RedCrab Math Tutorial


Complex numbers and polar coordinates

With the polar coordinates we can display complex numbers graphically. For this we uses the complex plane or z-plane. 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.
  • The real part of a complex number represented by a displacement along the x-axis

  • The imaginary part of a complex number represented by a displacement along the y-axis


Absolute value of a complex number

The representation with vectors always results in a right-angled 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:
  • If b > 0 use : arccos (a / |z|)

  • If b < 0 use: 2π - arccos (a / |z|)
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.
  • Real: a = |z| · cos φ

  • Imaginary: b = |z| · sin φ

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


Integer and Real Numbers
Complex Numbers
Roots and Power
Percentage Calculation
Interest Calculation
Absolute Value of a Number
Euclidean division
Modulo - Remainder of a division

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