Chapters

## Equal Complex Numbers

Two Complex numbers and are equal if and only if their Real  and their Imaginary parts are equal

and

Their arguments are also equal

#### Example

with argument and

with argument

We have equality of the Real and Imaginary parts

and

and their arguments are also equal

has traversed an angle larger than but its principle argument is in the 4th Quadrant of the Complex plane.

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## Conjugate Complex Numbers

Two Complex numbers and are conjugates if they have equal Real parts and opposite (negative) Imaginary parts

and with and

and

#### Example

If then

and

We will denote the conjugate of a Complex number as

### The Modulus of a Complex Number and its Conjugate

The modulus of is equal to the modulus of

and

and

Their product is equal to the square of their modulus

and

We saw from the example above that if a Complex number is located in the 1st Quadrant, then its conjugate is located in the 4th Quadrant. The opposite is also true. If a Complex number is located in the 4th Quadrant, then its conjugate lies in the 1st Quadrant.

The same relationship holds for the 2nd and 3rd Quadrants

If then

### The Argument of a Complex Number and its Conjugate

The arguments of a Complex number and its conjugate add up to .

If then because of the symmetry across the due to the Quadrant symmetries of 1 and 4 and 2 and 3

with

The sum of their arguments is .

## Opposite Complex Numbers

Two Complex numbers and are opposite if they have opposite (negative) Real and Imaginary parts

and or

#### Example

then its opposite is

### The Modulus of a Complex Number and its Opposite

The modulus of a Complex number and its opposite are equal in magnitude

#### Example

If then its opposite is

and

Then .

There is also quadrant symmetry between a Complex number and its opposite.

If a Complex number lies in the 1st Quadrant, then its opposite lies in the 3rd Quadrant and if a Complex number lies in the 4th Quadrant, then its opposite lies in the 2nd Quadrant.

#### Example

is in the 1st Quadrant because both and are positive, then lies in the 3rd Quadrant because both and are negative.

#### Example

is in the 2nd Quadrant because is negative and is positive, then its opposite is in the 4th Quadrant because is positive and is negative.

### The Argument of a Complex Number and its Opposite

The arguments of a Complex number and its opposite differ by a value of because the numbers lie along a straight line in the plane due to the Quadrant symmetries between 1 and 3 and between 2 and 4, and a straight line has a measure of , which is equivalent to radians.

## Reciprocal Complex Numbers

The reciprocal of a Complex number is with .

The reciprocal of a Complex number is its Multiplicative Inverse because .

#### Example

If then

We must rationalize the denominator to make the reciprocal into a form that we can Mathematically manipulate because having a Complex number in the denominator

is the conjugate of divided by the square of the modulus. It's just a scaled version of the conjugate of the original Complex number.

Multiplying a Complex number by its conjugate divided by the square of the modulus will yield because the product of the Complex number and its conjugate is just the square of the modulus

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