:: Complex
numbers

A complex
number is the sum of a real number and an imaginary number. 
A complex
number z
is written in the form of z
= x
+ yi,
where x
and y
are real numbers, and 

The real
number x
is called the real part of the
complex number, and the real number y
is the imaginary part. The real
part of z
is denoted Re(z)
=
x
and the imaginary part is denoted
Im(z)
=
y. 
Hence, an imaginary
number is a complex number whose real part is zero, while real
numbers may be considered to be complex numbers with an
imaginary part zero. 
That is,
the real number x
is equivalent to the complex number x
+ 0i. 
Equality
of complex numbers 
Two
complex numbers are equal if their real parts are equal and
their imaginary parts are equal. 

The
complex plane

Complex
numbers are represented by points or position vectors in the
coordinate plane called the complex plane (or the Gauss plane).
Where, the xaxis
is called the real axis and the yaxis
is called the imaginary axis. 
The
representation of a complex number by Cartesian coordinates is
called the rectangular form or algebraic form of the complex
number. The
standard symbol for the set of all complex numbers is
C. 

Absolute value,
modulus of a complex number 
The
absolute value  z
 of a complex number
is defined as the distance
from z
to the origin in the complex plane. 
Complex conjugates 
The
complex conjugate of the complex number z
= x
+ yi
is x

yi
that has the same real part x,
but differ in the sign of the imaginary part. That is,
the conjugate is the reflection of z
about the real axis,
as is shown in the above figure. 

Addition and
subtraction of complex numbers

To add or
subtract two complex numbers z_{1}
= a + bi
and z_{2}
= c + di, we add or subtract the real parts
and the imaginary parts.

addition:

z_{1} +
z_{2}
= (a + bi) + (c + di) = (a
+ c) + (b + d)i


subtraction:

z_{1}

z_{2}
= (a + bi) 
(c + di) = (a 
c) + (b 
d)i


Example: 
Given
are complex numbers, z_{1}
= 3 +
2i and
z_{2}
= 4 + 3i,
find z_{1} +
z_{2} and
z_{1}

z_{2}.

Solution: 
z_{1} +
z_{2}
= (3 +
2i) + (4 + 3i) =
(3 +
4) + (2 + 3)i = 1 + 5i

and 
z_{1}

z_{2}
= (4 + 3i) 
(1 + 5i) =
(4 
1) + (3 
5)i =
3 
2i 
Given
addition and subtraction are shown in the complex plane in
the figures below. 
z_{1} +
z_{2}
=
(3 +
2i) + (4 + 3i) = 1 + 5i 
z_{1}

z_{2}
=
(4 + 3i) 
(1 + 5i) = 3 
2i 





Multiplication and
division of
complex numbers

multiplication:

z_{1}·
z_{2}
=
(a + bi) · (c + di) = ac
+ bci + adi + bdi^{2
}= (ac 
bd) +
(ad + bc)i

division:


Example: 
Given
are complex numbers, z_{1}
= 3 +
2i and
z_{2}
= 4 + 3i,
find z_{1} ·
z_{2 } and
z_{1}
/
z_{2}.

Solution: 
z_{1} ·
z_{2}
= (3 +
2i) · (4 + 3i) = 3 ·
4 + 2 · 4i +
(3)
· 3i + 2 · 3 i^{2
}= 18

i

and 



Example:
For what real
number a the real part of the complex number 

equals
1.


Solution: 





Example:
Evaluate the
expression 

where
z = 1 
i.


Solution: 


