Complex numbers

Description

A complex number is a number of the form $a + b i$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, with $i^2 = -1$. We write $\mathbb{C}$ for the set of complex numbers. The real number $a$ is the real part and $b$ is the imaginary part of $a + b i$, see real and imaginary parts. Addition and multiplication follow the usual rules of algebra together with $i^2 = -1$: $$(a + b i) + (c + d i) = (a + c) + (b + d) i, \qquad (a + b i)(c + d i) = (a c - b d) + (a d + b c) i.$$ With these operations, $\mathbb{C}$ is a field that contains $\mathbb{R}$ as the numbers $a + 0 i$.