Complex Number Calculator
Arithmetic, polar form, modulus, argument, powers & more — with step-by-step working
Arithmetic Operations
Enter two complex numbers and choose an operation.
Operation
Quick Examples
Step-by-step
What is a Complex Number?
A complex number is a number of the form z = a + bi, where a and b are real numbers and i is the imaginary unit defined by i² = −1. The value a is called the real part (Re(z)) and b is called the imaginary part (Im(z)). Every real number is a complex number with b = 0, and every purely imaginary number has a = 0.
Geometrically, a complex number corresponds to a point (a, b) in the Argand plane (complex plane), where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This geometric interpretation makes concepts like modulus and argument intuitive.
Complex Number Operations
Addition and Subtraction
(a + bi) − (c + di) = (a−c) + (b−d)i
Add or subtract the real parts together and the imaginary parts together, independently.
Multiplication (FOIL Method)
= (ac − bd) + (ad + bc)i [since i² = −1]
Expand using the distributive property, then substitute i² = −1 to simplify.
Division (Conjugate Method)
Multiply numerator and denominator by the conjugate of the denominator to make the denominator real.
Polar Form of Complex Numbers
Any complex number z = a + bi can be written in polar form as z = r∠θ = r(cosθ + i sinθ), where:
- r = |z| = √(a²+b²) is the modulus (distance from origin)
- θ = arg(z) = atan2(b, a) is the argument (angle from positive real axis)
The conversion back to rectangular form is: a = r cosθ, b = r sinθ. Polar form is particularly powerful for multiplication (multiply moduli, add arguments) and for applying De Moivre’s theorem.
Euler’s Formula
Euler’s formula states: eⁱ𝛉 = cosθ + i sinθ. This means any complex number can be written as z = reⁱ𝛉. The most famous special case is Euler’s identity: eⁱπ + 1 = 0, connecting the five fundamental constants e, i, π, 1 and 0. Euler’s formula makes the exponential of a complex number easy to compute: e^(a+bi) = e^a × (cos b + i sin b).
Common Complex Number Values
| Expression | Rectangular | Modulus | Argument | Polar |
|---|---|---|---|---|
| i | 0 + 1i | 1 | 90° | 1∠90° |
| −i | 0 − 1i | 1 | −90° | 1∠−90° |
| −1 | −1 + 0i | 1 | 180° | 1∠180° |
| 1+i | 1 + 1i | √2 ≈ 1.4142 | 45° | √2∠45° |
| 3+4i | 3 + 4i | 5 | ≈53.13° | 5∠53.13° |
| eⁱπ | −1 + 0i | 1 | 180° | 1∠180° |
Applications of Complex Numbers
- AC Circuit Analysis: Impedance in electrical engineering is expressed as a complex number Z = R + jX, where R is resistance and X is reactance. Complex arithmetic simplifies the analysis of alternating current circuits.
- Fourier Transform: The Fourier transform uses complex exponentials eⁱ𝛉 to decompose signals into frequency components. Complex numbers are fundamental to signal processing and communications.
- Quantum Mechanics: Wavefunctions in quantum mechanics are complex-valued. The probability amplitude for a particle is a complex number whose squared modulus gives the probability density.
- Fractals (Mandelbrot Set): The famous Mandelbrot set is defined by iterating z → z² + c in the complex plane and checking whether the sequence remains bounded. Complex iteration generates stunning fractal patterns.
- Control Theory: The poles and zeros of transfer functions in control engineering are complex numbers. Their positions in the complex plane determine system stability.