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href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../537/algebra.html">Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Complex Numbers - The Basics</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/algebra/complex-numbers/basics/main.png" style="float: left; width: 100px; height: 100px;">This page will allow you to become confident in the basic principles of complex numbers. It is important to understand, and be able to use, the three different forms of a complex number: Cartesian, Polar and Euler. You will learn about the Argand diagram, how useful the complex conjugate is and also how to make good use of the properties of the modulus and argument of a complex number.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn to</p> <ul> <li>Understand that the number <span class="math-tex">\(i\)</span>, where <span class="math-tex">\(i^2=-1\)</span></li> <li>Use the notation for <ul> <li>real part, <span class="math-tex">\(Re(z)\)</span></li> <li>imaginary part, <span class="math-tex">\(Im(z)\)</span></li> <li>conjugate, <span class="math-tex">\(z^*\)</span></li> <li>modulus, <span class="math-tex">\(|z|\)</span></li> <li>argument, <span class="math-tex">\(arg(z)\)</span></li> </ul> </li> <li>Use the Argand diagram to represent the complex plane</li> <li>Use and convert between the three forms of a complex number <ul> <li>Cartesian, <span class="math-tex">\(z=a+bi\)</span></li> <li>Polar, <span class="math-tex">\(z=r(cos\theta+isin\theta)=rcis\theta\)</span></li> <li>Euler, <span class="math-tex">\(z=re^{i\theta}\)</span></li> </ul> </li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/basics/revision_notes-complex_numbers_the_basics.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/complex-numbers/basics/revision_notes-complex_numbers_the_basics.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz about Cartesian Form, Polar Form and Euler Form of a complex number</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#73cf353f"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="73cf353f"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Complex numbers - Cartesian, Polar and Euler Form <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-361-547" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Which of the following is the correct Cartesian Form for the complex number, z</p><p><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q1" style="width: 300px; height: 281px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(2(cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } )\)</span></label></p><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(2(cos\frac { 2\pi }{ 3 } +isin\frac {2 \pi }{ 3 } )\)</span></label></p><p><label class="radio"><input class="c" type="radio"> z = -1 + i<span class="math-tex">\(\sqrt{3}\)</span></label></p><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(-\sqrt{3}+i\)</span></label></p></div><div class="q-explanation"><p>z = <span class="math-tex">\(2(cos\frac {2 \pi }{ 3 } +isin\frac {2 \pi }{ 3 } )\)</span> is the correct answer in <strong>Polar Form</strong></p><p>It must be written in the <strong><em>z = a + ib</em></strong></p><p><strong><em><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q1a" style="width: 300px; height: 281px;"></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Which of the following is the correct Polar Form for the complex number, z</p><p><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q2" style="width: 300px; height: 281px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(3\sqrt { 2 } (cos\frac { -\pi }{ 4 } +isin\frac { -\pi }{ 4 } )\)</span></label></p><p><label class="radio"><input type="radio"> z = -3 - 3i</label></p><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(3(cos\frac { -3\pi }{ 4 } +isin\frac { -3\pi }{ 4 } )\)</span></label></p><p><label class="radio"><input class="c" type="radio"> z = <span class="math-tex">\(3\sqrt { 2 } (cos\frac { -3\pi }{ 4 } +isin\frac { -3\pi }{ 4 } )\)</span></label></p></div><div class="q-explanation"><p><label class="radio">z = -3 - 3i is the correct answer in <strong>Cartesian Form</strong></label></p><p>|z| = <span class="math-tex">\(\sqrt{(-3)^2+(-3)^2}=3\sqrt{2}\)</span></p><p>Arg(z) = <span class="math-tex">\(-\frac{3\pi}{4}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Which of the following is the correct Euler Form for the complex number, z</p><p><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q3" style="width: 300px; height: 