Complex Numbers: Arithmetic in Modulus-Argument Form/title> <meta http-equiv='Content-Type' content='text/html; charset=utf-8'/> <link href='../../css/metric-legacy.css' type='text/css' rel='stylesheet'/> <script src='../../jsMath/easy/load.js' type='text/javascript'></script> <script type="text/JavaScript">  </script> </head> <body> <h1>Complex Numbers: Arithmetic in Modulus-Argument Form</h1> <p>The complex number <span class='math'>z_1</span> with modulus <span class='math'>r_1</span> and argument <span class='math'>\theta_1</span> is $$r_1 (\cos \theta_1 + i \sin \theta_1), $$ and similarly for <span class='math'>z_2</span>, with modulus and argument <span class='math'>r_2</span> and <span class='math'>\theta_2</span> respectively. Now,</p> <p align="center"><span class="math">\begin{eqnarray}<br /> z_1\, z_2 & = &<br /> r_1 r_2 (\cos \theta_1 \cos \theta_2 + i \cos \theta_1 \sin \theta_2 + i \sin \theta_1 \cos \theta_2 + i2 \sin \theta_1 \sin \theta_2) \\<br /> & = & <br /> r_1 r_2 ([\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2] + i [\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2]) \\<br /> & = & <br /> r_1 r_2( \cos [\theta_1 + \theta_2] + i \sin [\theta_1 + \theta_2]). <br /> \end{eqnarray}</span></p> <p>The modulus of this complex number is <span class='math'>r_1 r_2</span> and its argument is <span class='math'>\theta_1 + \theta_2</span>. In other words, </p> <p align="center"><span class="math">\begin{eqnarray} |z_1\, z_2| & = & |z_1| \,|z_2|, \\ \arg (z_1\, z_2)& = & \arg (z_1) + \arg (z_2). \end{eqnarray}</span></p> <p> It can also be shown that </p> <p align="center"><span class="math">\begin{eqnarray} |z_1 / z_2| &=& |z_1| / |z_2|, \\ \arg (z_1 / z_2)& =& \arg (z_1) - \arg (z_2). \end{eqnarray} </span></p> <p>So, for example, consider <span class='math'>z_1 = 2 (\cos 7\pi/12 + i \sin 7\pi/12)</span> and <span class='math'>z_2 = 3 (\cos 2\pi/3 - i \sin 2p3)</span>. Now <span class='math'>|z_1| = 2</span> and <span class='math'>|z_2| = 3</span>, and therefore <span class='math'>|z_1\, z_2| = 6</span>. Also, <span class='math'>\arg z_1 = 7\pi/12</span> and <span class='math'>\arg z_2 = -2\pi/3</span>, and therefore <span class='math'>\arg (z_1\, z_2) = 7\pi/12 - 8\pi/12 = -\pi/12</span>. It follows that $$z_1 z_2 = 6(\cos \pi/12 - i \sin \pi/12). $$ Similarly, <span class='math'>|z_1 / z_2| = 2/3</span> and <span class='math'>\arg (z_1 / z_2) = 7\pi/12 + 8\pi/12 = 15 \pi/12 = 5\pi/4</span>. It follows that $$z_1 / z_2 = 2/3 (\cos 5\pi/4 + i \sin 5\pi/4) $$ ---or at least, it would do, except that by our convention the argument ought to lie between <span class='math'>-\pi</span> and <span class='math'>\pi</span>, so the above is more conventionally expressed as $$z_1 / z_2 = 2/3 (\cos 3\pi/4