February 28, 2020

Chapters

## Introduction

Division of Complex numbers is an undefined process by itself. There is no way to properly 'divide' a Complex number by another Complex number.

This means that if there is a Complex number that is a fraction that has something other than a pure Real number in the denominator, i.e. an Imaginary number or a Complex number, then we must *convert* that number into an equivalent fraction that we will be able to Mathematically manipulate.

When a Complex number appears in the denominator, we must * rationalize the denominator* to form an equivalent number that has a Complex number in the numerator only and a Real number in the denominator only.

We rationalize the denominator by multiplying both the top and the bottom of the fraction by the conjugate of the Complex number in the denominator, which is just equivalent to multiplication by

we must multiply it by

then

#### Example

If then the conjugate of the denominator is and

We turned an undefined operation of division problem into an equivalent multiplication problem by using the conjugate. Multiplying both the numerator and the denominator by the conjugate of the denominator makes the denominator the square of its modulus and the numerator a new Complex number. The new Complex number with just a Real number in the denominator is equivalent to the old Complex number because we only multiplied by an expression that is equivalent to .

#### Example

then

#### Example

then

## The Reciprocal of a Complex Number

If we have a certain Complex number , we can find the value of its reciprocal by rationalizing the denominator in the same way.

#### Example

If with and

This shows that

meaning the reciprocal of a Complex number is equal to the conjugate of that Complex number divided by the square of the modulus. The inverse is a scaled version of the conjugate.

## Visualizing the Division of Complex Numbers

The product of the division of a Complex numbers by another is a new, scaled Complex number that has a new angle . The numerator is multiplied by and the reciprocal of the squared modulus of the denominator , .

We subtract the angle during division because we are actually multiplying by the reflection of the denominator, which has an angle .

#### Example

and then

is a scaled by version of the regular product of

## Modulus and Argument

#### Example

and then

Dividing by subtracts an angle of because has an angle of .

This shows that

The new Complex number is located on the positive *x-axis* a distance of away from the origin.

This shows that

#### Example

and then

Dividing by adds an angle of because has an angle of . The new Complex number is located on the positive *y-axis* a distance of away from the origin.