// Numbas version: finer_feedback_settings
{"name": "Graham's copy of Complex Numbers: Modulus, Argument", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["ans1", "ans2", "ans3", "ans4", "b4", "b1", "b2", "b3", "d4", "d2", "q1", "q3", "q2", "q4", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "m4", "m1", "m3", "arg1", "z3", "arg2", "arg3", "tol", "arg4", "m2", "a1", "a3", "s8", "a4", "z4", "z5", "z6", "z1", "z2", "c4", "f", "n", "a2", "t", "c2"], "name": "Graham's copy of Complex Numbers: Modulus, Argument", "tags": ["arctan", "arg", "argument", "argument of complex numbers", "complex number", "complex numbers", "mas104220122013CBA3_1", "mod", "modulus", "modulus argument form", "modulus of complex numbers", "quadrants and complex numbers"], "preamble": {"css": "", "js": ""}, "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>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>$|\\var{z1}|=\\;\\;$[[0]], $\\arg(\\var{z1})=\\;\\;$[[1]] radians</p>\n        <p>Input both answers to 3 decimal places.</p>\n      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans1+tol", "minValue": "ans1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "arg1+tol", "minValue": "arg1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      <p>$|\\var{z2}|=\\;\\;$[[0]], $\\arg(\\var{z2})=\\;\\;$[[1]] radians</p>\n        <p>Input both answers to 3 decimal places.</p>\n      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "arg2+tol", "minValue": "arg2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      <p>$|\\var{z3}|=\\;\\;$[[0]], $\\arg(\\var{z3})=\\;\\;$[[1]] radians</p>\n        <p>Input both answers to 3 decimal places.</p>\n      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans3+tol", "minValue": "ans3-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "arg3+tol", "minValue": "arg3-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      <p>$|\\var{z4}|=\\;\\;$[[0]], $\\arg(\\var{z4})=\\;\\;$[[1]] radians</p>\n        <p>Input both answers to 3 decimal places.</p>\n      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans4+tol", "minValue": "ans4-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "arg4+tol", "minValue": "arg4-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n  <p>Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\\pi$ and $\\pi$.</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>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "variables": {"ans1": {"definition": "precround(abs(z1),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "ans2": {"definition": "precround(abs(z2),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2", "description": ""}, "ans3": {"definition": "precround(abs(z3),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans3", "description": ""}, "ans4": {"definition": "precround(abs(z4),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans4", "description": ""}, "b4": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b4", "description": ""}, "b1": {"definition": "s2*random(3..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "s5*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "b3": {"definition": "s8*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "d4": {"definition": "s5*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d4", "description": ""}, "d2": {"definition": "s7*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "q1": {"definition": "'The complex number is in the first quadrant.'", "templateType": "anything", "group": "Ungrouped variables", "name": "q1", "description": ""}, "q3": {"definition": "'The complex number is in the third quadrant.'", "templateType": "anything", "group": "Ungrouped variables", "name": "q3", "description": ""}, "q2": {"definition": "'The complex number is in the second quadrant.'", "templateType": "anything", "group": "Ungrouped variables", "name": "q2", "description": ""}, "q4": {"definition": "'The complex number is in the fourth quadrant.'", "templateType": "anything", "group": "Ungrouped variables", "name": "q4", "description": ""}, "s3": {"definition": "switch(t=1,1,t=2,-1,t=3,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "switch(t=1,-1,t=3,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "switch(t=1,1,t=4,1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s7": {"definition": "switch(t=2,-1,t=3,1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s7", "description": ""}, "s6": {"definition": "switch(t=1,-1,t=4,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s6", "description": ""}, "s5": {"definition": "switch(t=3,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "switch(t=1,-1,t=4,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}, "m4": {"definition": "switch(t=1,q1,t=2,q3,t=3,q2,q4)", "templateType": "anything", "group": "Ungrouped variables", "name": "m4", "description": ""}, "m1": {"definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "templateType": "anything", "group": "Ungrouped variables", "name": "m1", "description": ""}, "m3": {"definition": "switch(t=1,q3,t=2,q4,t=3,q1,q3)", "templateType": "anything", "group": "Ungrouped variables", "name": "m3", "description": ""}, "arg1": {"definition": "precround(arg(z1),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "arg1", "description": ""}, "c4": {"definition": "if(a4=f,f+1,f)", "templateType": "anything", "group": "Ungrouped variables", "name": "c4", "description": ""}, "arg2": {"definition": "precround(arg(z2),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "arg2", "description": ""}, "arg3": {"definition": "precround(arg(z3),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "arg3", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "arg4": {"definition": "precround(arg(z4),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "arg4", "description": ""}, "m2": {"definition": "switch(t=1,q2,t=2,q1,t=3,q4,q2)", "templateType": "anything", "group": "Ungrouped variables", "name": "m2", "description": ""}, "a1": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a3": {"definition": "s3*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "s8": {"definition": "switch(t=1,1,t=4,-1,t=3,1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s8", "description": ""}, "a4": {"definition": "s8*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a4", "description": ""}, "z4": {"definition": "a3+b3*i", "templateType": "anything", "group": "Ungrouped variables", "name": "z4", "description": ""}, "z5": {"definition": "a4+b4*i", "templateType": "anything", "group": "Ungrouped variables", "name": "z5", "description": ""}, "z6": {"definition": "c4+d4*i", "templateType": "anything", "group": "Ungrouped variables", "name": "z6", "description": ""}, "z1": {"definition": "a1+b1*i", "templateType": "anything", "group": "Ungrouped variables", "name": "z1", "description": ""}, "z2": {"definition": "a2+b2*i", "templateType": "anything", "group": "Ungrouped variables", "name": "z2", "description": ""}, "z3": {"definition": "c2+d2*i", "templateType": "anything", "group": "Ungrouped variables", "name": "z3", "description": ""}, "f": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "n": {"definition": "random(3..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "a2": {"definition": "s4*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "t": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "c2": {"definition": "s6*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>5/07/2012:</strong></p>\n    \t\t                <p>Added tags.</p>\n    \t\t                <p>Changed some of the grammar in the advice section.</p>\n    \t\t                <p>Question appears to be working correctly.</p>\n    \t\t                <p>The presentation in IE on using Test Run is not good.</p>\n    \t\t                <p><strong>9/07/2012:</strong></p>\n    \t\t                <p>Display in Advice set out properly.</p>\n    \t\t                <p><strong>13/07/2009:</strong></p>\n    \t\t                <p>Set new tolerance variable tol=0.001 for all numeric input.</p>\n    \t\t", "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>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Graham Wynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/602/"}]}]}], "contributors": [{"name": "Graham Wynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/602/"}]}