

Imaginary
and Complex Numbers 
Multiplication and
division of
complex numbers 
Polar or trigonometric
notation of complex numbers 
Multiplication
and division of complex numbers in the polar form 





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:




Polar or trigonometric
notation of complex numbers 
A point (x,
y)
of the complex plane that represents the complex number z
can also be specified by its distance r
from the origin and the angle j
between the line joining the point to the origin and the
positive xaxis. 
Cartesian
coordinates expressed by polar coordinates: 
x
= r cosj 
y
= r sinj 
plugged
into z
= x
+
yi
give 

z
= r
(cosj
+
isinj), 

where 







Thus,
obtained is the polar or trigonometric form of a complex number
where polar coordinates are r,
called the absolute value or modulus, and j,
that is called the argument, written j
= arg(z). 
By using
Euler's formula e^{ij}
= cosj
+
isinj,
a complex number can also be
written as 

z
= r
e^{ij} 
which
is called the exponential form. 


To show
the equivalence between the algebraic and the trigonometric form of a complex number, 
z
= r
e^{ij}
= r
(cosj
+
isinj) 
express
the sine and the cosine functions in terms of the tangent 

and substitute
into above expression 


Example: Given
the complex number z
= 1

Ö3i,
express z
= x
+
yi
in the trigonometric form.

Solution:
The modulus 


the argument 




the trigonometric form is





Multiplication
and division of complex numbers in the polar form 
If
given z_{1}
= r_{1}(cosj_{1}
+ isinj_{1})
and z_{2}
= r_{2}(cosj_{2}
+ isinj_{2})
then 

z_{1
}·_{ }z_{2} = r_{1
}r_{2 }·_{ }[cos(j_{1}
+ j_{2})
+ isin(j_{1}
+ j_{2})] 


and 











Functions
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