

Imaginary
and Complex Numbers 
Imaginary
numbers basic definitions 
Imaginary unit 
Complex numbers 
Real and imaginary
parts 
The complex plane 
The set of all complex
numbers C 
Absolute value,
modulus of a complex number 
Complex conjugates 
Addition and
subtraction of complex numbers 
Multiplication
and division of complex numbers 





Imaginary
numbers basic definitions 
Imaginary
numbers are introduced to enable us to take the square root of
negative numbers. 
Thus, for
example 

so, the square roots of negative numbers are called imaginary
numbers since they do not lie on the real number line. 
Therefore,
the square of any imaginary number (except 0) is a negative
number. 
That
is, any number
of the form 

yi,
where 


the imaginary unit and y
is any real number, 

we call imaginary
number. 
By using
the definition derived are powers of the imaginary unit i: 

note that the powers of
i
repeat in a cycle, so that 


Examples:
Reduce, i^{15},
i^{26}
and i^{149}. 
Solutions:
i^{15}
= i^{4
· 3}^{ }· i^{3} = i^{3}
= i,
i^{26}
= i^{4
· 6}^{ }· i^{2} = i^{2}
= 1
and i^{149}
= i^{4
· 37 }· i = i. 

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 of 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 of a complex number z
is defined as the distance
from z
to the origin in the complex 
plane,
i.e., 




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 



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

z_{2}.

Solutions:
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: 




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

Solutions:
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:











Intermediate
algebra contents 



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