## Complex numbers

## Background

Until the 16th century mathematicians knew that solutions to equations could be integers, rational (fractions) or irrationals. However, some equations seemingly had solutions which involved taking square-roots of negative quantities. These solutions are are all ultimately based on \(\sqrt{-1}\). Mathematicians gradually came to accept that this was a new species of number. Rene Descartes mocked the idea by calling \(\sqrt{-1}\) "imaginary". The name stuck and by implication the familiar integers, fractions and irrationals became known as "real" numbers. The symbol \( i \) was used to denote \(\sqrt{-1}\) so that the definition of the imaginary unit was \[i^2 = -1\]

Imaginary numbers often turned up in solutions bonded to real numbers, in a form like \(x+yi\). This generalised form is a complex number.

- Real numbers are complex numbers in which the imaginary part is zero.
- Imaginary numbers are complex numbers in which the real part is zero.

Many mathematicians examined complex numbers in the 16th to 18th centuries and applied concepts from algebra, geometry and calculus to discover properties of complex numbers. In return complex numbers showed surprising links between unrelated areas of mathematics. We'll discuss some of these discoveries below.

At the end of the 18th century Carl Gauss proved the Fundamental theorem of Algebra and complex numbers became a part of mainstream mathematics thereafter.

## Arithmetic

Complex numbers