The IMPRODUCT function calculates the product of complex numbers 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.

1. IMPRODUCT Function Syntax

IMPRODUCT(inumber1, [inumber2], ...)

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2. IMPRODUCT Function Arguments

inumber1	Required. A complex number in x+yi or x+yj text format.
[inumber2]	Optional. Up to 255 complex numbers to multiply.

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3. IMPRODUCT function example

The image above demonstrates a formula in cell D3 that calculates the product of two complex numbers specified in cells C3 and B3.

Cell B3 contains this complex number "1+2i" and cell C3 contains the following complex number "-1-2i". The IMPRODUCT function in cell E3 calculates the complex product.

Formula in cell C3:

The image above shows a chart displaying the complex plane, the vertical axis is the imaginary axis, and the horizontal axis is the real axis. The light blue line C₁ is the complex number "1+2i" and the green line C₂ is the complex number "-1-2i". The product of C₁ and C₂ is the dark blue line.

3.1 Explaining formula

Step 1 - Populate arguments

IMPRODUCT(inumber, number)

becomes

IMPRODUCT(B3, C3)

Step 2 - Evaluate the IMPRODUCT function

IMPRODUCT(B3, C3)

becomes

IMPRODUCT("1+2i", "-1-2i")

and returns

3-4i

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4. How is the IMPRODUCT function calculated in detail?

A product of two complex numbers is calculated like this:

C₁ = x + yi
C₂ = z + wi

IMPRODUCT(C₁,C₂) = (x+y)(z+w) = (xz - yw) + (xw + yz)i

4.1 Detailed calculation

C₁ = 1 + 2i
C₂ = -1 - 2i

IMPRODUCT(C₁,C₂) = (x+y)(z+w) = (1*(-1) - 2*(-2)) + (1*(-2)+ 2*(-1))i

Step 1 - Multiply first part (xz - yw)

(1*(-1) - 2*(-2))

becomes

(-1-(-4))

Step 2 - Calculate the difference for the first part (xz - yw)

(-1-(-4))

equals 3.

Step 3 - Multiply second part (xw + yz)i

(1*(-2) + (2*(-1))i

becomes

(-2 + -2)i

Step 4 - Calculate the addition for the second part (xw + yz)i

(-2 + -2)i

equals

-4i

Step 5 - Construct complex product - add parts

(xz - yw) + (xw + yz)i

equals

3-4i

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5. Calculate the complex determinant of a 2x2 matrix

The image above shows a complex 2 by 2 matrix in cells F23:G24, they are [[1+2i, 3-4i], [5+6i, 7-8i]]. The formula in cell B37 calculates the complex determinant of a 2 by 2 matrix based on complex numbers.

Formula in cell B37:

The image above demonstrates all four complex values in a chart and their corresponding determinant.

5.1 Explaining the formula in cell B37

a : 1+2i
b : 3-4i
c : 5+6i
d : 7-8i

det(A) = a*d - b*c = (1+2i)*(7-8i) - (3-4i)*(5+6i) = (23+6i) - (39-2i) = -16+8i

Step 1 - Calculate complex product

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

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

IMPRODUCT(B25,B34)

becomes

IMPRODUCT("1+2i","7-8i")

and returns

"23+6i"

Step 2 - Calculate complex product

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

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

IMPRODUCT(B28,B31)

becomes

IMPRODUCT("3-4i","5+6i")

and returns

"39-2i"

Step 3 - Calculate the difference

The IMSUB function calculates the difference between two complex numbers in x + yi or x + yj text format.

Function syntax: IMSUB(inumber1, inumber2)

IMSUB(IMPRODUCT(B25,B34),IMPRODUCT(B28,B31))

becomes

IMSUB("23+6i","39-2i")

and returns

-16+8i

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6. Multiply a complex number by i

Multiplying a complex number by i is equivalent to rotating it by 90 degrees counterclockwise on the complex plane. (x + yi)*i = -y + xi

For example, if you multiply 1 + 2i (green line in the chart above) by i you get -2 + i (blue line in the chart above) which is 90 degrees counterclockwise from 1 + 2i.

Formula in cell B28:

You can also simplify i² = -1 to multiply complex numbers by i. For example, if you multiply (-2 + i) by i you get -2i+i² equals -1-2i.

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Useful links

IMPRODUCT function - Microsoft
Multiplying Complex Numbers - Cuemath
Multiplying complex numbers - Clark university

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Functions in 'Engineering' category

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

Excel function categories

Excel categories

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