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) = z̄ = (x-yi)
The complex conjugate is often denoted as z̄.
Z = x+yi
z̄ = 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*z̄ = |z|^2
z*z̄ = (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 z̄ has the same modulus as the complex number z.
|z̄| = |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
The IMCONJUGATE function calculates the complex conjugate of a complex number in x + yi or x + yj text format.
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 many functions in the 'Engineering' category.
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