# Complex Conjugate Calculator

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

> The complex conjugate calculator flips the sign of the imaginary part of a + bi to give a − bi, and shows the modulus |z| and the real product z · z̄ = a² + b².

## Inputs

- Real part (a)
- Imaginary part (b)

## Results

- Conjugate z̄
- Absolute value |z|
- z · z̄
- Argument of z̄

## Formula

```
z̄ = a − bi
|z| = √(a² + b²)
z · z̄ = a² + b²
```

## Explanation

The conjugate reflects a complex number in the real axis. Multiplying a number by its conjugate gives a real number, which is how division by complex numbers is done: multiply the top and bottom by the conjugate of the denominator.

### How to use the Complex Conjugate Calculator

- **Real part (a)**
- **Imaginary part (b)**

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

### How it works

The real part stays the same and the imaginary part changes sign. The modulus is the distance from the origin; the argument of the conjugate is the negative of the original argument.

## Worked example

Inputs — Real part (a): 3; Imaginary part (b): 4.

Results — **Conjugate z̄ = 3 − 4i**; Absolute value |z| = 5; z · z̄ = 25; Argument of z̄ = −53.13010235°.

## Key terms

- **Complex conjugate**: a − bi for z = a + bi.
- **Modulus**: The distance |z| of z from 0.

## FAQ

**What is the conjugate of 3 + 4i?**
3 − 4i.

**What is the conjugate of a real number?**
The number itself, because its imaginary part is 0.

**Why multiply by the conjugate?**
z · z̄ = a² + b² is real, so it removes i from a denominator.

**Do roots of real polynomials come in conjugate pairs?**
Yes – non-real roots of a polynomial with real coefficients always come in conjugate pairs.


## Sources

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