# conj() Function & Examples

Use the conj() complex function in Calcul.io. Review its syntax, edit working examples, understand the result, and explore related math functions.

## conj

#### conj(x)

Try it yourself:

```calculio
conj(2 + 3i)
conj(2 - 3i)
conj(-5.2i)
```

[re](https://calcul.io/function/re/index.md)

[im](https://calcul.io/function/im/index.md)

[abs](https://calcul.io/function/abs/index.md)

[arg](https://calcul.io/function/arg/index.md)

---

## Find a complex conjugate with conj

**conj(x)** returns the complex conjugate of `x`. For `a + bi`, the conjugate is `a - bi`: the real part stays unchanged and the sign of the imaginary part reverses. Real inputs are unchanged.

## Why conjugates are useful

Conjugation is central to complex division, signal processing, and inner products. Multiplying a number by its conjugate gives a real non-negative value: `z * conj(z)` equals the squared magnitude. Use [abs](https://calcul.io/function/abs/index.md) when you want the magnitude itself, [re](https://calcul.io/function/re/index.md) for the real component, and [im](https://calcul.io/function/im/index.md) for the imaginary component.

## Example

For `z = 3 + 4i`, `conj(z)` is `3 - 4i`, and `z * conj(z)` is 25. In a complex vector calculation, conjugation belongs on one vector before taking a dot product. For matrices, use [ctranspose](https://calcul.io/function/ctranspose/index.md) to transpose and conjugate together rather than applying the operations piecemeal.

## Input notes

The function accepts numeric values, including complex numbers; it does not turn arbitrary text into a complex value. Be careful not to confuse conjugation with negation: `conj(3 + 4i)` is not `-3 - 4i`. For polar representations, [arg](https://calcul.io/function/arg/index.md) gives the angle, whose sign is reversed by conjugation.

[All functions](https://calcul.io/functions/index.md)