

Imaginary
and complex numbers 
Exponentiation
and root extraction of complex numbers in the polar form 
Powers and roots of
complex numbers, use of de Moivre’s formulas examples 
Euler’s
formula,
relationship between trigonometric functions and the complex exponential
function 





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 




Powers and roots of
complex numbers, use of de Moivre’s formulas 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 







Euler’s formula,
relationship between trigonometric functions and the complex exponential
function 
Euler's
formula 
e^{i}^{j}
= cosj
+
isinj,
where 
e
is the base of the natural logarithm,

i is
the imaginary unit, and 
j is
the angle between xaxis
and the vector pointing to the complex number z
measured counter clockwise, that is, j
is the argument of z, 
describes the unit circle in the complex plane. 



That is,
on the unit circle lie points of the complex plane that correspond to the
complex numbers each of which is one unit far from the origin. 
Thus, by
plugging the angles, 

into Euler's
formula respectively 



obtained
are the four complex numbers that lie on the unit circle, the
two of which lie on the real axis and the two on the imaginary
axis as shows the above picture. 
The expression e^{i}^{p}
+ 1 = 0 is called Euler's
equation or identity.


Euler's formula shows relations
between trigonometric functions and complex
exponentials. 
Thus, by adding and subtracting
Euler's formulas:

(1) e^{i}^{j}
= cosj
+
isinj 
and
(2) e^{}^{i}^{j}
= cos(j)
+
isin(j)
= cosj

isinj 
and solving for
cosine and sine 






We use
Euler's formula to write complex number z
= x
+
yi
in polar coordinates, 

z
= x
+
yi = r
(cosj
+
isinj)
= r
e^{ij} 


where 











College
algebra contents




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