IMARGUMENT Function

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+4i") // returns 0.927295218

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

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

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

Explanation

Given a complex number, many equivalent angles correspond to the line on which the number lies. For example, consider the complex number "-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 "-5-5i".

The IMARGUMENT function always returns angles in the range from -π to π radians. For the complex number "-5-5i" 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. Given a complex number just below the negative real axis, the function returns an angle 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.