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Complex Numbers and Quadratic Equations
Introduction
Complex Numbers
Algebra of Complex Numbers
The Modulus and the Conjugate of a Complex Number
Practice: Complex Number Arithmetic
Argand Plane and Polar Representation
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Glossary

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Complex Numbers and Quadratic EquationsComplex Numbers

Reading time: ~15 min

Time to give the missing number a name. Write i for \sqrt{-1}, the imaginary unit. Then i^2 = -1. This one rule powers the whole chapter: wherever i^2 turns up, you may it with -1. It also means that i is a of the equation x^2 + 1 = 0.

Now mix i with ordinary numbers. A number of the form a + ib, where a and b are numbers, is called a complex number. For example, 2 + i3, (-1) + i\sqrt{3} and 4 + i\left(\frac{-1}{11}\right) are complex numbers. Writing i3 or 3i means the same thing.

For z = a + ib, the number a is the real part of z, written Re z, and b is the imaginary part, written Im z. Careful: the imaginary part is the real number b that multiplies i, not the term ib. So for z = 2 + i5, Re z = and Im z = .

One colour code runs through the whole chapter: the real part is blue, the imaginary part green, a complex number drawn as a point purple, and its mirror twin, the conjugate, orange.

Two complex numbers z_1 = a + ib and z_2 = c + id are when a = c and b = d: both parts must match.

Sort these numbers into the three buckets. For each one, ask: is its real part zero? Is its imaginary part zero?

2 + 5i
7
−3i
i²
1 − i
i/2
−4 + 0i
−1 + i√3
Real: Im z = 0
Purely imaginary: Re z = 0
Both parts non-zero

So the real numbers are not left behind: every real number a is the complex number a + i0. A number like 3i = 0 + i3, whose real part is zero, is called purely imaginary.

Tip

One equation between two complex numbers is secretly two real equations: the real parts must match, and the imaginary parts must match. This "compare the parts" move unlocks Example 1.

Example 1 : If 4x + i(3x - y) = 3 + i(-6), where x and y are real numbers, then find the values of x and y.

Solution : We have

4x + i(3x - y) = 3 + i(-6) \qquad \ldots (1)

Equating the real parts of (1), 4x = , so x = .

Equating the imaginary parts of (1), 3x - y = .

Putting x = \frac{3}{4}: \frac{9}{4} - y = -6, so y = \frac{9}{4} + 6 = .

So x = \frac{3}{4} and y = \frac{33}{4}. Check the imaginary part: 3 \times \frac{3}{4} - \frac{33}{4} = \frac{9 - 33}{4} = -6 ✓.

Speed Round: Real Part, Imaginary Part

Everyone falls into this trap once: Im z is the number that multiplies i, never the i-term itself. Spot the parts, whatever order they are written in.

Get five right to move on.

Sina