RedCrab Math Tutorial

The polar form of a complex number

In the article on the geometric representation of complex numbers, it has been described that every complex number z in the Gaussian plane of numbers can be represented as a vector. This vector is uniquely determined by the real part and the imaginary part of the complex number z.

A vector emanating from the zero point can also be used as a pointer. This pointer is clearly defined by its length and the angle φ to the real axis.

The following figure shows the vector with the length r = 2 and the angle φ = 45 °

Polar form of  complex number

Positive angles are measured counterclockwise, negative angles are clockwise.

A complex number can thus be uniquely defined in the polar form by the pair (|z|, φ). φ is the angle belonging to the vector. The length of the vector r corresponds to the absolute value |z| the complex number.

The general spelling is called normal form z = a + bi. We write in polar form r = |z| and φ = arg(z)

Convert Polar form to Normal form

This article describes the conversion from the polar form to the normal form of a complex number.

If the magnitude and angle of a complex number are known, the real and imaginary values can be calculated.

The representation by means of vectors always results in a right triangle, which consists of the two catheters a and b and the hypotenuse z. The conversion can therefore be performed using trigonometric functions. With reference to the figure below:

Reale Einheit Re der Polarform einer komplexen Zahl and Imaginäre Einheit der Polarform einer komplexen Zahl

calculate polar of  complex number

To convert a complex number from polar to normal, the following applies:

convert a complex number from polar to normal

In the RedCrab Calculator you use the function:

z = FromPolar (r, φ)

Conversion from coordinates to polar coordinates

This article describes the determination of the polar coordinates of a complex number by calculating the angle φ and the length of the vector z.

The radius r of the polar form is identical to the magnitude |z| of the complex number. The formula for calculating the radius is thus the same as that described in the article of the absolute value of a complex number.

For the length r of the vector results:

Conversion to polar coordinates

If the vector is in the 1st or 2nd quadrant, the angle φ applies:

the angle vector of komplex number


the angle of komplex number

When calculating the angle, it must be taken into account in which quadrant the vector is located. Consider the following figure:

Polar form with different  Quadrant

For the complex number 3 + 4i in the picture above, the ,agnitude is:

Betrag einer komplexen Zahl

The angle is:

angle in 1. quadrant of polar form of komplex number


For the complex number 3 - 4i the magnitude is also:

The calculation of the angle also gives:


In this case 180 ° must be added to the calculated angle to get into the right quadrant.

After calculating the angle φ with the aid of the arc sine, a test of the quadrant must always be carried out. For a negative imaginary part, the angle must be corrected.

For a complex number a + bi applies

If b ≥ 0 is:    
If b < 0 is:   or:
      when calculated in radians

In the calculations above, the angle between 0 ° and 360 ° is given as the angle φ to the real axis. The angle can also be specified between 0 ° and ± 180 °.

If you calculate the angle in the RedCrab Calculator with the function Arg (), you get the following results:

Arg (3 + 4i) = 53.1

Arg (3 − 4i) = −53.1

Arg (−3 + 4i) = 127

Arg (−3 − 4i) = −127

Multiplication of complex numbers in polar form

The representation of complex numbers in polar form also simplifies the multiplication of complex numbers. In multiplication, the angles are added and the length of the vectors multiplied.
The figure below shows the geometric multiplication of the complex numbers: 2 + 2i and 3 + 1i

Multiplication of complex numbers in polar

For multiplication in polar form the following applies:

geometric multiplication in polar form of complex number


The division of complex numbers in polar form

From the handling of multiplication, the division of two complex numbers in polar form can be derived. Complex numbers are divided by dividing their absolute values and subtracting their angles. The following applies:

The division of complex numbers in polar form


Complex numbers

Basics of complex numbers
Computing with complex numbers
Addition and subtraction
Multiplication of complex numbers
Conjugate complex numbers
Division of complex numbers
Quadratic equations, complex numbers
Geometric representation
Geometric addition and subtraction
Absolute value of a complex number
The polar form of complex numbers
Calculate polar form
Convert polar form to normal
Multiplication in polar form
Division in polar form

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