281px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(4(cos\frac { -\pi }{ 6 } +isin\frac { -\pi }{ 6 } )\)</span></label></p><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(4(cos\frac { -\pi }{ 3 } +isin\frac { -\pi }{ 3 } )\)</span></label></p><p><label class="radio"><input class="c" type="radio"> z = <span class="math-tex">\(4e^{i(-\frac{\pi}{6}) }\)</span></label></p><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(4e^{i(-\frac{\pi}{3}) }\)</span></label></p></div><div class="q-explanation"><p>Arg(z) = <span class="math-tex">\(arctan(\frac{-2}{2\sqrt{3}})\)</span></p><p>|z| = <span class="math-tex">\(\sqrt{(-2)^2+(2\sqrt{3})^2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Which of the following Argand diagrams represents the complex number, z = <span class="math-tex">\(5e^{i(-\frac{\pi}{2})}\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p style="text-align: center;"><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q4a" style="width: 300px; height: 282px;"></p><p style="text-align: center;">A</p></td><td><p style="text-align: center;"><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q4b" style="width: 300px; height: 282px;"></p><p style="text-align: center;">B</p></td></tr><tr><td><p style="text-align: center;"><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q4c" style="width: 300px; height: 282px;"></p><p style="text-align: center;">C</p></td><td><p style="text-align: center;"><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q4d" style="width: 300px; height: 282px;"></p><p style="text-align: center;">D</p></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><hr class="hidden-separator"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><hr class="hidden-separator"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> C</label></p><hr class="hidden-separator"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>Arg(z) = <span class="math-tex">\(-\frac{\pi}{2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Which of the following is true about the complex number z = <span class="math-tex">\(-4\sqrt{3}-4i\)</span>?</p><p>Find <strong><u>all</u></strong><em> correct answers</em></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> |z| = 8</label></p><p><label class="checkbox"><input type="checkbox"> z* = <span class="math-tex">\(4\sqrt{3}+4i\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> arg(z) = <span class="math-tex">\(\frac { 7\pi }{ 6 } \)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> Arg(z) = <span class="math-tex">\(-\frac { 5\pi }{ 6 } \)</span></label></p></div><div class="q-explanation"><p><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q5a" style="width: 300px; height: 281px;"></p><p>Remember that Arg(z) is the principal argument, <span class="math-tex">\(-\pi<arg(z)></arg(z)></span></p><p>z* = <span class="math-tex">\(-4\sqrt{3}+4i\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Which of the following is true about the complex number z</p><p><span class="math-tex">\(z=2{ e }^{ -\frac { 3\pi }{ 4 } }\)</span></p><p>Find <strong><u>all</u></strong> of the correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> z* = <span class="math-tex">\(-\sqrt { 2 } +\sqrt { 2 } i\)</span></label></p><p><label class="checkbox"><input type="checkbox"> z = <span class="math-tex">\(\sqrt { 2 } -\sqrt { 2 } i\)</span></label></p><p><label class="checkbox"><input type="checkbox"> Arg(z) = <span class="math-tex">\(\frac{5\pi}{4}\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> arg(z) = <span class="math-tex">\(\frac{5\pi}{4}\)</span></label></p></div><div class="q-explanation"><p>z* is the complex conjugate of z</p><p>Arg(z) represents the principal argument, therefore -<span class="math-tex">\(\pi\)</span><span class="math-tex">\(<\)</span> Arg(z) <span class="math-tex">\(\leqslant\)</span> <span class="math-tex">\(\pi\)</span><label class="checkbox">Arg(z) <span class="math-tex">\(\neq \frac{5\pi}{4}\)</span></label> , but as arg(z) is not the principal argument arg(z) =<span class="math-tex">\(\frac{5\pi}{4}\)</span></p><p><img alt="" src="../../files/algebra/complex-numbers/properties-of-mod-and-arg/quiz/q6a1" style="width: 250px; height: 234px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z is a complex number such that</p><p>Arg(z) = <span class="math-tex">\(-\frac{3\pi}{4}\)</span></p><p>|z| = <span class="math-tex">\(5\sqrt{2}\)</span></p><p><em>If z = <strong>a</strong> + i</em><strong><em>b </em></strong>, work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><em><strong>a </strong></em> = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span></p><p><strong><em>b </em></strong>= <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q7" style="width: 300px; height: 282px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z is a complex number such that</p><p>|z| = <span class="math-tex">\(4\sqrt{3}\)</span></p><p>Re(z) = -6</p><p>0 < Arg(z) < <span class="math-tex">\(\pi\)</span></p><hr class="hidden-separator"><p>If Arg(z) = <span class="math-tex">\(\frac{a}{b}\pi\)</span> , find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(arccos(\frac{-6}{4\sqrt{3}})=\frac{5}{6}\pi\)</span></p><p><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q8" style="width: 300px; height: 282px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z is a complex number such that</p><p>|z| = 10</p><p>Im(z) = -5</p><p><span class="math-tex">\(-\frac{\pi}{2}<Arg(z)<\frac{\pi}{2}\)</span></p><p>If Arg(z*) = <span class="math-tex">\(\frac{a}{b}\pi\)</span>, find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q9" style="width: 300px; height: 268px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z is a complex number such that <strong><em>iz = -3 -4i</em></strong></p><p>If <strong><em>z = a + ib</em></strong> , find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Multiplying a complex number by <strong><em>i</em></strong> represents a 90° anticlockwise rotation.</p><p>Hence, to find <strong><em>z</em></strong>, we need to perform a 90° clockwise rotation on the complex number <strong><em>-3 -4i</em></strong></p><p><strong><em><img alt="" src="../../files/algebra/complex-numbers/different-forms-of-complex-numbers/quiz/q10" style="width: 300px; height: 257px;"></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz about the properties of of the modulus and arugment of a complex number</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#c752620b"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="c752620b"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Complex Numbers - Mod and Arg <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-362-547" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z is a complex number</p><p>|z| = <span class="math-tex">\(\sqrt{3}\)</span></p><p>Work out |z<sup>6</sup>|</p></div><div class="q-answer"><p>|z<sup>6</sup>| = <input type="text" style="height: auto;" data-c="27"> <span class="review"></span></p></div><div class="q-explanation"><p>|z<sup>6</sup>| = <span class="math-tex">\(|z|^6\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z and w are complex numbers</p><p>|z| = <span class="math-tex">\(\sqrt{2}\)</span> and |w| = <span class="math-tex">\(3\sqrt{2}\)</span></p><p>Work out |zw|</p></div><div class="q-answer"><p>|zw| = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>|zw| = |z|<span class="math-tex">\(\times\)</span>|w| = <span class="math-tex">\(\sqrt{2}\times3\sqrt{2}=6\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z and w are complex numbers</p><p>arg(z) = 80° and arg(w) = 120°</p><p>Work out Arg(zw) in degrees</p></div><div class="q-answer"><p>Arg(zw) = <input type="text" style="height: auto;" data-c="-160"> <span class="review"></span></p></div><div class="q-explanation"><p>arg(zw) = arg(z)+arg(w) = 80° + 120° = 200°</p><p>Remember that the principal argument for -180°<span class="math-tex">\(<\)</span> Arg(zw) <span class="math-tex">\(\leqslant\)</span> 180°</p><p>Arg(zw) = -160°</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z is a complex number</p><p>arg(z) = <span class="math-tex">\(\frac{\pi}{3}\)</span></p><p>If arg(z<sup>n</sup>) = <span class="math-tex">\(-\frac{\pi}{3}\)</span> and n>0, what is the smallest possible value of n</p></div><div class="q-answer"><p>n = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(-\frac{\pi}{3}=\frac{5\pi}{3}\)</span></p><p>arg(z<sup>n</sup>) = 5arg(z)</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z is a complex number</p><p>arg(z) = <span class="math-tex">\(\frac{5\pi}{8}\)</span></p><p>If Re(z<sup>n</sup>) = 0 and n>0, find the smallest value of n</p></div><div class="q-answer"><p>n = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>arg(z²) = <span class="math-tex">\(\frac{10\pi}{8}\)</span></p><p>arg(z<sup>3</sup>) = <span class="math-tex">\(\frac{15\pi}{8}\)</span></p><p>arg(z<sup>4</sup>) = <span class="math-tex">\(\frac{20\pi}{8}\)</span> = <span class="math-tex">\(\frac{5\pi}{2}\)</span></p><p><img alt="" src="../../files/algebra/complex-numbers/properties-of-mod-and-arg/quiz/q5.png" style="width: 