## Purpose

## Return value

## Syntax

`=IMARGUMENT(complex_num)`

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

## How to use

The Excel IMARGUMENT function returns the angle of a complex number measured in radians. For example, given the complex number "3+4i" the function returns the angle 0.927295218.

```
=IMARGUMENT(3,4) // returns 0.927295218
```

Geometrically, this value represents the angle between the positive real axis and the line the complex number lies on in the complex plane.

To convert the angle to degrees, use the DEGREES function.

`=DEGREES(IMARGUMENT(COMPLEX(3,4))) // returns 53.13°`

### Explanation

For any given point *z* in the complex plane, many equivalent angles correspond to the line that *z* lies on. For example, consider the point "-5-5i" in the complex plane. Starting from the positive real axis, you can rotate 225 degrees in the positive direction, or you can also rotate negative 135 degrees to get to *z*.

The IMARGUMENT function deals with this ambiguity by always returning angles in the range from -π to π radians. For the complex number "-5-5i" this means the function returns the angle -3/4 π.

`=IMARGUMENT(COMPLEX(-5, -5)) // returns -3/4π`

For complex numbers that lie on the negative real axis, the function returns π radians.

`=IMARGUMENT(COMPLEX(-5, 0)) // returns π`

In other words, the range excludes -π and includes π. The function still returns angles really close to, but not quite equal to -π.

```
=IMARGUMENT(COMPLEX(-5, -0.01)) // returns -3.131592987
```

### Notes

- Given zero the function returns a #DIV/0! error.