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"

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.