Purpose
Return value
Syntax
=IMAGINARY(inumber)
- inumber - The string representing a complex number.
Using the IMAGINARY function
The Excel IMAGINARY function returns the imaginary part of a complex number. For example:
=IMAGINARY("4+3i") // returns 3
Excel handles complex numbers as strings formatted like “x+yi” or “x+yj”. Use the COMPLEX function to get the string representing a complex number.
Explanation
A complex number is drawn as an arrow in the complex plane, where the horizontal axis corresponds to the real part of the number and the vertical axis corresponds to the imaginary part of the number. For example, the complex number “2-5i” is drawn below.
The IMAGINARY function returns the vertical component of the complex number.
=IMAGINARY("2-5i") // returns -5
Notes:
- IMAGINARY returns a #NUM error when the input does not represent a valid complex number.
Images courtesy of wumbo.net .
Purpose
Return value
Syntax
=IMARGUMENT(inumber)
- inumber - The complex number in the form “x+yi”.
Using the 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
Excel handles complex numbers as strings formatted like “x+yi” or “x+yj”. Use the COMPLEX function to get the string representing a complex number.
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.
