Complex Numbers

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Complex Numbers</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="95%" align="left" valign="top"><p class="word_list"> <a href="#complex numbers">complex number</a><p class="word_list"> <a href="#Euler’s formula">Euler's formula</a><p class="word_list"> <a href="#fundamental theorem of algebra">fundamental theorem of algebra</a><p class="word_list"> <a href="#imaginary">imaginary number</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="left" id="table5"> <tr> <td width="300" height="14"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"></td> </tr> <tr> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">Many people find <b>complex numbers</b> disturbing. Before delving into their mathematics, let’s consider why we might be interested in these constructs. </p> <p class="body">Complex numbers act much like a bridge between two villages that are located on opposite sides of a river. If the nearest ford is 10 miles upstream, a bridge may provide a more direct path between the two villages. In traveling between the two villages, we might take the ford or the bridge. Either way, our destination is the same. </p> <p class="body">Similarly, we may have a mathematical problem that is expressed entirely with <a href="../articles/set.htm">real numbers</a> and has a solution that depends only on real numbers. However, using complex numbers to reach that solution may provide a convenient shortcut compared to techniques that only involve real numbers. We use the complex numbers to bridge the gap between the problem and its solution. Doing so does not change the solution. It merely provides a convenient means—a bridge—for obtaining the solution. </p> <p class="body"><span class="header_section_green">The Number i <br> </span>The real numbers <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16"> contain no solution to the equation <img border="0" src="../formulas/imaginary_numbers_i^2_1.gif" align="absbottom" width="52" height="20">. We remedy this by extending the real numbers through the inclusion of an “imaginary” number <i>i</i> that <a name="[1]">satisfies</a> </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <p class="body"> <img border="0" src="../formulas/imaginary_numbers_[1].gif" width="44" height="18"></td> <td width="5%" align="right"> <p class="body">[1]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table7"> <tr> <td align="right" width="9" height="14"></td> <td width="336" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body">As with any number, we can add, multiply, take roots and perform other operations with this new number <i>i</i>. Multiplying <i>i</i> by 5 results in the number 5<i>i</i>. Adding 3 to this yields 3 + 5<i>i</i>. Squaring this yields 9 + 30<i>i</i> + 25<i>i</i><sup>2</sup>. </p> <p class="body">At this point, our imaginary number may be starting to seem like a Pandora’s box. By adding it to <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16">, we have actually added many numbers, and the expressions for these numbers seem to be getting more and more complicated. What would happen now if we were to divide our number 9 + 30<i>i</i> + 25<i>i</i><sup>2</sup> into 23? </p> <p class="body">In fact, such concerns are unfounded. Although the addition of <i>i</i> to <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16"> does add many numbers to <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16">, expressions for these numbers always simplify to the <a name="[2]">form</a> </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table8"> <tr> <td width="100%" align="center"> <p class="body"><i>a + bi</i></td> <td width="5%" align="right"> <p class="body">[2]</td> </tr> </table> <p class="body">where <i>a</i> and <i>b</i> are real. Using [<a href="#[1]">1</a>], we can simplify our number 9 + 30<i>i</i> + 25<i>i</i><sup>2</sup> as follows: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <p class="body">9 + 30<i>i</i> + 25<i>i</i><sup>2</sup> = 9 + 30<i>i</i> + 25(–1) = –16 + 30<i>i</i></td> <td width="5%" align="right"> <p class="body">[3]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table10"> <tr> <td align="right" width="345" colspan="3" height="14"></td> </tr> <tr> <td align="right" width="9" rowspan="2"> </td> <td width="168" height="15"> <p class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a>  </td> <td width="168" height="15" align="right"> <p class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Advertise on this site</font></a></td> </tr> <tr> <td width="336" colspan="2">  <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=84506;type=iframe" allowtransparency="true" background-color="transparent" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I1"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=84506;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=84506"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=84506;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>  </td> </tr> </table> <p class="body">which has the form [<a href="#[2]">2</a>].