Why Complex Numbers Show up in Real Life
Ever wondered why engineers talk about "imaginary" numbers when designing bridges or why video games use them for rotations? Those "imaginary" tricks are just complex numbers doing their magic.
Complex numbers let us handle equations like x^2+1=0 that have no ordinary (real) answer. Think of them as a two‑dimensional number line: one axis for the usual numbers, another for the mysterious "i" that stands for √‑1.
What is a Complex Number?
A complex number is any expression written as a + bi. Here a is called the real part (it lives on the ordinary number line) and b is the imaginary part (it lives on the new direction that we call the imaginary axis). The letter i is defined as the square root of –1, so i² = –1.
Standard Form (a + bi)
Whenever you see a number like 3 – 4i, 0.5 + 2i, or –7i, it is already in standard form. The real part may be zero (as in –5i) and the imaginary part may be zero (as in 6), which simply means the number lives on one of the axes.
Visualising on the Argand Plane
Imagine a city map where the east‑west street is the real axis and the north‑south street is the imaginary axis. The point (a, b) tells you how far to go east (or west) and then north (or south). That point represents the complex number a + bi.
Basic Operations
Just like you add or subtract ordinary numbers, you can do the same with complex numbers. The trick is to treat the real and imaginary parts separately.
- Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
- Subtraction: (a + bi) – (c + di) = (a – c) + (b – d)i
- Multiplication: (a + bi)(c + di) = (ac – bd) + (ad + bc)i (remember i² = –1)
- Division: Divide by multiplying numerator and denominator by the conjugate of the denominator (see below).
Worked Example – Multiplication
Find (3 + 2i)(1 – 4i).
Step 1: Multiply as if i were a variable.
(3)(1) + (3)(‑4i) + (2i)(1) + (2i)(‑4i) = 3 – 12i + 2i – 8i².
Step 2: Replace i² with –1.
‑8i² = ‑8(‑1) = 8.
Now combine real and imaginary parts: (3 + 8) + (‑12i + 2i) = 11 – 10i.
Conjugate and Modulus
The conjugate of a + bi is a – bi. It flips the sign of the imaginary part. Conjugates are handy for division because (a + bi)(a – bi) = a² + b², a pure real number.
The modulus (or absolute value) of a + bi, written |a + bi|, is the distance from the origin on the Argand plane. Use the Pythagorean theorem: |a + bi| = √(a² + b²).
Worked Example – Division
Divide (5 + i) by (2 – 3i).
Multiply numerator and denominator by the conjugate of the denominator (2 + 3i):
[(5 + i)(2 + 3i)] / [(2 – 3i)(2 + 3i)]
Numerator: 5·2 + 5·3i + i·2 + i·3i = 10 + 15i + 2i + 3i² = 10 + 17i – 3 = 7 + 17i.
Denominator: 2² + 3² = 4 + 9 = 13.
Result: (7/13) + (17/13)i.
Polar Form – Turning the Plane into a Circle
Sometimes it’s easier to think of a complex number as a length (radius) and an angle, just like giving directions on a compass. The length is the modulus r, and the angle (called the argument) is θ.
From rectangular a + bi to polar:
- r = √(a² + b²) (the distance from origin)
- θ = tan⁻¹(b / a) (the angle measured from the positive real axis)
Then we write the number as r(cos θ + i sin θ), or using Euler’s formula, re^{iθ} (where e is the natural exponential constant).
Worked Example – Polar Conversion
Convert 1 + √3 i to polar form.
r = √(1² + (√3)²) = √(1 + 3) = 2.
θ = tan⁻¹(√3 / 1) = tan⁻¹(√3) = 60° (or π/3 radians).
Polar form: 2(cos 60° + i sin 60°) or 2e^{iπ/3}.
Quick Summary Table
| Operation | Rectangular Result | Polar Shortcut |
|---|---|---|
| Add / Subtract | Combine real parts and imaginary parts separately | Not convenient; stay rectangular |
| Multiply | (ac‑bd) + (ad+bc)i | Multiply moduli, add arguments: r₁r₂∠(θ₁+θ₂) |
| Divide | Use conjugate of denominator | Divide moduli, subtract arguments: (r₁/r₂)∠(θ₁‑θ₂) |
📝 Likely Exam Questions
- Find the modulus and argument of 4 – 3i.
Modulus = √(4²+(-3)²)=5. Argument = tan⁻¹((-3)/4)= -36.87° (or 323.13°). - Simplify (2 + i)(2 – i).
Result = (2)² – (i)² = 4 – (‑1) = 5. - Write (1 + i)⁴ in a + bi form.
First convert to polar: r=√2, θ=45°. Raising to 4th power: r⁴=4, θ·4=180°. So result = 4(cos 180° + i sin 180°)= -4. - Divide (7 – 5i) by (1 + 2i) and give the answer in a + bi form.
Multiply numerator and denominator by conjugate (1‑2i): [(7‑5i)(1‑2i)]/(1²+2²)= (7‑14i‑5i+10i²)/5 = (7‑19i‑10)/5 = (-3‑19i)/5 = -3/5 – (19/5)i. - Express 3(cos 30° + i sin 30°) in rectangular form.
cos30°=√3/2, sin30°=1/2. So 3(√3/2 + i·1/2)= (3√3/2) + (3/2)i.