Complex numbers in a nutshell
A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1. You can picture it as the point (a, b) on an Argand diagram: real numbers run across, imaginary numbers run up. The same point can be described by its distance from 0, the modulus r = |z|, and its angle from the positive real axis, the argument θ. Complex numbers turn up in AC circuits, signal processing, quantum mechanics and in solving polynomial equations.
How each operation works
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
(a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)
|z| = √(a² + b²), θ = atan2(b, a), z = r(cos θ + i sin θ) = re^(iθ)
De Moivre: zⁿ = rⁿ(cos nθ + i sin nθ)
nth roots: r^(1/n)[cos((θ + 2πk) ÷ n) + i sin((θ + 2πk) ÷ n)], k = 0, 1, …, n − 1
To divide, multiply the top and bottom by the conjugate of the bottom, c − di. That turns the bottom into the real number c² + d². The argument used here is the principal value, between −180° and 180°, so a number on the negative real axis has argument 180°.
Worked example
Take z₁ = 3 + 4i and z₂ = 1 − 2i.
- Multiply: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 3 − 2i + 8 = 11 − 2i.
- Divide: multiply top and bottom by 1 + 2i. The top is (3 + 4i)(1 + 2i) = −5 + 10i, and the bottom is 1² + 2² = 5, so the answer is -1 + 2i.
- Modulus and argument: |3 + 4i| = √(9 + 16) = 5, and θ = atan2(4, 3) ≈ 53.1301°, so 3 + 4i = 5(cos 53.13° + i sin 53.13°).
- Power: 1 + i has r = √2 and θ = 45°, so (1 + i)⁸ = (√2)⁸(cos 360° + i sin 360°) = 16.
- Roots: −8 has r = 8 and θ = 180°, so its cube roots have modulus 2 and angles 60°, 180° and 300°: 1 + √3 i, −2 and 1 − √3 i.
Powers of i
| i¹ | i² | i³ | i⁴ | i⁵ |
|---|---|---|---|---|
| i | −1 | −i | 1 | i |
The pattern repeats every 4, so iⁿ depends only on the remainder of n ÷ 4.
Limitations
- Answers are decimals rounded to 10 significant figures. Exact surds like √3 are shown as 1.732050808.
- Powers must be whole numbers. A non-integer power of a complex number has many values, and this page doesn't pick one for you.
- Angles follow the principal value convention. Some textbooks use 0° to 360° instead, which just adds 360° to negative angles.