## How to use the IMPRODUCT function

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.

### Table of Contents

## 1. IMPRODUCT Function Syntax

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

## 2. IMPRODUCT Function Arguments

inumber1Â |
Required. A complex number in x+yi or x+yj text format. |

[inumber2] |
Optional. Up to 255 complex numbers to multiply. |

## 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 C_{1} is the complex number "1+2i" and the green line C_{2} is the complex number "-1-2i". The product of C_{1} and C_{2} 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

## 4. How is the IMPRODUCT function calculated in detail?

A product of two complex numbers is calculated like this:

C_{1} = x + yi

C_{2} = z + wi

IMPRODUCT(C_{1},C_{2}) = (x+y)(z+w) = (xz - yw) + (xw + yz)i

### 4.1 Detailed calculation

C_{1} = 1 + 2i

C_{2} = -1 - 2i

IMPRODUCT(C_{1},C_{2}) = (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

## 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

## 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 i^{2} = -1 to multiply complex numbers by i. For example, if you multiply (-2 + i) by i you get -2i+i^{2} equals -1-2i.

### Useful links

IMPRODUCT function - Microsoft

Multiplying Complex Numbers - Cuemath

Multiplying complex numbers - Clark university

### Functions in 'Engineering' category

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

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