# How to use the IMCONJUGATE function

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

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

### Table of Contents

- IMCONJUGATE Function Syntax
- IMCONJUGATE Function arguments
- IMCONJUGATE function example
- How is the complex conjugate calculated?
- How to calculate the product of a complex number and its complex conjugate
- How to calculate the modulus of a complex conjugate
- How to calculate the conjugate of a conjugate complex number
- Get Excel *.xlsx file

## 1. IMCONJUGATE Function Syntax

IMCONJUGATE(*inumber*)

## 2. IMCONJUGATE Function Arguments

inumber |
Required. A complex number in x+yi or x+yj text format. |

## 3. IMCONJUGATE function Example

The image above demonstrates a formula in cell D3 that calculates the complex conjugate of a complex number specified in cell C3. The chart shows the complex number 3+4i on the complex plane, it also shows the complex conjugate of 3+4i which is 3-4i.

Formula in cell D3:

### 3.1 Explaining formula

#### Step 1 - Populate arguments

IMCONJUGATE(*inumber*)

becomes

IMCONJUGATE(C3)

#### Step 2 - Evaluate IMCONJUGATE function

IMCONJUGATE(C3)

becomes

IMCONJUGATE("3+4i")

and returns

"3-4i".

## 4. How is the complex conjugate calculated?

The complex conjugate is calculated by changing the sign of the imaginary value of a complex number. The real part and the imaginary part are equal in magnitude however the imaginary part is opposite in sign.

IMCONJUGATE(x+yi) = zÌ„ = (x-yi)

The complex conjugate is often denoted as zÌ„.

Z = x+yi

zÌ„ = x-yi

## 5. How to calculate the product of a complex number and its complex conjugate

The product of a complex number and its conjugate is a real number.

z*zÌ„ = |z|^2

z*zÌ„ = (3+4i)*(3-4i) = 9-12i+12i+16 = 25

|z|^2 = 5^2 = 25

Formula in cell D3:

The image above shows a chart that has a complex number, its complex conjugate and the product plotted on a complex plane. The light blue line is the complex number, the green line is its complex conjugate and the orange line is the product of those two complex numbers.

### 5.1 Explaining formula in cell D3

#### Step 1 - Calculate the complex conjugate

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

Function syntax: IMCONJUGATE(inumber)

IMCONJUGATE(B25)

becomes

IMCONJUGATE("3+4i")

and returns

"3-4i"

#### Step 2 - Calculate the product of a complex number and its complex conjugate

The IMPRODUCT function calculates the product of complex numbers in x + yi or x + yj text format.

Function syntax: IMPRODUCT(inumber1, [inumber2], ...)

IMPRODUCT(B25,IMCONJUGATE(B25))

becomes

IMPRODUCT("3+4i","3-4i")

and returns 25.

## 6. How to calculate the modulus of a complex conjugate

The conjugate of a complex number zÌ„ has the same modulus as the complex number z.

|zÌ„| = |z|

The image above shows the modulus as a grey dashed circle on the chart, both the complex number and its complex conjugate has the same modulus.

Formula in cell C24:

IMABS(B24)

Formula in cell B25:

Formula in cell C25:

The formulas above also show that the modulus of a complex number is the same as the modulus of its complex conjugate. Cell C24 contains "3+4i", the complex conjugate is "3-4i" displayed in cell B25. The modulus is 5 for both the complex number and its complex conjugate.

## 7. How to calculate the conjugate of a conjugate complex number

The conjugate of a conjugate of a complex number is the original complex number,

Formula in cell D3:

The above formula demonstrates that it is the original complex number. Cell C3 contains the original complex number, cell D3 contains a formula that calculates the conjugate of a conjugate of the complex number in cell C3. The result is the exact same complex number.

### Explaining formula

#### Step 1 - Calculate the complex conjugate

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

Function syntax: IMCONJUGATE(inumber)

IMCONJUGATE(C3)

becomes

IMCONJUGATE("3+4i")

and returns

"3-4i"

#### Step 2 - Calculate the complex conjugate

Function syntax: IMCONJUGATE(inumber)

IMCONJUGATE(IMCONJUGATE(C3))

becomes

IMCONJUGATE("3-4i")

and returns "3+4i"

### Useful links

IMCONJUGATE function - Microsoft

Complex Conjugate

Complex conjugate - wikipedia

### Functions in 'Engineering' category

The IMCONJUGATE function function is one of 42 functions in the 'Engineering' category.

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