Begin with the real numbers, $\RR$. Add “the” square root of $-1$; call it $i$. You have just constructed the complex numbers, $\CC$, in the form \begin{equation} \CC = \RR \oplus \RR \, i \end{equation} That is, a complex number $z$ is a pair of real numbers ($a$,$b$), which is usually written as \begin{equation} z = a + bi \end{equation} and which can be thought of as either a point in the (complex) plane with coordinates $(a,b)$ or as a vector with components $a$ and $b$.