## Purpose

## Return value

## Syntax

`=IMCONJUGATE(complex_num)`

*complex_num*- The complex number in the form "x+yi".

## How to use

The Excel IMCONJUGATE function returns the complex conjugate of a complex number. For example, given the complex number "3+4i" as input, the function returns "3-4i" as output.

```
=IMCONJUGATE("3+4i") // returns "3-4i"
```

### Explanation

The conjugate of a complex number has the same real part and flips the sign of the imaginary part. If a complex number is written as "z = x + yi", its conjugate equals "x - yi". Typically, the conjugate appears in text with a horizontal bar over the complex number.

The conjugate can be used to define what it means to divide a complex number by another. For example, let's say you want to divide the complex number "x+yi" by another complex number "a+bi".

This expression can be converted into a multiplication problem, which can be more easily interpreted using the defining property of complex numbers.

In other words, the divisor is converted into a real number, and we already know how to divide by a real number. This is equal to the following formula in Excel, which divides "x+yi" by "a+bi".

```
=IMPRODUCT(
"x+yi",
IMCONJUGATE("a+bi"),
COMPLEX(1/IMREAL(IMPRODUCT("a+bi", IMCONJUGATE("a+bi"))), 0)
)
```

In practice, Excel provides the IMDIV function, which is a simpler way to divide two complex numbers.

`=IMDIV(COMPLEX(-11,29),COMPLEX(2,3)) // returns 5+7i`

The conjugate is still useful because it allows one to divide two complex numbers by hand.

### Notes

- In math, the complex conjugates help with factoring and solving polynomials.