

Imaginary
and Complex Numbers 
Polar or trigonometric
notation of complex numbers 
Multiplication
and division of complex numbers in the polar form 
Exponentiation
and root extraction of complex numbers in the polar form 
Powers and roots of
complex numbers, use of de Moivre’s formulas examples 





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 




Exponentiation
and root extraction of complex numbers in the polar form  de
Moivre's formula 
We use the polar form
for exponentiation and root extraction of complex numbers that
are known as de Moivre's formulas. 

z^{n}_{
} = r^{n}_{ }·_{ }[cos(nj)
+ isin(nj)] 


and 




Exponentiation
and root extraction of complex numbers in the polar form
examples

Example: 
Compute 



Solution: 


or
in the polar form, 


and 


since
exponentiation with integer exponent 


then 



Example: 
Compute 



Solution:
As square root of a
complex number is a complex number, then 

and, two
complex numbers are equal if their real parts are equal and
their imaginary parts are equal, that is 




Example: 
Calculate 

using de Moivre's
formula. 




These complex numbers satisfy the equation z^{3}
= 8
and by the Fundamental theorem of algebra, since this equation
is of degree 3, there must be 3 roots. 
Thus, for
example to check the root z_{k=}_{2}
we cube this solution, 

then 



Example: 
Calculate 




r
= 64 and
j =
p 




thus, 






These complex numbers satisfy the equation z^{6}
= 64
and by the Fundamental theorem of algebra, since this equation
is of degree 6, there must be 6 roots. 


Example: 
Calculate 







Example: 
Calculate 














Intermediate
algebra contents 



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