// Numbas version: exam_results_page_options {"name": "Complex numbers Ex Sheet 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Complex numbers Ex Sheet 4", "preamble": {"css": "", "js": ""}, "parts": [{"gaps": [{"expectedvariablenames": [], "vsetrange": [0, 1], "checkvariablenames": false, "showFeedbackIcon": true, "answersimplification": "std", "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "notallowed": {"partialCredit": 0, "message": "

Make sure that you input the real and imaginary parts as fractions and not as decimals

", "showStrings": false, "strings": ["."]}, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "vsetrangepoints": 5, "answer": "{round((z1*conj(z1))^c1)}", "showpreview": true, "variableReplacements": [], "type": "jme"}], "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "prompt": "

\\[z=(\\var{z1})^{\\var{c1}}(\\var{conj(z1)})^{\\var{c1}}\\]
$z=\\;\\;$[[0]]

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\\[z=(\\var{z2})^4\\]
$z=\\;\\;$[[0]]

Make sure that you input the real and imaginary parts as fractions and not as decimals

", "showStrings": false, "strings": ["."]}, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "vsetrangepoints": 5, "answer": "{re(conj(z3^d1))}/{abs(z3^(2*d1))}+({im(conj(z3^d1))}/{round(abs(z3)^(2*d1))})*i", "showpreview": true, "variableReplacements": [], "type": "jme"}], "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "prompt": "

\\[z=(\\var{z3})^{\\var{-d1}}\\]
$z=\\;\\;$[[0]]

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\\[z=i^{\\var{n}}\\]
$z=\\;\\;$[[0]]

", "type": "gapfill", "showCorrectAnswer": true}], "variables": {"z1": {"name": "z1", "definition": "s1*random(1..5)+s3*random(1..5)*i", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "d1": {"name": "d1", "definition": "random(2,3)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "rem": {"name": "rem", "definition": "random(0..3)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "definition": "random(1,-1)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "m": {"name": "m", "definition": "random(2000..5000)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "definition": "random(3..5)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "z3": {"name": "z3", "definition": "s2*random(1..5)+s3*random(1..5)*i", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "definition": "s2*random(1..5)+s4*random(1..5)*i", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "n": {"name": "n", "definition": "4*m+rem", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "definition": "random(1,-1)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "s4": {"name": "s4", "definition": "random(1,-1)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "definition": "random(1,-1)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}}, "functions": {}, "statement": "

Express the following complex numbers $z$ in the form $a+bi$.

Input $a$ and $b$ as fractions and not as decimals.

", "advice": "

Note that for a complex number $z=a+bi$ we have:

$z\\overline{z}=|z|^2=a^2+b^2$.

But since $\\var{conj(z1)}=\\overline{\\var{z1}}$ we have:
\\[\\begin{eqnarray*}z&=&(\\var{z1})^{\\var{c1}}(\\var{conj(z1)})^{\\var{c1}}\\\\ &=&((\\var{z1})(\\var{conj(z1)}))^{\\var{c1}}\\\\ &=&\\simplify[]{({re(z1)}^2+{im(z1)}^2)^{c1}}\\\\ &=&\\var{(re(z1)^2+im(z1)^2)^c1} \\end{eqnarray*}\\]

Note that $(\\var{z2})^4=((\\var{z2})^2)^2$.

Since $(\\var{z2})^2=\\simplify[std]{{z2^2}}$ we have:
\\[(\\var{z2})^4=(\\simplify[std]{{z2^2}})^2=\\simplify[std]{{z2^4}}\\]

c)
We have
\\[ \\begin{eqnarray*} z&=&(\\var{z3})^{\\var{-d1}}\\\\ &=&\\frac{1}{(\\var{z3})^{\\var{d1}}}\\\\ &=&\\frac{(\\var{conj(z3)})^{\\var{d1}}}{(\\var{z3})^{\\var{d1}}(\\var{conj(z3)})^{\\var{d1}}}\\\\ &=&\\frac{\\var{conj(z3)^d1}}{\\var{abs(z3)^(2*d1)}}\\\\ &=&\\simplify[std]{{re(conj(z3^d1))}/{(abs(z3)^(2*d1))}+({im(conj(z3^d1))}/{round(abs(z3)^(2*d1))})*i} \\end{eqnarray*}\\]
d)
We have $i^2=-1,\\;\\;i^3=-i,\\;\\;i^4=1$.

So if $n=4m+r,\\;\\;0\\le r\\le 3$ we have \\[i^n=i^{4m+r}=(i^4)^m \\times i^r=i^r\\]
Hence since $\\var{n}=4\\times\\var{m}+\\var{rem}$ we have:
\\[i^{\\var{n}}=i^{\\var{rem}}=\\simplify{{i^rem}}\\]

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Direct calculation of low positive and negative powers of complex numbers. Calculations involving a complex conjugate. Powers of $i$. Four parts.

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