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