250px; height: 241px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Which of the following is true about the complex numbers z and w</p><p>z = 3 + 3i and w = <span class="math-tex">\(4(cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } )\)</span></p><p>Find <strong><u>all</u></strong> of the correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> arg(zw) = <span class="math-tex">\(\frac{7\pi}{12}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> |zw| = <span class="math-tex">\(3\sqrt{2}+4\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> w* = 2 - 2<span class="math-tex">\(\sqrt{3}i\)</span></label></p><p><label class="checkbox"><input type="checkbox"> [z*| = 3</label></p></div><div class="q-explanation"><p>arg(z) = <span class="math-tex">\(\frac{\pi}{4}\)</span></p><p>arg(w) = <span class="math-tex">\(\frac{\pi}{3}\)</span></p><p>arg(zw) = <span class="math-tex">\(\frac{\pi}{4}+\frac{\pi}{3}\)</span></p><hr class="hidden-separator"><p>|z| = <span class="math-tex">\(3\sqrt{2}\)</span></p><p>|w| = 4</p><p>|zw| = <span class="math-tex">\(3\sqrt{2}\times4=12\sqrt{2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z and w are complex numbers</p><p>z = <span class="math-tex">\(2{ e }^{ i\frac { \pi }{ 2 } }\)</span> and w = -1 - i</p><p>Arg(zw) = <span class="math-tex">\(\frac{a}{b}\pi\)</span></p><p>Work out <em><strong>a</strong></em> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>arg(z) = <span class="math-tex">\(\frac{\pi}{2}\)</span></p><p>arg(w) = <span class="math-tex">\(-\frac{3\pi}{4}\)</span></p><p>arg(zw) = <span class="math-tex">\(\frac{\pi}{2}+(-\frac{3\pi}{4})=-\frac{\pi}{4}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z and w are complex numbers</p><p>z = -i and arg(w) = <span class="math-tex">\(\frac{\pi}{3}\)</span></p><p>Arg(<span class="math-tex">\(\frac{z}{w*}\)</span>) = <span class="math-tex">\(\frac{a}{b}\pi\)</span></p><p>Work out <em><strong>a</strong></em> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>arg(z) = <span class="math-tex">\(-\frac{\pi}{2}\)</span></p><p>arg(w*) = <span class="math-tex">\(-\frac{\pi}{3}\)</span></p><p>arg(<span class="math-tex">\(\frac{z}{w*}\)</span>) = <span class="math-tex">\(-\frac{\pi}{2}-(-\frac{\pi}{3})=-\frac{\pi}{6}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z is a complex number</p><p>Im(z<sup>5</sup>) = 0</p><p>Which of the following is definitely FALSE?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Arg(z) = 90°</label></p><p><label class="radio"><input type="radio"> Arg(z) = 36°</label></p><p><label class="radio"><input type="radio"> Arg(z<sup>5</sup>) = 180°</label></p><p><label class="radio"><input type="radio"> Arg(z*) = -72°</label></p></div><div class="q-explanation"><p>If Im(z<sup>5</sup>) = 0, then Arg(z<sup>5</sup>) = 0° or 180°</p><p>arg(z<sup>5</sup>) = ..., -360°, -180°, 0°,180°, 360°, ...</p><p>arg(z) = ..., -72°, -36°, 0°,36°, 72, ...</p><p>arg(z*) = ..., 72°, 36°, 0°,-36°, -72, ...</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> z and w are complex numbers</p><p>z = 2(cos20° + i sin20°)</p><p>w = 3e<sup>i(-30°)</sup></p><p><span class="math-tex">\(Im(\frac { { z }^{ 3 } }{ { w }^{ n } } )=0\)</span></p><p>Given that <strong><em>n</em></strong> is positive, find the smallest possible value of <strong><em>n</em></strong></p></div><div class="q-answer"><p><strong><em>n</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation">arg(z) = 20°<p>arg(w) = -30°</p><p><span class="math-tex">\(arg(\frac { { z }^{ 3 } }{ { w }^{ n } } )\)</span> = 60°-(-30n°)</p><hr class="hidden-separator"><p>If <span class="math-tex">\(Im(\frac { { z }^{ 3 } }{ { w }^{ n } } )=0\)</span>, then <span class="math-tex">\(arg(\frac { { z }^{ 3 } }{ { w }^{ n } } )=\)</span>..., -360°, -180°, 0°,180°, 360°, ...</p><hr class="hidden-separator">If n=1, 60+30n = 90<p>If n=2, 60+30n = 120</p><p>If n=3, 60+30n = 150</p><p>If n=4, 60+30n = 180</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="831"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the possible values of <em><strong>a</strong></em>, given that <span class="math-tex">\(|\frac{z_1}{z_2}|=2\)</span> and <span class="math-tex">\(z_1=a+2i\)</span> and <span class="math-tex">\(z_2=1-2i\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Find <span class="math-tex">\(\frac{z_1}{z_2}\)</span></p> <p>To make the denominator real, multiply the numerator and denominator by the <strong>complex conjugate </strong>of <span class="math-tex">\(z_2\)</span> <content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/basics/esq1_complex_basics.