IMCONJUGATE function

The IMCONJUGATE function in Excel returns the complex conjugate of a given complex number. The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. For a complex number in the form a + bi (or a + bj), its complex conjugate is a - bi (or a - bj).

Syntax

IMCONJUGATE(inumber)

Parameters

• inumber: The complex number for which you want to find the conjugate. The complex number can be entered as a string in the form a + bi or a + bj, where a is the real part and b is the imaginary part.

Examples

1. Basic Example: To find the complex conjugate of the complex number 3 + 4i:

   =IMCONJUGATE("3+4i")
   

   The result will be "3-4i", as the sign of the imaginary part is inverted.

2. Complex Number with Negative Imaginary Part: To find the complex conjugate of the complex number 2 - 5i:

   =IMCONJUGATE("2-5i")
   

   The result will be "2+5i", as the sign of the imaginary part changes from -5i to +5i.

3. Purely Real Number: For a real number like 7 (which has no imaginary part):

   =IMCONJUGATE("7")
   

   The result will be "7", as there is no imaginary part to change.

4. Purely Imaginary Number: To find the complex conjugate of 0 + 6i:

   =IMCONJUGATE("0+6i")
   

   The result will be "0-6i", as the sign of the imaginary part changes.

5. Using a Cell Reference: If cell A1 contains the complex number "4-9i", you can find its conjugate using:

   =IMCONJUGATE(A1)
   

   The result will be "4+9i".

Important Notes

• The function works only with valid complex numbers. If the input is not a valid complex number, Excel will return a #VALUE! error.
• Complex numbers in Excel are typically written as a + bi or a + bj, where a is the real part, and b is the imaginary part. Excel accepts both i and j as the imaginary unit.
• The IMCONJUGATE function is useful in operations involving complex numbers, such as in electrical engineering, physics, and applied mathematics.

Summary

The IMCONJUGATE function returns the complex conjugate of a complex number by changing the sign of its imaginary part. This function is particularly useful in various mathematical and engineering applications that involve complex numbers, where the conjugate is needed for operations like simplifying expressions or calculating magnitudes.