// Numbas version: finer_feedback_settings
{"name": "Polar form of a complex number", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"absz": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z),3)", "description": "", "name": "absz"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)*sign(random(-1,1))", "description": "", "name": "a"}, "argz": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z),3)", "description": "", "name": "argz"}, "z": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+b*i", "description": "", "name": "z"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5 except a)*sign(random(-1,1))", "description": "", "name": "b"}}, "ungrouped_variables": ["a", "argz", "b", "absz", "tol", "z"], "name": "Polar form of a complex number", "functions": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "<p>$r=$ [[0]] (Enter your answer to 3 d.p.)</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "absz-tol", "maxValue": "absz+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "<p>$\\theta=$ [[0]] (Enter your answer to 3 d.p.)</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "argz-tol", "maxValue": "argz+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>Write the complex number $z=\\var{z}$ in polar form $z=r\\mathrm{e}^{i\\theta}$, with $r&gt;0$, $-\\pi&lt;\\theta\\leqslant\\pi$, by calculating $r$ and $\\theta$.</p>", "tags": ["checked2015"], "rulesets": {}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Polar form of a complex number.</p>"}, "advice": "<p>To write a complex number $z=a+bi$ in polar form $z=r\\mathrm{e}^{i\\theta}$, we calculate the modulus $r = \\lvert z \\rvert$ and argument $\\theta = \\arg(z)$.</p>\n<p>Hence</p>\n<p>\\[r=\\lvert z \\rvert=\\sqrt{a^2+b^2}=\\sqrt{(\\var{a})^2+(\\var{b})^2}=\\var{absz}\\;\\text{to 3d.p.}\\]</p>\n<p>and, in general,</p>\n<p>\\[\\theta=\\arg(z)=\\arctan\\left(\\frac{b}{a}\\right).\\]</p>\n<p>If $a=0$, however, then $\\mathrm{Re}(z)=0$, so $\\arg(z)=\\pm\\frac{\\pi}{2}$, depending on whether $\\mathrm{Im}(z)$ is positive or negative.</p>\n<p>In this case $a=\\var{a}$, and $b=\\var{b}$, so $\\arg(z)=\\var{argz}$.</p>", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}