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
and
with
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. 
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
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
then
is located in Quadrant 3, so
is located in Quadrant 2 
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
.
Quadrant Location
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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