# 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 **i **(sometimes denoted **j**) which is defined by its property **i ^{2}** = −1. In general, the square of an imaginary number

**bi**is

**−b**. For example, 9

^{2}**i**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 **rea**l 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, **a ^{2} + b^{2}**.

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. *