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.
Purpose
Return value
Syntax
=IMCONJUGATE(inumber)
- inumber - The complex number in the form “x+yi”.
Using the IMCONJUGATE function
The Excel IMCONJUGATE function returns the 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"
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
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 “x + yi”, its conjugate equals “x - yi”. Typically, the conjugate appears in text with a horizontal bar over the complex number.
The conjugate is used 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”.
We can convert this expression into a multiplication problem by multiplying the numerator and numerator by the conjugate of “a+bi”.
In other words, the divisor is converted into a real number which we know how to divide by. This is equal to the following formula in Excel.
=IMPRODUCT(
"x+yi",
IMCONJUGATE("a+bi"),
COMPLEX(1/IMREAL(IMPRODUCT("a+bi", IMCONJUGATE("a+bi"))), 0)
)
In practice, Excel provides the IMDIV function to perform complex division.
=IMDIV(COMPLEX(-11,29),COMPLEX(2,3)) // returns 5+7i
The conjugate is still useful to know because, aside from being the key to defining complex division, it also appears in other contexts in math, like factoring and solving polynomials.
