// Numbas version: finer_feedback_settings
{"name": "Modulus and argument of a complex number II", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Modulus and argument of a complex number II", "tags": ["checked2015"], "metadata": {"description": "<p>Modulus and argument of a single complex number $z=z_1/z_2$, where $\\mathrm{Re}(z_1)=\\mathrm{Im}(z_1)$ and $\\mathrm{Re}(z_2)=-\\mathrm{Im}(z_2)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the modulus $\\lvert z \\rvert$ and argument $\\theta$ (with $-\\pi&lt;\\theta\\leqslant\\pi$) of the complex number</p>\n  <p>\\[\\var{z}=\\frac{\\var{z1}}{\\var{z2}}.\\]</p>", "advice": "<h4>a)</h4>\n<p>For a complex number $z=\\frac{a+ai}{b+bi}$, the modulus is given by</p>\n<p>\\[\\lvert z \\rvert=\\frac{\\sqrt{a^2+a^2}}{\\sqrt{b^2+b^2}}=\\frac{a\\sqrt{2}}{b\\sqrt{2}}=\\frac{a}{b}\\]</p>\n<p>In this part $a=\\var{a}$ and $b=\\var{b}$, so $\\lvert z \\rvert=\\frac{\\var{a}}{\\var{b}}=\\var{absz}$ to 3 d.p.</p>\n<h4>b)</h4>\n<p>To calculate the argument of a complex number $z=\\frac{a+ai}{b-bi}$, with $a&gt;0$, $b&gt;0$, first write $z$ in the form $z=c+di$.</p>\n<p>To do this, multiply&nbsp;$z=\\frac{a+ai}{b-bi}$ by&nbsp;$\\frac{b+bi}{b+bi}$, so that</p>\n<p>\\[z=\\frac{(a+ai)(b+bi)}{(b-bi)(b+bi)}=\\frac{2abi}{2b^2}=\\frac{a}{b}i\\]</p>\n<p>So, given that $a$ and $b$ are both positive, $\\mathrm{Im}(z)&gt;0$, and because $\\mathrm{Re}(z)=0$, the argument is $\\theta=\\frac{\\pi}{2}$.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"absz": {"name": "absz", "group": "Ungrouped variables", "definition": "precround(abs(z1/z2),3)", "description": "", "templateType": "anything", 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