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# A to Z of Excel Functions: The IMCONJUGATE Function

13 July 2020

Welcome back to our regular A to Z of Excel Functions blog. Today we look at the IMCONJUGATE function.

The IMCONJUGATE function An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit (sometimes denoted j) which is defined by its property i2 = −1.  In general, the square of an imaginary number bi is −b2.  For example, 9i is an imaginary number, and its square is −81.  Zero is considered to be both real and imaginary.

An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.  For example, (if a and b are real, then) the complex conjugate of a + bi is a – bi.

The product of a complex number and its conjugate is a real number, a2 + b2

Complex conjugates are important for finding the roots of polynomials.  According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as the quadratic equation or the cubic equation), so must its conjugate.

The IMCONJUGATE function returns the complex conjugate of a complex number in x + yi or x + yj text format.

The IMCONJUGATE function employs the following syntax to operate:

IMCONJUGATE(inumber)

The IMCONJUGATE function has the following argument:

• inumber: this is required and represents the complex number for which you want to calculate the conjugate.

It should be further noted that:

• you should use COMPLEX to convert real and imaginary coefficients into a complex number
• IMCONJUGATE recognises either the i or j notation
• if the complex number ends in +i or -i (or j), i.e. there is no coefficient between the operator and the imaginary unit, there must be no space, otherwise IMCONJUGATE will return an #NUM! error
• the conjugate of a complex number is calculated as Please see my example below: We’ll continue our A to Z of Excel Functions soon. Keep checking back – there’s a new blog post every business day.

A full page of the function articles can be found here.