01 What are complex numbers?


The complex numbers extend the number range of the real numbers in such a way that the equation x * x = -1 becomes solvable.

Since all real square numbers are non-negative, the solution of this equation cannot be real. So we need a new number, it is called i, with the property i * i = -1.

This number i is called imaginary unit.

Representation of complex numbers

While real numbers can be represented by points on a number line, complex numbers can be represented as points in a plane.

Cartesian coordinates

Complex numbers can be represented in the form z = a + bi, where a and b are real numbers and i is the imaginary unit. The coefficients a and b are called the real part and imaginary part, respectively.

Polar coordinates

Alternatively, a complex number z can be represented as a vector with radius r and angle ϕ.