# Polar form of complex numbers

Description of the polar form of complex numbers

## 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

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