Historical Context & Motivation
For centuries, mathematicians were perfectly comfortable with whole numbers, fractions, and even irrational numbers like √2. But a persistent problem haunted algebra: certain equations, such as x² + 1 = 0, simply had no solution among real numbers. No matter what real number you square, the result is always zero or positive — never −1. Rather than accept this dead end, mathematicians invented an entirely new kind of number to break through the barrier.
The story of complex numbers is one of gradual acceptance. Early algebraists encountered square roots of negative numbers while solving cubic equations, treated them with suspicion, and even called them "imaginary." Over several centuries, however, these numbers proved so useful that they became an essential part of modern mathematics, physics, and engineering.
The central question this lesson addresses is straightforward: How do we define, represent, and compute with numbers of the form a + bi? Understanding this opens the door to solving every quadratic equation, visualising numbers in two dimensions, and connecting algebra with geometry in powerful new ways.
Core Principles & Definitions
A complex number is any number that can be written in the form z = a + bi, where a and b are real numbers and i is defined by the property i² = −1. The value a is called the real part, written Re(z), and b is called the imaginary part, written Im(z). Notice that the imaginary part itself is a real number — it is the coefficient of i, not the entire term bi.
The Imaginary Unit i
Real & Imaginary Parts
Complex Conjugate
Equality of Complex Numbers
The Modulus |z|
The Argand Diagram
The most powerful way to understand complex numbers is to visualise them on the Argand diagram (also called the complex plane). The horizontal axis represents the real part and the vertical axis represents the imaginary part. Every complex number z = a + bi corresponds to the point (a, b), and every point on the plane corresponds to a complex number.
Notice how the conjugate z* is simply a mirror image of z across the real axis. This geometric insight makes many algebraic properties of conjugates feel natural. For example, the product z × z* = (3 + 2i)(3 − 2i) = 9 + 4 = 13, which equals |z|² — a result that always holds and is the basis for dividing complex numbers.
Arithmetic of Complex Numbers
The arithmetic of complex numbers follows from the standard rules of algebra combined with the single new rule that i² = −1. Let z₁ = a + bi and z₂ = c + di be two complex numbers. The four basic operations are defined below.
One particularly useful identity connects the modulus and the conjugate: z × z* = |z|². This means that if z = a + bi, then (a + bi)(a − bi) = a² + b², which is always a non-negative real number. This identity is the engine behind rationalising complex denominators and appears frequently in IB exam questions.
Powers of i & Classification of Complex Numbers
The powers of i follow a repeating cycle of length 4. Understanding this cycle is essential for simplifying expressions like i⁴⁷ without tedious multiplication. The pattern is: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then it repeats. To find iⁿ for any positive integer n, divide n by 4 and look at the remainder.
Types of Complex Numbers
| Type | Form | Example | Notes |
|---|---|---|---|
| Real number | a + 0i | 5, −3, π | The imaginary part is zero. Every real number is a complex number. |
| Purely imaginary | 0 + bi (b ≠ 0) | 4i, −7i | The real part is zero. These numbers lie on the imaginary axis. |
| Complex (general) | a + bi (b ≠ 0) | 3 + 2i, −1 − 4i | Both real and imaginary parts are non-zero. |
| Conjugate pair | a ± bi | 3 + 2i and 3 − 2i | Always appear in pairs as roots of quadratics with real coefficients. |
Worked Example — Division of Complex Numbers
Division is the most challenging of the four operations, so let's walk through a complete example. We will express the quotient (5 + 3i) ÷ (2 − i) in the form a + bi.
Comparing Real and Complex Number Systems
Complex numbers extend the real number system, but the extension comes with trade-offs. Some familiar properties are preserved, while others change. The table below highlights the most important comparisons for IB students.
| Property | Real Numbers ℝ | Complex Numbers ℂ |
|---|---|---|
| Ordering | Can always say a < b or a > b. Numbers have a natural order on the number line. | No ordering. You cannot say 3 + 2i < 1 + 5i — it's meaningless. Complex numbers live on a plane, not a line. |
| Square roots of negatives | Not possible. √(−4) is undefined in ℝ. | Always possible. √(−4) = 2i. Every negative number has two complex square roots. |
| Polynomial roots | A degree-n polynomial may have fewer than n real roots. x² + 1 = 0 has none. | Every degree-n polynomial has exactly n roots (counted with multiplicity) — the Fundamental Theorem of Algebra. |
| Addition & Multiplication | Commutative, associative, distributive — all standard. | Same properties hold: commutative, associative, and distributive. Arithmetic is fully consistent. |
| Geometric representation | Points on a one-dimensional number line. | Points on a two-dimensional plane (the Argand diagram). Distance from origin is the modulus. |
Connection to Polar Form & Advanced Topics
The rectangular form a + bi is just one way to represent a complex number. In later topics (AHL 1.13–1.14), you will encounter the polar form and Euler form, which express a complex number using its modulus and argument (angle). These alternative representations make multiplication, division, and finding powers dramatically easier.
| Feature | Rectangular: a + bi | Polar: r(cos θ + i sin θ) |
|---|---|---|
| Addition / Subtraction | Very easy — add real and imaginary parts directly. | Difficult — no simple rule for adding moduli and arguments. |
| Multiplication | Requires FOIL expansion and simplification. | Elegant — multiply moduli, add arguments: r₁r₂ cis(θ₁ + θ₂). |
| Powers (De Moivre) | Tedious for large exponents. | Straightforward: zⁿ = rⁿ cis(nθ). |
| Best used for | Addition, subtraction, equating real/imaginary parts. | Multiplication, division, powers, roots. |
The rectangular form you've learned in this lesson is the essential foundation for all of these advanced techniques. Mastering addition, subtraction, multiplication, division, and the concept of conjugates ensures you're ready for the polar and Euler representations. Complex numbers also appear in later IB topics including solving polynomial equations, transformations in the complex plane, and applications in electrical engineering and signal processing.
Practice Problems
Lesson Summary
A complex number has the form z = a + bi, where a is the real part, b is the imaginary part, and i² = −1. Complex numbers are added and subtracted by combining real and imaginary parts separately. Multiplication uses FOIL expansion followed by replacing i² with −1, and division is performed by multiplying numerator and denominator by the complex conjugate of the denominator.
Geometrically, complex numbers are plotted on the Argand diagram, where the horizontal axis is the real axis and the vertical axis is the imaginary axis. The modulus |z| = √(a² + b²) gives the distance from z to the origin, and the conjugate z* = a − bi is the reflection across the real axis. The powers of i cycle with period 4: i, −1, −i, 1. These foundational ideas prepare you for polar form, De Moivre's theorem, and the Fundamental Theorem of Algebra — the guarantee that every polynomial of degree n has exactly n complex roots.