The IMDIV function calculates the quotient of two complex numbers in x + yi or x + yj text format.

The quotient is the result of dividing one complex number inumber1 by another complex number inumber2. The numerator is inumber1  and the denominator is inumber2.

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

1. IMDIV Function Syntax

IMDIV(inumber1, inumber2)

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2. IMDIV Function Arguments

inumber1	Required. The complex numerator in x+yi or x+yj text format.
inumber2	Required. The complex denominator in x+yi or x+yj text format.

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3. IMDIV Function Example

The image above demonstrates a formula in cell F3 that calculates the quotient of two complex numbers specified in cell C3 and D3 respectively.

The complex number in cell C3 is the numerator and the value in D3 is the denominator. The numerator and denominator are the top and bottom numbers of a fraction.

Formula in cell F3:

The IMDIV function divides one complex number by another complex number, the formula above divides the complex number in cell C3 by the complex number in cell D3.

The chart above shows complex numbers "-2+2i" and "2-4i" on the complex plane, the y-axis is the imaginary axis and the x-axis is the real axis.

3.1 Explaining formula

Step 1 - Populate arguments

IMDIV(inumber1, inumber2)

becomes

IMDIV(C3, D3)

Step 2 - Evaluate the IMDIV function

IMDIV(C3, D3)

becomes

IMDIV("-2+2i","2-4i")

and returns

-0.6-0.2i

Use the rectangular form when you want to perform addition, subtraction, multiplication, and division of complex numbers. Section 5 below demonstrates how to convert complex numbers in polar form to rectangular form.

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4. How to perform division between two complex numbers

This example demonstrates how Excel calculates in detail the quotient between two complex numbers in rectangular form.

z₁ is the light blue line on the chart,  z₂ is green line, and the quotient is the dark blue line.

z₁ = a+ib

z₂ = c+id

IMDIV(z₁,z₂) = (a+ib)/(c+id) = ((ac+bd) + (bc-ad)i)/(c²+d²)

To perform a complex division of two complex numbers we need to divide the dividend and the divisor to calculate the quotient.

z₁ = -2+2i

z₂ = 2-4i

IMDIV(z₁,z₂) = (-2+2i)/(2-4i) = ((-2*2+2*(-4)) + (2*2-(-2)*(-4))i)/(2²+(-4)²) = (-12 + -4i)/20 = -0.6 - 0.2i

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5. How to convert complex numbers from polar form to rectangular form

The polar form has an absolute value or modulus which is the distance from the origin to a +bi.  The θ is the angle of direction.

The following math formula allows us to calculate the complex value in rectangular form using the modulus and the θ:

Z = r(cos θ + isin θ)

This part calculates the real value: r*cos θ and this part calculates the imaginary value: r*i*sin θ

Formula in cell E9 calculates the complex values in rectangular form with Excel functions:

The result is obtained with a two-digit approximation, you can change the argument in the ROUND functions or remove the ROUND functions altogether from the formula to get a more accurate result.

Explaining formula in cell E9

Step 1 - Convert degrees to radians

The RADIANS function converts degrees to radians.

Function syntax: RADIANS(angle)

RADIANS(D9)

becomes

RADIANS(60)

and returns

1.0471975511966

Step 2 - Calculate cosines for θ

The COS function calculates the cosine of an angle.

Function syntax: COS(number)

COS(RADIANS(D9))

becomes

COS(1.0471975511966)

and returns

0.5

Step 3 - Multiply magnitude with cosines for θ

C9*COS(RADIANS(D9))

becomes

5*0.5

equals 2.5

Step 4 - Round to two digits

The ROUND function rounds a number based on the number of digits you specify.

Function syntax: ROUND(number, num_digits)

ROUND(C9*COS(RADIANS(D9)),2)

becomes

ROUND(2.5, 2)

and returns

2.5

Step 5 - Calculate sine for θ

The SIN function calculates the sine of an angle.

Function syntax: SIN(number)

SIN(RADIANS(D9))

becomes

SIN(1.0471975511966)

and returns

0.866025403784439

Step 6 - Multiply magnitude with sine for θ

C9*SIN(RADIANS(D9))

becomes

5*0.866025403784439

and returns

4.33012701892219

Step 7 - Round to two digits

The ROUND function rounds a number based on the number of digits you specify.

Function syntax: ROUND(number, num_digits)

ROUND(C9*SIN(RADIANS(D9)),2)

becomes

ROUND(4.33012701892219,2)

and returns

4.33

Step 8 - Calculate complex numbers

The COMPLEX function returns a complex number based on a real and imaginary number.

Function syntax: COMPLEX(real_num, i_num, [suffix])

COMPLEX(ROUND(C9*COS(RADIANS(D9)),2),ROUND(C9*SIN(RADIANS(D9)),2))

becomes

COMPLEX(2.5,4.33)

and returns

2.5+4.33i

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Useful links

IMDIV function - Microsoft
Complex number
Dividing Complex Numbers

Functions in 'Engineering' category

The IMDIV function function is one of many functions in the 'Engineering' category.

Excel function categories

Excel categories

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