Complex Number Operations
Formulas for adding, multiplying, and converting complex numbers.
Polar form, modulus, and conjugate with worked examples.
Standard Form
A complex number has a real part (a) and an imaginary part (b). The imaginary unit i is defined as the square root of −1.
Addition and Subtraction
Add the real parts together and the imaginary parts together. Subtraction works the same way — subtract each part separately.
Multiplication
Multiply using the distributive property (like FOIL), then replace i² with −1. This is why the real part has a minus sign — the i² term becomes negative.
Modulus (Absolute Value)
The modulus gives the distance from the complex number to the origin in the complex plane. It is always a non-negative real number.
Complex Conjugate
The conjugate flips the sign of the imaginary part. Multiplying a complex number by its conjugate gives a real number: z × z̄ = a² + b².
Variables
| Symbol | Meaning |
|---|---|
| z | A complex number |
| a | Real part of the complex number, Re(z) |
| b | Imaginary part of the complex number, Im(z) |
| i | Imaginary unit, where i² = −1 |
| |z| | Modulus (absolute value) of z |
| z̄ | Complex conjugate of z |
Example 1: Multiplication
Multiply (3 + 2i) × (1 − 4i).
Use the formula: (ac − bd) + (ad + bc)i
a = 3, b = 2, c = 1, d = −4
Real part: (3)(1) − (2)(−4) = 3 + 8 = 11
Imaginary part: (3)(−4) + (2)(1) = −12 + 2 = −10
(3 + 2i)(1 − 4i) = 11 − 10i
Example 2: Division
Divide (5 + 3i) by (2 − i).
Multiply numerator and denominator by the conjugate of the denominator: (2 + i)
Numerator: (5 + 3i)(2 + i) = (10 − 3) + (5 + 6)i = 7 + 11i
Denominator: (2 − i)(2 + i) = 4 + 1 = 5
Divide each part by 5: (7/5) + (11/5)i
(5 + 3i) / (2 − i) = 1.4 + 2.2i
When to Use These
Complex numbers appear throughout mathematics, physics, and engineering.
- Solving quadratic equations with negative discriminants
- Analyzing alternating current (AC) circuits using impedance
- Signal processing and Fourier transforms
- Quantum mechanics wave functions
- Fractal generation (Mandelbrot and Julia sets)