Division of two complex numbers can be performed by mutiplying both the numerator and denominator by the conjugate of the denominator.
Suppose that \\[ z = \\frac{a+bi}{c+di},\\;\\; c+di \\neq 0\\] then we have:
\\[\\begin{eqnarray*}\n \n z&=&\\frac{a+bi}{c+di}\\\\\n \n &=&\\frac{(a+bi)(c-di)}{(c+di)(c-di)}\\\\\n \n &=&\\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}\\\\\n \n &=&\\frac{ac+bd}{c^2+d^2}+\\frac{bc-ad}{c^2+d^2}i\n \n \\end{eqnarray*}\n \n \\]
Although this is a formula for the inverse, the best way to find these complex numbers is to remember to multiply top and bottom by the conjugate of the denominator.
(a)
\\[\\begin{eqnarray*}\\simplify[std]{{c1}/{z1}} &=&\\simplify[std]{({c1}*{conj(z1)})/({z1}*{conj(z1)})}\\\\\n \n &=&\\simplify[std]{{c1*conj(z1)}/{abs(z1)^2}}\\\\\n \n &=& \\simplify[std]{{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i}\n \n \\end{eqnarray*} \\]
(b)
\\[\\begin{eqnarray*}\\simplify[std]{{c2}/{z2}} &=&\\simplify[std]{({c2}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n \n &=&\\simplify[std]{{c2*conj(z2)}/{abs(z2)^2}}\\\\\n \n &=& \\simplify[std]{{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i}\n \n \\end{eqnarray*} \\]
(c)
\\[\\begin{eqnarray*}\\simplify[std]{{z1}/{z3}} &=&\\simplify[std]{({z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\\n \n &=&\\simplify[std]{{z1*conj(z3)}/{abs(z3)^2}}\\\\\n \n &=& \\simplify[std]{{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i}\n \n \\end{eqnarray*} \\]
(d)
\\[\\begin{eqnarray*}\\simplify[std]{{z3}/{z2}} &=&\\simplify[std]{({z3}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n \n &=&\\simplify[std]{{z3*conj(z2)}/{abs(z2)^2}}\\\\\n \n &=& \\simplify[std]{{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i}\n \n \\end{eqnarray*} \\]
$\\displaystyle \\simplify[std]{{c1}/{z1}}\\;=\\;$[[0]].
\nDo not include brackets in your answer.
\n\n ", "gaps": [{"notallowed": {"message": "
Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.
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\nDo not include brackets in your answer.
\n ", "gaps": [{"notallowed": {"message": "Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.
", "showstrings": false, "strings": [".", "(", ")"], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n$\\displaystyle \\simplify[std]{{z1}/{z3}}\\;=\\;$[[0]].
\nDo not include brackets in your answer.
\n ", "gaps": [{"notallowed": {"message": "Make sure that you input the real and imaginary parts as fractions and not as decimals. Also do not include brackets in your answer.
", "showstrings": false, "strings": [".", "(", ")"], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n$\\displaystyle \\simplify[std]{{z3}/{z2}}\\;=\\;$[[0]].
\nDo not include brackets in your answer.
\n ", "gaps": [{"notallowed": {"message": "Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.
", "showstrings": false, "strings": [".", "(", ")"], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\nExpress the following in the form $a+bi$.
\nInput $a$ and $b$ as fractions or integers and not as decimals.
\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "s4": {"definition": "random(1,-1)", "name": "s4"}, "a3": {"definition": "s3*random(1..9)", "name": "a3"}, "rz3": {"definition": "if(a3=re(z1),a3+random(1,-1),a3)", "name": "rz3"}, "c2": {"definition": "random(1..5)", "name": "c2"}, "c1": {"definition": "s3*random(1..9)", "name": "c1"}, "z1": {"definition": "s2*random(1..9)+s1*random(1..9)*i", "name": "z1"}, "z2": {"definition": "re(z1)+s2*random(1,2)+s4*random(1..9)*i", "name": "z2"}, "z3": {"definition": "rz3+s1*random(1..9)*i", "name": "z3"}}, "metadata": {"notes": "\n \t\t4/07/2012:
\n \t\tAdded tags
\n \t\tQuestion a - sometimes the complex number is generated as a/(b+i*c) but sometimes the complex number is displayed as a decimal, i.e. 0.0975609756+0.1219512195i if this happens then the question is invalid.
\n \t\t16/07/2012:
\n \t\tThe above issue has been resolved,
\n \t\tAlso forbid brackets in the answers as otherwise can repeat the question and be marked as correct.
\n \t\t0.0975609756+0.1219512195i
\n \t\t\n \t\t", "description": "
Inverse and division of complex numbers. Four parts.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}