</p> <p class="body">We call the <a href="../articles/set.htm">set</a> of numbers of the form [<a href="#[2]">2</a>] the <b> <a name="complex numbers">complex numbers</a></b> and denote this set <img border="0" src="../symbols/complex_numbers.gif" align="absbottom" width="11" height="16">. Given a complex number <i>z</i> = <i>a + bi</i>, we call the real number <i>a</i> the real part of <i>z</i>. We call the real number <i>b</i> the imaginary part of <i>z</i>. This motivates the <i>Re</i> and <i>Im</i> functions that map a complex number <i>z</i> = <i>a + bi</i> to its real and imaginary parts <i>a</i> and <i>b</i>, respectively: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table11"> <tr> <td width="100%" align="center"> <p style="margin-top: 12; margin-bottom: 12"><i>Re</i>(<i>a + bi</i>) = <i>a</i></td> <td width="5%" align="right">[4]</td> </tr> <tr> <td width="100%" align="center"> <p class="body" style="margin-top: 12; margin-bottom: 12"><i>Im</i>(<i>a + bi</i>) = <i>b</i></td> <td width="5%" align="right"> <p class="body">[5]</td> </tr> </table> <p class="body">If a complex number's real part <i>a</i> equals 0—so it has the form <i>bi</i> for some real <i>b</i>—we say the number is purely imaginary (or, more simly, <b><a name="imaginary">imaginary</a></b>).</p> <p class="body"><span class="header_section_green">Complex Operations </span><br> Operations on complex numbers are extensions of the familiar operations for real numbers. Indeed, we have already performed complex addition and multiplication. We now formally define the operations of complex addition, subtraction, multiplication, division, and the taking of square roots. Let <i>a + bi</i> and <i>c + di</i> be complex numbers where <i>a</i>,<i>b</i>,<i>c</i>,<i>d</i> <img border="0" src="../symbols/contained_in_element.gif" align="absbottom" width="8" height="14"> <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16">. Then</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table12"> <tr> <td width="100%" align="center"> <p class="body" style="margin-top: 18; margin-bottom: 18"> <img border="0" src="../formulas/imaginary_numbers_[5].gif" width="255" height="24"></td> <td width="5%" align="right"> <p class="body" style="margin-top: 18; margin-bottom: 18">[5]</td> </tr> <tr> <td width="100%" align="center"> <p style="margin-top: 18; margin-bottom: 18"> <img border="0" src="../formulas/imaginary_numbers_[6].gif" width="251" height="20"></td> <td width="5%" align="right"> <p style="margin-top: 18; margin-bottom: 18">[<a name="[7]">6</a>]</td> </tr> <tr> <td width="100%" align="center"> <p style="margin-top: 18; margin-bottom: 18"> <img border="0" src="../formulas/imaginary_numbers_[7].gif" width="267" height="21"></td> <td width="5%" align="right"> <p style="margin-top: 18; margin-bottom: 18">[<a name="[8]">7</a>]</td> </tr> <tr> <td width="100%" align="center"> <p style="margin-top: 12; margin-bottom: 12"> <img border="0" src="../formulas/imaginary_numbers_[8].gif" width="304" height="43"></td> <td width="5%" align="right"> <p style="margin-top: 12; margin-bottom: 12">[<a name="[9]">8</a>]</td> </tr> <tr> <td width="100%" align="center"> <p style="margin-top: 12; margin-bottom: 12"> <img border="0" src="../formulas/imaginary_numbers_[9].gif" width="417" height="68"></td> <td width="5%" align="right"> <p style="margin-top: 12; margin-bottom: 12">[9]</td> </tr> </table> <p class="body">With the exception of 0, every real or complex number has two square roots. For example, the square roots of 4 are 2 and –2. The square roots of –1 are <i>i</i> and –<i>i</i>. </p> <p class="body">In [5] through [9], the formulas reduce to the corresponding operations for real numbers if they are applied to real numbers. Also, the right side of each formula is always defined and corresponds to a complex number of the form [<a href="#[2]">2</a>]. The only exception is division by zero, which is undefined with regard to real as well as complex numbers.