# Polar form of complex numbers

Description of the polar form of complex numbers with examples

## 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 $$φ = 45°$$

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 $$r = |z|$$ und $$φ = arg(z)$$

## 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.

$$Re=r·cos(φ)$$   $$Im=r·sin(φ)$$

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

$$z=r·cos(φ)+ir·sin(φ)=a+bi$$

## Conversion from coordinates to polar coordinates

This article describes the determination of the polar coordinates of a complex number by calculating the angle $$&phi$$ 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

$$r=|z|=\sqrt{a^2+b^2}=\sqrt{Re^2+Im^2}$$

If the vector is in the 1. or 2. quadrant, the angle $$φ$$ applies

$$\displaystyle φ=arccos\left(\frac{a}{r}\right)=arccos\left(\frac{Re}{|z|}\right)$$

or

$$\displaystyle φ=arctan\left(\frac{b}{a}\right)=arctan\left(\frac{Im}{Re}\right)$$

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

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

$$|z|=\sqrt{3^2+4^2}=5$$

The angle is

$$\displaystyle φ=arccos\left(\frac{Re}{|z|}\right)=arccos\left(\frac{3}{5}\right)=53.1°$$

For the complex numbe $$3 - 4i$$ the magnitude is also

$$|z|=\sqrt{3^2-4^2}=5$$

The calculation of the angle also gives $$53.1°$$. 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   $$\displaystyle φ=arccos\left(\frac{a}{|z|}\right)$$

If $$b < 0$$ is   $$\displaystyle φ= 360 - arccos\left(\frac{a}{|z|}\right)$$

or $$\displaystyle φ= 2π - arccos\left(\frac{a}{|z|}\right)$$ 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 $$± 180°$$.

$$Arg (3 + 4i) = 53.1$$

$$Arg (3 − 4i) = −53.1$$

$$Arg (−3 + 4i)=127$$

$$Arg (−3 − 4i)=−127$$