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/basics/esq1_complex_basics.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="832"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>It is given that <span class="math-tex">\(z_1=2+3i\)</span> and <span class="math-tex">\(z_2=4+ai\)</span></p> <p>Find <strong><em>a</em></strong> if <span class="math-tex">\(Im(z_1z_2^*)=0\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>Work out <span class="math-tex">\(z_1z_2^*\)</span> and find the imaginary part.</p> <p>Solve where this equals zero.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/basics/esq2_complex_basics.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/basics/esq2_complex_basics.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="833"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(z=a+bi\)</span></p> <p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong> if <span class="math-tex">\(z^2=\left|z\right|^2-4\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content><span class="math-tex">\(z^2=(a+bi)(a+bi)\)</span></p> <p>Use Pythagoras' Theorem to find the modulus of a complex number:</p> <p><span class="math-tex">\(|z^2|=a^2+b^2\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/basics/esq3_complex_basics.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/basics/esq3_complex_basics.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="834"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Show that <span class="math-tex">\((1-\frac{1}{3} e^{2iθ})(1-\frac{1}{3} e^{-2iθ})=\frac{1}{9}(10-6\cos 2θ)\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>sinx is an odd function: sin(-x) = -sinx<br> <content> </content>cosx is an even function: cos(-x) = cosx</p> <p><img alt="" src="../../files/algebra/complex-numbers/basics/esq6_hint1" style="width: 300px; height: 104px;"></p> <p><img alt="" src="../../files/algebra/complex-numbers/basics/esq6_hint2" style="width: 300px; height: 104px;"></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/basics/esq4_complex_basics.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/basics/esq4_complex_basics.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="835"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Given that z is a complex number and <span class="math-tex">\(\frac{3z-4}{5}=\frac{p-2i}{3-i}\)</span>, where <span class="math-tex">\(p \in\mathbb{R} \)</span></p> <p>a) Express <em>z</em> in the form <span class="math-tex">\(a+bi\)</span>, <span class="math-tex">\(a,b\in\mathbb{R}\)</span></p> <p>Given that <span class="math-tex">\(arg(z)=-\frac{\pi}{2}\)</span></p> <p>b) Find the value of <strong><em>p</em></strong></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content>If <span class="math-tex">\(arg(z)=-\frac{\pi}{2}\)</span>, then <span class="math-tex">\(Re(z)=0\)</span></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/basics/esq5_complex_basics.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/basics/esq5_complex_basics.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="836"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Given that <span class="math-tex">\(z=a+bi\)</span>, show that <span class="math-tex">\(z+\frac{1}{z}=2cos\theta\)</span></p> <p>b) Hence, show that <span class="math-tex">\(z^n+\frac{1}{z^n}=2cosn\theta\)</span></p> <p>c) Use the Binomial expansion to expand <span class="math-tex">\((z+{1\over z})^5\)</span></p> <p>d) Hence, show that <span class="math-tex">\(cos^5 θ={1 \over 16}cos5θ+{5 \over 16} cos3θ+{5\over 8} cosθ\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) sinx is an odd function, so sin(-x) = -sinx</p> <p>cosx is an even function, so cos(-x) = cosx</p> <p><img alt="" src="../../files/algebra/complex-numbers/basics/esq6_hint1" style="width: 300px; height: 104px;"></p> <p><img alt="" src="../../files/algebra/complex-numbers/basics/esq6_hint2" style="width: 300px; height: 104px;"></p> <hr class="hidden-separator"> <p>b) <span class="math-tex">\(arg(z^n )=nθ\)</span> , so <span class="math-tex">\(z^n=cos(nθ)+isin(nθ)\)</span></p> <hr class="hidden-separator">d) <span class="math-tex">\((z+{1\over z})^5=(2cos\theta)^5\)</span> <p><content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/basics/esq6_complex_basics.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/basics/esq6_complex_basics.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="547"> <div 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