Complex Numbers as Vectors

A complex number can be drawn as a vector, the tip of which is at the point , where

$\displaystyle x$ $\displaystyle \triangleq$ the horizontal coordinate--the real part $\displaystyle ,$

$\displaystyle y$ $\displaystyle \triangleq$ the vertical coordinate--the imaginary part $\displaystyle .$

Figure 2: Cartesian and polar representations of complex numbers in the complex plane.
$\begin{figure}\centerline{% \input{complexplane.pstex_t}}\end{figure}$
Thus, the - and -axes may be referred to as the real and imaginary axes, respectively.
A multiplication by may be seen as an operation meaning ``rotate counterclockwise $90^\circ$ or $\pi/2$ radians''.
Two successive rotations by $\pi/2$ bring us to the negative real axis (), and thus $j =\sqrt{-1}$ .

**Figure 2:** Cartesian and polar representations of complex numbers in the complex plane.
$\begin{figure}\centerline{% \input{complexplane.pstex_t}}\end{figure}$

``Music 270a: Complex Exponentials and Spectrum Representation'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).

$\displaystyle x$	$\displaystyle \triangleq$	the horizontal coordinate--the real part $\displaystyle ,$
$\displaystyle y$	$\displaystyle \triangleq$	the vertical coordinate--the imaginary part $\displaystyle .$