Chapters

Equal Complex Numbers
Conjugate Complex Numbers
Opposite Complex Numbers
Reciprocal Complex Numbers

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Equal Complex Numbers

Two Complex numbers $\text{[math]}$ and $\text{[math]}$ are equal if and only if their Real and their Imaginary parts are equal

$\text{[math]}$ and $\text{[math]}$

Their arguments are also equal

$\text{[math]}$

Example

$\text{[math]}$ with $\text{[math]}$ and

$\text{[math]}$ with $\text{[math]}$

We have equality of the Real and Imaginary parts

$\text{[math]}$ and $\text{[math]}$

and their arguments are also equal

$\text{[math]}$

$\text{[math]}$ has traversed an angle larger than $\text{[math]}$ but its principle argument is $\text{[math]}$ in the 4th Quadrant of the Complex plane.

Conjugate Complex Numbers

Two Complex numbers $\text{[math]}$ and $\text{[math]}$ are conjugates if they have equal Real parts and opposite (negative) Imaginary parts

$\text{[math]}$ and $\text{[math]}$ with $\text{[math]}$ and $\text{[math]}$

$\text{[math]}$ and $\text{[math]}$

Example

If $\text{[math]}$ then $\text{[math]}$

$\text{[math]}$ and $\text{[math]}$

We will denote the conjugate of a Complex number $\text{[math]}$ as $\text{[math]}$

The Modulus of a Complex Number and its Conjugate

The modulus of $\text{[math]}$ is equal to the modulus of $\text{[math]}$

$\text{[math]}$ and $\text{[math]}$

$\text{[math]}$ and

$\text{[math]}$

Their product $\text{[math]}$ is equal to the square of their modulus

$\text{[math]}$ and

$\text{[math]}$

Quadrant Location

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

Example

If $\text{[math]}$ then $\text{[math]}$

$\text{[math]}$ is located in Quadrant 3, so $\text{[math]}$ is located in Quadrant 2

The Argument of a Complex Number and its Conjugate

The arguments of a Complex number $\text{[math]}$ and its conjugate $\text{[math]}$ add up to $\text{[math]}$ .

If $\text{[math]}$ then $\text{[math]}$ because of the symmetry across the $\text{[math]}$ due to the Quadrant symmetries of 1 and 4 and 2 and 3

with $\text{[math]}$

The sum of their arguments is $\text{[math]}$ .

Opposite Complex Numbers

Two Complex numbers $\text{[math]}$ and $\text{[math]}$ are opposite if they have opposite (negative) Real and Imaginary parts

$\text{[math]}$ and $\text{[math]}$ or

$\text{[math]}$

Example

$\text{[math]}$ then its opposite is $\text{[math]}$

The Modulus of a Complex Number and its Opposite

The modulus of a Complex number $\text{[math]}$ and its opposite $\text{[math]}$ are equal in magnitude

Example

If $\text{[math]}$ then its opposite is $\text{[math]}$

$\text{[math]}$ and

$\text{[math]}$

Then $\text{[math]}$ .

Quadrant Location

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

If a Complex number $\text{[math]}$ lies in the 1st Quadrant, then its opposite $\text{[math]}$ lies in the 3rd Quadrant and if a Complex number $\text{[math]}$ lies in the 4th Quadrant, then its opposite $\text{[math]}$ lies in the 2nd Quadrant.

Example

$\text{[math]}$ is in the 1st Quadrant because both $\text{[math]}$ and $\text{[math]}$ are positive, then $\text{[math]}$ lies in the 3rd Quadrant because both $\text{[math]}$ and $\text{[math]}$ are negative.

Example

$\text{[math]}$ is in the 2nd Quadrant because $\text{[math]}$ is negative and $\text{[math]}$ is positive, then its opposite $\text{[math]}$ is in the 4th Quadrant because $\text{[math]}$ is positive and $\text{[math]}$ is negative.

The Argument of a Complex Number and its Opposite

The arguments of a Complex number $\text{[math]}$ and its opposite $\text{[math]}$ differ by a value of $\text{[math]}$ 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 $\text{[math]}$ , which is equivalent to $\text{[math]}$ radians.

Reciprocal Complex Numbers

The reciprocal of a Complex number $\text{[math]}$ is $\text{[math]}$ with $\text{[math]}$ .

The reciprocal of a Complex number is its Multiplicative Inverse because $\text{[math]}$ .

Example

If $\text{[math]}$ then $\text{[math]}$

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

$\text{[math]}$

$\text{[math]}$ is the conjugate of $\text{[math]}$ 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 $\text{[math]}$ because the product of the Complex number and its conjugate is just the square of the modulus

$\text{[math]}$

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Equal, Conjugate, Opposite and Reciprocal Complex Numbers

Equal Complex Numbers

Example

Conjugate Complex Numbers

Example

The Modulus of a Complex Number and its Conjugate

Quadrant Location

Example

The Argument of a Complex Number and its Conjugate

Opposite Complex Numbers

Example

The Modulus of a Complex Number and its Opposite

Example

Quadrant Location

Example

Example

The Argument of a Complex Number and its Opposite

Reciprocal Complex Numbers

Example

Adding and Subtracting Complex Numbers

Complex Numbers