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"showFeedbackIcon": true, "mustBeReduced": false, "mustBeReducedPC": 0, "variableReplacements": [], "marks": 1, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "maxValue": "ans3+tol", "scripts": {}}, {"type": "numberentry", "minValue": "arg3-tol", "showFeedbackIcon": true, "mustBeReduced": false, "mustBeReducedPC": 0, "variableReplacements": [], "marks": 1, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "maxValue": "arg3+tol", "scripts": {}}], "showFeedbackIcon": true}, {"prompt": "\n      <p>$|\\var{z4}|=\\;\\;$[[0]], $\\arg(\\var{z4})=\\;\\;$[[1]] radians</p>\n        <p>Input both answers to 3 decimal places.</p>\n      ", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "gaps": [{"type": "numberentry", "minValue": "ans4-tol", "showFeedbackIcon": true, "mustBeReduced": false, "mustBeReducedPC": 0, "variableReplacements": [], "marks": 1, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "maxValue": "ans4+tol", "scripts": {}}, {"type": "numberentry", "minValue": "arg4-tol", "showFeedbackIcon": true, "mustBeReduced": false, "mustBeReducedPC": 0, "variableReplacements": [], "marks": 1, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "maxValue": "arg4+tol", "scripts": {}}], "showFeedbackIcon": true}], "statement": "<p>Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between <strong>$-\\pi$ and $\\pi$</strong>. If your answer is outside this range, simply add or subtract $2\\pi$ to your answer.</p>\n<p>When calculating the argument pay particular attention to the quadrant in which the complex number lies.</p>\n<p>Input all answers to 3 decimal places.</p>", "advice": "<p>Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi &lt; \\theta \\le \\pi$.</p>\n<p>You have to be careful with using a standard calculator when you are finding the argument of a complex number.</p>\n<p>If $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.</p>\n<p>Using a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi &lt; \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants &ndash; you get the correct value.</p>\n<p>However, The calculator gives the wrong value for complex numbers in the other quadrants.</p>\n<h4>Complex number in the Second Quadrant.</h4>\n<p>Since $\\arctan(b/a)$ does not distinguish between the second and fourth quadrants and the calculator gives the argument for the fourth quadrant you have to add $\\pi$ onto the calculator value.</p>\n<h4>Complex number in the Third Quadrant.</h4>\n<p>Since $\\arctan(b/a)$ does not distinguish between the first and third quadrants and the calculator gives the argument for the first quadrant you have to take away $\\pi$ from the calculator value.</p>\n<h4>a)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z1}|&amp;=&amp;\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &amp;=&amp; \\var{abs(z1)}\\\\ &amp;=&amp;\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m1}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z1}) &amp;=&amp; \\var{arg(z1)}\\\\ &amp;=&amp; \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>b)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z2}|&amp;=&amp;\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &amp;=&amp; \\var{abs(z2)}\\\\ &amp;=&amp;\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m2}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z2}) &amp;=&amp; \\var{arg(z2)}\\\\ &amp;=&amp; \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>c)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z3}|&amp;=&amp;\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &amp;=&amp; \\var{abs(z3)}\\\\ &amp;=&amp;\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m3}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &amp;=&amp; \\var{arg(z3)}\\\\ &amp;=&amp; \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>d)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z4}|&amp;=&amp;\\sqrt{(\\var{a3})^2+(\\var{b3})^2}\\\\ &amp;=&amp; \\var{abs(z4)}\\\\ &amp;=&amp;\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m4}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z4}) &amp;=&amp; \\var{arg(z4)}\\\\ &amp;=&amp; \\var{arg4}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>", "variables": {"arg1": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "arg1", "definition": "precround(arg(z1),3)"}, "m4": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "m4", "definition": "switch(t=1,q1,t=2,q3,t=3,q2,q4)"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "f", "definition": "random(1..9)"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "s2", "definition": "switch(t=1,-1,t=3,-1,1)"}, "s8": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "s8", "definition": 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"a3", "s8", "a4", "z4", "z5", "z6", "z1", "z2", "c4", "f", "n", "a2", "t", "c2"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p><span style=\"color: #222222; font-family: sans-serif;\"><span style=\"line-height: 20px;\">Finding&nbsp;</span></span>the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!</p>"}, "functions": {}, "name": "Complex Numbers: Modulus, Argument", "extensions": [], "type": "question", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}