</p> <p class="body">Recall that we were motivated to introduce complex numbers by the equation <img border="0" src="../formulas/imaginary_numbers_i^2_1.gif" align="absbottom" width="52" height="20">, which has no solution in <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16">. Is it possible that there is some equation that has no solution in <img border="0" src="../symbols/complex_numbers.gif" align="absbottom" width="11" height="16">? If this were the case, we might feel compelled to extend the complex numbers through the addition of still another “imaginary” number to solve this new equation. This will never happen, due to the <b> <a name="fundamental theorem of algebra">fundamental theorem of algebra</a></b>. Consider a polynomial equation of the form </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table13"> <tr> <td width="100%" align="center"> <p class="body"> <img border="0" src="../formulas/imaginary_numbers_[13].gif" width="129" height="129"></td> <td width="5%" align="right"> <p class="body">[10]</td> </tr> </table> <p class="body">where the <img border="0" src="../formulas/imaginary_numbers_w_i.gif" align="absbottom" width="14" height="12"> <img border="0" src="../symbols/contained_in_element.gif" align="absbottom" width="8" height="14"> <img border="0" src="../symbols/complex_numbers.gif" align="absbottom" width="11" height="16"> are constants. The theorem states that every such equation has exactly <i> n</i> solutions <i>z</i> <img border="0" src="../symbols/contained_in_element.gif" align="absbottom" width="8" height="14"> <img border="0" src="../symbols/complex_numbers.gif" align="absbottom" width="11" height="16"> (including repeated solutions).</p> <p class="body"><span class="header_section_green">Complex Functions</span><br> We extend the exponential function to <img border="0" src="../symbols/complex_numbers.gif" align="absbottom" width="11" height="16"> with </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table14"> <tr> <td width="100%" align="center"> <p class="body"> <img border="0" src="../formulas/imaginary_numbers_[11].gif" width="191" height="26"></td> <td width="5%" align="right"> <p class="body">[11]</td> </tr> </table> <p class="body">This is the famous <b><a name="Euler’s formula">Euler’s formula</a></b> that links the exponential function with the sine and cosine functions. We extend the sine and cosine functions to <img border="0" src="../symbols/complex_numbers.gif" align="absbottom" width="11" height="16"> with </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table15"> <tr> <td width="100%" align="center" rowspan="2"> <p class="body" style="margin-bottom: 0"> <img border="0" src="../formulas/imaginary_numbers_[13].gif" width="129" height="129"></td> <td width="5%" align="right"> <p class="body">[12]</td> </tr> <tr> <td width="5%" align="right">[13]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black" style="margin-top: 24"><a name="Exercises">Exercises</a></td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table17"> <tr> <td width="100%"> <p class="body" style="margin-left: 10; margin-top: 8; margin-bottom: 8"> <img border="0" src="../images/bullet_exercise.gif" align="absbottom" width="19" height="17">Express the following in <i>a</i> + <i>bi</i> form:<p style="margin-left: 31">a. (5 + 3<i>i </i>)(2 – <i>i</i> )</p> <p style="margin-left: 31">b. 5/(2 + <i>i </i>)</p> <p style="margin-left: 31">c. <i>e</i><sup><i>i</i></sup><p style="margin-left: 31; margin-bottom: 6">[<a href="../solutions/imaginary_numbers_1.htm">solution</a>]</td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Internal Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table18"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/eigenvalue.htm">eigenvalue, eigenvector</a> Concepts from linear algebra.</strong></p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/fast_fourier_transform.htm">fast Fourier transform</a> An algorithm for rapidly valuing Fourier transforms.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/polynomial.htm">linear and quadratic polynomials</a> Two basic forms of polynomials.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/set.htm">set</a> A</strong>n unordered collection of elements.</td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_1.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Courses</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table19"> <tr> <td width="100%"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table20"> <tr> <td width="127" bgcolor="#669933" align="left" valign="top"> <img border="0" src="../images/picture_course_math.jpg" width="125" height="125"></td> <td width="100%" align="left" valign="top"> <p class="explanatory_text"><b> <font size="2"> <a target="_blank" href="http://www.risklearning.com/link/math_3.htm"> Financial Math 3</a></font></b></p> <p class="explanatory_text">This is the third in a series of three-day courses on financial mathematics that take participants from pre-calculus to stochastic calculus. 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