Complex Numbers
Add, subtract, multiply, and divide complex numbers — plus modulus and conjugate, step by step.
z₁ = 3 + 4i, z₂ = 1 + 2i
Add the real parts and the imaginary parts separately.
A complex number has the form a + bi, where i is the imaginary unit and i² = −1. Add or subtract the real and imaginary parts separately; multiply with FOIL. So (3 + 4i) + (1 + 2i) = 4 + 6i, (3 + 4i)(1 + 2i) = −5 + 10i, and the modulus |3 + 4i| = 5.
What a complex number is
A complex number combines a real part and an imaginary part: z = a + bi. The imaginary unit i is defined by the rule i² = −1, which is the one fact that drives every operation below. Real numbers are just complex numbers with b = 0, so everything here generalises ordinary arithmetic.
Addition and subtraction act on the two parts independently. Multiplication uses the distributive (FOIL) rule and then replaces i² with −1, which mixes the parts together.
multiply with FOIL, then substitute i² = −1
Worked example
Multiply z₁ = 3 + 4i by z₂ = 1 + 2i, so a = 3, b = 4, c = 1, d = 2:
- 1 Expand with FOIL. (3 + 4i)(1 + 2i) = 3×1 + 3×2i + 4i×1 + 4i×2i = 3 + 6i + 4i + 8i².
- 2 Replace i² with −1. 8i² = 8×(−1) = −8, so the expression becomes 3 + 6i + 4i − 8.
- 3 Collect real and imaginary parts. Real: 3 − 8 = −5. Imaginary: 6i + 4i = 10i. Result: −5 + 10i. ✓
Complex-number operations
For z₁ = a + bi and z₂ = c + di.
| Operation | Formula |
|---|---|
| Add | (a + c) + (b + d)i |
| Subtract | (a − c) + (b − d)i |
| Multiply | (ac − bd) + (ad + bc)i |
| Divide | ((ac + bd) + (bc − ad)i) ÷ (c² + d²) |
| Modulus |z₁| | √(a² + b²) |
| Conjugate z̄₁ | a − bi |
Conjugates, division, and the modulus
i² = −1 is the rule everything rests on: it is why multiplying two imaginary parts produces a real number, and why the parts mix when you multiply.
To divide, multiply the numerator and denominator by the conjugate of z₂. The conjugate a − bi turns the denominator (c + di)(c − di) into the real number c² + d², so the answer can be written cleanly as a real part plus an imaginary part.
The modulus |z| = √(a² + b²) is the distance of z from the origin when you plot it in the complex plane (real axis horizontal, imaginary axis vertical). For 3 + 4i that distance is √(9 + 16) = √25 = 5 — the hypotenuse of a 3–4–5 right triangle.