Products

Let z = r(cos $\theta$ + j sin $\theta$ ) and w = s(cos $\varphi$ + j sin $\varphi$ ) be two complex numbers in polar form. Thus r = | z| and $\theta$ = arg(z), while s = | w| and $\varphi$ = arg(w).

Consider the product of z and w:

zw	= rs(cos $\displaystyle \theta$ + j sin $\displaystyle \theta$ )(cos $\displaystyle \varphi$ + j sin $\displaystyle \varphi$ )
	= rs((cos $\displaystyle \theta$ cos $\displaystyle \varphi$ - sin $\displaystyle \theta$ sin $\displaystyle \varphi$ ) + j(sin $\displaystyle \theta$ cos $\displaystyle \varphi$ + cos $\displaystyle \theta$ sin $\displaystyle \varphi$ ))
	= rs(cos( $\displaystyle \theta$ + $\displaystyle \varphi$ ) + j sin( $\displaystyle \theta$ + $\displaystyle \varphi$ ))

This tells us that the modulus of zw is just the product of the moduli of z and w:

| zw| = | z| | w|

and, provided we adjust the angles to the correct range by adding or subtracting multiples of 2 $\pi$ , the argument of the product is the sum of the arguments:

arg(zw) = arg(z) + arg(w) (modulo 2 $\displaystyle \pi$ ).

For example, if z has argument 120^o and w has argument 150^o then an argument of zw is 120 + 150 = 270, which is not in the right range, so we subtract 360^o and get the principal argument, which is -90^o (or - $\pi$ /2 radians).

Similarly, for w $\neq$ 0,

$\displaystyle \left\vert\vphantom{\frac{z}{w}}\right.$ $\displaystyle {\frac{{z}}{{w}}}$ $\displaystyle \left.\vphantom{\frac{z}{w}}\right\vert$ = $\displaystyle {\frac{{\vert z\vert}}{{\vert w\vert}}}$

and

arg $\displaystyle \left(\vphantom{\frac{z}{w}}\right.$ $\displaystyle {\frac{{z}}{{w}}}$ $\displaystyle \left.\vphantom{\frac{z}{w}}\right)$ = arg(z) - arg(w) (modulo 2 $\displaystyle \pi$ ).

Ian Craw 2003-12-15