IMDIV function

The IMDIV function in Excel divides one complex number by another and returns the result as a complex number.

Syntax

IMDIV(inumber1, inumber2)

Parameters

• inumber1: The complex number to be divided (the numerator). This can be a string in the form a + bi or a + bj, where a is the real part and b is the imaginary part.
• inumber2: The complex number that divides inumber1 (the denominator). This can also be provided as a string in the form a + bi or a + bj.

How It Works

For two complex numbers inumber1 = a + bi and inumber2 = c + di, the division is performed using the formula:

$$\frac{a + bi}{c + di} = \frac{(a + bi) \times (c – di)}{(c + di) \times (c – di)}$$

Which results in:

$$\frac{(ac + bd) + (bc – ad)i}{c^2 + d^2}$$

Examples

1. Dividing Two Complex Numbers: To divide the complex number 5 + 3i by 2 - 1i:

   =IMDIV("5+3i", "2-1i")
   

   The result will be approximately 1.4 + 1.8i.

2. Real Numbers: If both numbers are real (no imaginary parts), the division will work as standard real number division. For example:

   =IMDIV(6, 2)
   

   The result will be 3.

3. Complex Number by a Real Number: If you want to divide a complex number by a real number, such as 6 + 4i by 2:

   =IMDIV("6+4i", 2)
   

   The result will be 3 + 2i.

4. Using a Cell Reference: If cell A1 contains the complex number "4+3i" and cell B1 contains the complex number "1+2i", you can divide them using:

   =IMDIV(A1, B1)
   

   The result will be approximately 2 - 1i.

Important Notes

• The IMDIV function works only with valid complex numbers. If the input is not a valid complex number or results in division by zero, Excel will return a #NUM! or #VALUE! error.
• If the second argument (inumber2) is zero, Excel will return a #DIV/0! error because division by zero is not allowed.
• Complex numbers in Excel can use either i or j to represent the imaginary part.

Summary

The IMDIV function in Excel divides two complex numbers and returns the result as a complex number. It is useful for mathematical and engineering applications involving complex arithmetic.