Let $j$ denote the imaginary number $\sqrt{-1}$.

A complex number $z$ is any number of the form $z = x+jy$, where $x$ is called the real part of $z$ and $y$ is called the imaginary part of $z$. Note: The imaginary part is not $jy$, rather it is only $y$. It is important to stick to this terminology, otherwise computations can go wrong.

Often, it is useful to think of a complex number $z = x + j y$ as a vector in a two-dimensional plane as shown in the figure below, where $x$ is the $X$-coordinate and $y$ is $Y$-coordinate of the vector. Such a plane is called the complex plane or the Argand diagram. Due to this relationship between a complex number and the corresponding vector, we will abuse the terminology and use the terms complex number and vector interchangeably, if the context should resolve any possible ambiguity. For example, a complex number is said to lie in the first quadrant (or, second quadrant etc) if the corresponding vector lies in the first quadrant (or, second quadrant etc).

A complex number is any number of the form

A complex number can be visually represented as a pair of numbers `math|(''a'',''b'')` forming a vector on a diagram called an [[Argand diagram]], representing the [[complex plane]]. ''Re'' is the real axis, ''Im'' is the imaginary axis, and `math|''i''` is the [[imaginary unit]], satisfying `math|1=''i''<sup>2</sup> = −1`.]]