What complex numbers are
A complex number has a real part and an imaginary part: z = a + bi, where i is the imaginary unit, defined by i² = −1. Complex numbers let every polynomial equation have solutions, and they are essential in electrical engineering, signal processing, quantum mechanics and fluid dynamics. This calculator performs the four basic operations and gives the modulus and argument of the result.
The operations
- 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
- Division: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)
- Modulus: |z| = √(a² + b²)
- Argument: the angle of z from the positive real axis, between −180° and 180°
Worked example
Multiply z₁ = 3 + 4i by z₂ = 1 − 2i.
- Real part: 3 × 1 − 4 × (−2) = 3 + 8 = 11.
- Imaginary part: 3 × (−2) + 4 × 1 = −6 + 4 = −2.
- Result: 11 − 2i, with a modulus of √125 = 11.18 and an argument of −10.30°.
- Check: |z₁| × |z₂| = 5 × 2.236 = 11.18. Moduli multiply, and arguments add: 53.13° + (−63.43°) = −10.30°.
Dividing instead gives (3 + 4i) ÷ (1 − 2i) = −1 + 2i.
Polar form
Every complex number can also be written as z = r(cos θ + i sin θ), where r is the modulus and θ the argument. In this form, multiplication multiplies the moduli and adds the arguments, and division divides the moduli and subtracts the arguments. Use the polar coordinates calculator to switch between the two forms.
Powers of i
| Power | Value |
|---|---|
| i¹ | i |
| i² | −1 |
| i³ | −i |
| i⁴ | 1 |
The pattern repeats every four powers, so i¹⁰ = i² = −1.
Where they are used
- Electrical engineering: impedances of resistors, capacitors and inductors combine as complex numbers, which makes AC circuit analysis much simpler. Engineers write j instead of i to avoid confusion with current.
- Equations: x² + 1 = 0 has the solutions i and −i; the quadratic formula gives complex roots whenever the discriminant is negative.
- Signals: the Fourier transform describes sounds and images as sums of complex exponentials.
- Fractals: the Mandelbrot set is drawn by repeatedly squaring complex numbers.
Conjugate
The conjugate of a + bi is a − bi. A number multiplied by its conjugate gives a real number, a² + b², which is the trick used to divide complex numbers.