Imaginary and Complex Numbers
      Polar or trigonometric notation of complex numbers
      Eulerís formula, relationship between trigonometric functions and the complex exponential function
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 x-axis. 
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  eij = cosj + isinj,  a complex number can also be written as
  z = r eij which is called the exponential form.
To show the equivalence between the algebraic and the trigonometric form of a complex number,
 z = r eij = 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 - ÷3i,  express  z = x + yi  in the trigonometric form.
Solution:  The modulus  
                 the argument  
 
    the trigonometric form is  
Eulerís formula, relationship between trigonometric functions and the complex exponential function
Euler's formula
eij = cosj + isinj,  where
e is the base of the natural logarithm,
i is the imaginary unit, and
j is the angle between x-axis 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  eip + 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)   eij = cosj + isinj
and                              (2)   e-ij = 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 eij  
where
Intermediate algebra contents
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