Polar coordinates

Introduction to the basics of polar coordinates of complex numbers.
More detailed descriptions can be found in the chapter on complex numbers.


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 \((real axis)\)

The imaginary part of a complex number represented by a displacement along the y-axis \((imaginäre Achse)\)


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 \(|z| = \sqrt{a^{2}+b^{2}}\)

The figure below shows the graphical representation of the complex number \(3+4i\)


Calculation of the absolute value of the complex number \(z = 3 - 4i\)

\(|z|= \sqrt{3+8}=\sqrt{3^{2}+(-42)}=\sqrt{25}=5\)

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 \(2+2i\) und \(3+1i\).


Convert 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 \(z=-\sqrt{2}+i\sqrt{2}\).

Calculation of the absolute value: \(|z|=\sqrt{(-\sqrt{2})^{2}+(\sqrt{2})^{2}}=\sqrt{2+2}=2\)

Calculation of the angle: \(φ =arccos(a / |z|) = arccos(-\sqrt{2}/2)=135\)


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: \( FromPolar\)\((2, 135) = -1.41 + 1.41i\)