
Complex numbers and polar coordinates 
With the polar coordinates we can display complex numbers graphically. For this we uses the complex plane or zplane. It is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane. 


Absolute value of a complex number 
The representation with vectors always results in a rightangled triangle consisting of the two catheters a and b and the hypotenuse z. The absolute value of a complex number corresponds to the length of the vector. 
The absolute value of a complex number z = a + bi is: 
The figure below shows the graphical representation of the complex number 3 + 4i. 

Calculation of the absolute value of the complex number z = 3  4i: 


The position of a point (a, b) can also be determined by the angle φ and the length of the vector z. For this you use the cosine and sine function at the right triangle: 
 z = a + bi = z · cos φ + i · z · sin φ = z · ( cos φ + i · sin φ)

A complex number can be defined by the pair ( z , φ). φ is the angle belonging to the vector. 
This representation of complex numbers also simplifies the geometric representation of a multiplication of complex numbers. In multiplication, the angles are added and the length of the vectors is multiplied. 
The figure below shows the example of a geometric representation of a multiplication of the complex numbers:
2+2i and 3+1i 

Conversion from coordinates to polar coordinates 
The following description shows the determination of the polar coordinates of a complex number by the calculation of the angle φ and the length of the vector z. 
For the calculation of the angle of the complex number z = a + bi the following trigonometric formulas apply: 

The following example calculates the polar coordinates of the complex number 
 Calculation of the absolute value:
 Calculation of the angle:

The following example shows the calculation with the RedCrab Calculator 


Conversion of polar coordinates into coordinates 
If the magnitude and angle of a complex number are known, the real and imaginary values can be calculated using the following formulas. 

If the values from the example above are used, the complex number results 1.41 + 1.41i 

a = 2 · cos(135) = 1.41

b = 2 · sin(135) = 1.41

The RedCrab Calculator provides the FromPolar function: FromPolar (2, 135) = 1.41+1.41i 

