Maths
Complex numbers — quick study summary
A-Level Further MathsAP PrecalculusIB Maths HL
A complex number has a real part and an imaginary part: z = a + bi, where i² = −1. They extend the real numbers to solve equations like x² = −1. Complex arithmetic works like normal algebra with i² replaced by −1. Modulus |z| = √(a² + b²); argument arg(z) is the angle from the positive real axis. Euler's formula: e^(iθ) = cosθ + i·sinθ, leading to the famous e^(iπ) + 1 = 0.
Key points
- i² = −1; z = a + bi where a, b are real
- Conjugate of a + bi is a − bi; |z|² = z·z̄
- Modulus |z| = √(a² + b²)
- Polar form: z = r(cosθ + i sinθ) = r·e^(iθ)
- Multiplication in polar: moduli multiply, arguments add
Practice quiz
Click each question to reveal the answer.
1. What is (2 + 3i) + (4 − i)?
- 6 + 2i
- 6 + 4i
- 2 + 3i
- 8 − 3i
Answer: 6 + 2i
Add real and imaginary parts separately: (2+4) + (3i−i) = 6 + 2i.
2. What is the modulus of 3 + 4i?
Answer: 5
|z| = √(3² + 4²) = √25 = 5.
3. What does Euler's formula state?
Answer: e^(iθ) = cosθ + i sinθ
It links exponential and trigonometric functions via complex numbers. Setting θ = π gives the famous e^(iπ) + 1 = 0.
Last reviewed: May 2026