# Complex Number Calculator

URL: https://speedycalculator.com/calculators/math/complex-number/
Last updated: 2026-10-30

> The complex number calculator adds, subtracts, multiplies and divides two complex numbers z₁ = a + bi and z₂ = c + di, and also gives their moduli and arguments, the power z₁^z₂ and the natural logarithm ln z₁.

## Inputs

- z₁ real part (a)
- z₁ imaginary part (b)
- z₂ real part (c)
- z₂ imaginary part (d)

## Results

- z₁ + z₂
- z₁ − z₂
- z₁ × z₂
- z₁ ÷ z₂
- |z₁|
- Argument φ₁
- |z₂|
- Argument φ₂
- z₁^z₂
- ln z₁

## Formula

```
Sum = (a + c) + (b + d)i
Product = (ac − bd) + (ad + bc)i
Quotient = [(ac + bd) + (bc − ad)i] / (c² + d²)
ln z = ln|z| + i·arg z
```

## Explanation

Complex numbers extend the real numbers with i, where i² = −1. They are essential in electrical engineering (impedance), signal processing (Fourier transforms), quantum mechanics and anywhere rotations and oscillations appear.

### How to use the Complex Number Calculator

- **z₁ real part (a)**
- **z₁ imaginary part (b)**
- **z₂ real part (c)**
- **z₂ imaginary part (d)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Addition and subtraction work component by component. Multiplication uses i² = −1, and division multiplies by the conjugate of the denominator. Powers and logarithms use polar form, z = |z|e^{iφ}, with the principal argument between −180° and 180°.

## Worked example

Inputs — z₁ real part (a): 3; z₁ imaginary part (b): 4; z₂ real part (c): 1; z₂ imaginary part (d): -2.

Results — **z₁ + z₂ = 4 + 2i**; z₁ − z₂ = 2 + 6i; z₁ × z₂ = 11 − 2i; z₁ ÷ z₂ = −1 + 2i; |z₁| = 5; Argument φ₁ = 53.13010235° (0.92729522 rad).

## Key terms

- **Imaginary unit i**: A number with i² = −1.
- **Argument**: The angle of a complex number in the complex plane.

## FAQ

**How do I multiply complex numbers?**
Expand like binomials and use i² = −1: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i.

**How do I divide complex numbers?**
Multiply the top and bottom by the conjugate of the bottom so the denominator becomes real.

**What is the argument?**
The angle φ of the point (a, b) measured from the positive real axis.

**Is z₁^z₂ unique?**
No – complex powers have many values; the calculator gives the principal value.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [Wolfram MathWorld – Complex number](https://mathworld.wolfram.com/ComplexNumber.html)