$|\\var{z1}|=\\;\\;$[[0]], $\\arg(\\var{z1})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg1+tol", "minValue": "arg1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$|\\var{z2}|=\\;\\;$[[0]], $\\arg(\\var{z2})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg2+tol", "minValue": "arg2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$|\\var{z3}|=\\;\\;$[[0]], $\\arg(\\var{z3})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg3+tol", "minValue": "arg3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$|\\var{z4}|=\\;\\;$[[0]], $\\arg(\\var{z4})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans4+tol", "minValue": "ans4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg4+tol", "minValue": "arg4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\\pi$ and $\\pi$.
\nWhen calculating the argument pay particular attention to the quadrant in which the complex number lies.
\nInput all answers to 3 decimal places.
", "tags": ["arctan", "arg", "argument", "argument of complex numbers", "checked2015", "complex number", "complex numbers", "mas1602", "MAS1602", "mod", "modulus", "modulus argument form", "modulus of complex numbers", "quadrants and complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/7/2015:
\nAdded tags.
\n5/07/2012:
\nAdded tags.
\nChanged some of the grammar in the advice section.
\nQuestion appears to be working correctly.
\nThe presentation in IE on using Test Run is not good.
\n9/07/2012:
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\n13/07/2009:
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", "licence": "Creative Commons Attribution 4.0 International", "description": "Finding the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi < \\theta \\le \\pi$.
\nYou have to be careful with using a standard calculator when you are finding the argument of a complex number.
\nIf $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.
\nUsing a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi < \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants – you get the correct value.
\nHowever, The calculator gives the wrong value for complex numbers in the other quadrants.
\nComplex number in the Second Quadrant.
\nSince $\\arctan(b/a)$ 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.
\nComplex number in the Third Quadrant.
\nSince $\\arctan(b/a)$ 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.
\na)Modulus.
\n\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.
\nArgument.
\n{m1}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\nb)Modulus.
\n\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.
\nArgument.
\n{m2}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\nc)Modulus.
\n\\[ \\begin{eqnarray*} |\\var{z3}|&=&\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &=& \\var{abs(z3)}\\\\ &=&\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.
\nArgument.
\n{m3}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &=& \\var{arg(z3)}\\\\ &=& \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\nd)Modulus.
\n\\[ \\begin{eqnarray*} |\\var{z4}|&=&\\sqrt{(\\var{a3})^2+(\\var{b3})^2}\\\\ &=& \\var{abs(z4)}\\\\ &=&\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.
\nArgument.
\n{m4}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z4}) &=& \\var{arg(z4)}\\\\ &=& \\var{arg4}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}