#math/linear-algebra A **Complex Number** is an ordered pair $(a,b)$ where $a,b\in \mathbb{R}$, denoted as $a+bi$. The set of all complex numbers is $\mathbb{C} = \{ a + bi : a, b \in \mathbb{R}\}$. The motivation for complex numbers was to formalize arithmetic in order to take square roots of negative numbers. The symbol $i$ is used to denote $\sqrt{-1}$. If we think of $(a,b)$ with $b=0$ as $\mathbb{R}$, then we can surmise that $\mathbb{R}\subset \mathbb{C}$.