternating currents and impedances with them (they write $j$ instead of $i$, because $i$ is already the current), the signal processing in your phone's camera and music player runs on them, and quantum mechanics cannot even be stated without them. In this chapter you will meet $i$, learn to add, subtract, multiply and divide complex numbers, measure their size with the modulus, reflect them with the conjugate, and finally see them as points of a plane — the Argand plane — where addition becomes a parallelogram and multiplication by $i$ becomes a quarter turn.

Invent: a number $i$ with $i^2=-1$→Build: $z=a+ib$ and its algebra→Measure and divide: $\bar z$, $|z|$, $z^{-1}$→Picture: the Argand plane
🎯 By the end of this chapter you can…
  • explain why $i$ is needed, write $\sqrt{-a}=i\sqrt a$ for $a\gt0$, and write any complex number in the form $a+ib$ with its real and imaginary parts;
  • add, subtract and multiply complex numbers, use equality of complex numbers to find unknowns, and simplify any power of $i$;
  • find the conjugate, the modulus and the multiplicative inverse of a complex number, and divide one complex number by another;
  • use the properties of $\bar z$ and $|z|$, and the algebraic identities, to simplify expressions and prove results;
  • plot complex numbers in the Argand plane, and read $\bar z$, $-z$, $z_1+z_2$, $|z|$ and $|z_1-z_2|$ geometrically.
Teaser — Where Do the Roots Go?

On the left, the parabola $y=x^2+c$ slides up as $c$ grows. Its roots are where it crosses the $x$-axis — the solutions of $x^2=-c$. On the right is a plane of numbers: real numbers along the horizontal line, multiples of $i$ along the vertical one. While $c\lt0$ the two roots sit on the real line. At $c=0$ they meet. Once $c\gt0$ the parabola lifts off the axis — and the roots do not disappear: they step off the real line onto the vertical line, as $\pm i\sqrt c$.

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equation—
roots—
real roots?—

Sections in this chapter

4.1

Complex Numbers and Their Algebra

The imaginary unit $i$, the form $a+ib$, equality, addition, subtraction, multiplication, powers of $i$ and the algebraic identities.

$i^2=-1,\ \ i^{4k}=1$
Open section →4.2

Conjugate, Modulus and Division

The conjugate $\bar z$, the modulus $|z|$, the multiplicative inverse and the quotient of two complex numbers, with their properties.

$z\bar z=|z|^2,\ \ z^{-1}=\dfrac{\bar z}{|z|^2}$
Open section →4.3

The Argand Plane

Complex numbers as points; reflections, the parallelogram for $z_1+z_2$, distance $|z_1-z_2|$, and multiplication by $i$ as a quarter turn.

$|z|=\sqrt{a^2+b^2}=OP$
Open section →
✅ Before you start
You need the number systems $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$ and set notation from Section 1.1, and the Cartesian plane $\mathbb{R}\times\mathbb{R}$ with ordered pairs from Section 2.1 — a complex number $a+ib$ will turn out to be the ordered pair $(a,b)$ in disguise. From Class 10 Mathematics you need surds and irrational numbers (Chapter 1), the quadratic formula and the discriminant $b^2-4ac$ (Chapter 4, "The Quadratic Formula and the Nature of Roots"), and the distance formula of coordinate geometry (Chapter 7). Fluent expansion of brackets — $(a+b)^2$, $(a+b)(a-b)$, $(a+b)^3$ — is used on every page. The optional JEE box in Section 4.3 also uses $\sin$ and $\cos$ of an angle from Chapter 3.
← 3.4 Multiple-Angle, Product and Sum Formulae4.1 Complex Numbers and Their Algebra →