a)
The formula for multiplying complex numbers is
\\[\\begin{eqnarray*}\\simplify[]{Re((a + ib)(c + id))} &=& ac -bd \\\\ \\simplify[]{Im((a + ib)(c + id))} &=& ad +bc \\end{eqnarray*} \\]
So we have:
\\[\\begin{eqnarray*}\\simplify[]{Re({a}*{b})} &=& \\simplify[]{{Re(a)}*{Re(b)} - {Im( a)}*{Im(b)} = {Re(a*b)}}\\\\ \\simplify[]{Im({a}*{b})} &=& \\simplify[]{{Re(a)}*{Im(b)} + {Im( a)}*{Re(b)} = {Im(a*b)}} \\end{eqnarray*} \\]
Hence the solution is :
\\[(\\simplify[std]{{a}})(\\simplify[std]{{b}})=\\var{a*b}\\]
b)
This is calculated in a similar way once the expression is written as:
\n$(\\simplify[std]{{a1}})^2= (\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})$ then we find:
\n\\[\\begin{eqnarray*}(\\simplify[std]{{a1}})^2&=& (\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})\\\\ &=& \\simplify[]{({Re(a1)}*{Re(a1)} - {Im(a1)}*{Im(a1)})+ ({Re(a1)}*{Im(a1)} + {Im(a1)}*{Re(a1)})i}\\\\ &=& \\simplify[std]{{a1^2}} \\end{eqnarray*} \\]
c)
We know that $i^2=-1$ which gives $i^3=i^2i=-i$.
Hence:
\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}&=&\\simplify[std]{{a3} + {b3} * i -{c3} -({d3} * i)}\\\\ &=&\\simplify[std]{ {a3} -{c3} + ({b3} -{d3}) * i}\\\\ &=&\\simplify[std]{{a3 -c3} + {b3 -d3} * i} \\end{eqnarray*} \\]
d)
This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses,
and then to multiply the result of that calculation by the third set of parentheses.
So we obtain:
\\[ \\begin{eqnarray*} (\\var{z1})(\\var{z2})(\\var{z3})&=&((\\var{z1})(\\var{z2}))(\\var{z3})\\\\ &=&(\\var{z1*z2})(\\var{z3})\\\\ &=&\\var{z1*z2*z3} \\end{eqnarray*} \\]
$(\\simplify[std]{{a}})(\\simplify[std]{{b}})\\;=\\;$[[0]].
\n\n
\n ", "gaps": [{"notallowed": {"message": "
Do not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.
", "showstrings": false, "strings": [".", "(", ")"], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 1.0, "answer": "{a*b}", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "$(\\simplify[std]{{a1}})^2\\;=\\;$[[0]].
", "gaps": [{"notallowed": {"message": "Do not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.
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", "gaps": [{"notallowed": {"message": "\nDo not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.
\n\n \n ", "showstrings": false, "strings": [".", ")", "("], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 1.0, "answer": "{z1*z2*z3}", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "\n
Express the following in the form $a+bi\\;$ where $a$ and $b$ are real.
\nDo not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.
\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(1..9)+s2*random(1..9)*i", "name": "a"}, "b": {"definition": "s3*random(1..9)+s4*random(1..9)*i", "name": "b"}, "f6": {"definition": "s4*random(1..4)", "name": "f6"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "z1": {"definition": "s3*random(1..9)+f6*i", "name": "z1"}, "s6": {"definition": "random(1,-1)", "name": "s6"}, "s4": {"definition": "random(1,-1)", "name": "s4"}, "a1": {"definition": "s1*random(1..9)+ s4*random(1..9)*i", "name": "a1"}, "d6": {"definition": "s1*random(1..4)", "name": "d6"}, "a3": {"definition": "s1*random(1..9)", "name": "a3"}, "b3": {"definition": "s2*random(1..9)", "name": "b3"}, "c3": {"definition": "s3*random(1..9)", "name": "c3"}, "e6": {"definition": "s3*random(1..4)", "name": "e6"}, "d3": {"definition": "s4*random(1..9)", "name": "d3"}, "z2": {"definition": "s2*random(1..9)+d6*i", "name": "z2"}, "z3": {"definition": "s6*random(1..9)+e6*i", "name": "z3"}}, "metadata": {"notes": "\n \t\t \t\t4/07/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\t16/07/2012:
\n \t\t \t\tAdded forbidden strings and warnings about not including decimal points or brackets in the answers as otherwise can just repeat the question and be marked correct.
\n \t\t \n \t\t", "description": "Elementary examples of multiplication and powers of complex numbers. Four parts.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Complex Numbers_2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["complex numbers", "conjugate of a complex number", "division of complex numbers", "inverse of complex numbers", "mas104220122013CBA1_2", "multiplication of complex numbers"], "advice": "\n \n \nDivision 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.
", "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": "{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n$\\displaystyle \\simplify[std]{{c2}/{z2}}\\;=\\;$[[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": "{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}], "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": []}]}, {"name": "Complex Numbers_3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["addition of complex numbers", "algebra of complex numbers", "complex numbers", "mas104220122013CBA1_3", "multiplication of complex numbers"], "advice": "\na)
\nThe solution is given by:
\n
$\\simplify[std]{{e6*i}}(\\simplify[std]{{a}})=\\simplify[std]{{a*e6*i}}$
b)
$\\simplify[std]{{a}*{z4}={a*z4}}$
\n
c)
\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}&=&\\simplify[std]{{a}*{a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3}}\\\\ &=&\\simplify[std]{{a*(a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3)}} \\end{eqnarray*} \\]
d)
This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses,
\nand then to multiply the result of that calculation by the third set of parentheses.
\nSo we obtain:
\\[ \\begin{eqnarray*} (\\var{a})(\\var{z1})(\\var{z3})&=&((\\var{a})(\\var{z1}))(\\var{z3})\\\\ &=&(\\var{a*(z1)})(\\var{z3})\\\\ &=&\\var{a*(z1)*(z3)} \\end{eqnarray*} \\]
$\\var{e6*i}(\\simplify[std]{{a}})\\;=\\;$[[0]].
", "gaps": [{"notallowed": {"message": "Input all numbers as fractions or integers. Also do not include brackets in your answers.
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", "gaps": [{"notallowed": {"message": "Input all numbers as fractions or integers. Also do not include brackets in your answers.
", "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": "({a*z4})", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "$\\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}\\;=\\;$[[0]].
", "gaps": [{"notallowed": {"message": "Input all numbers as fractions or integers. Also do not include brackets in your answers.
", "showstrings": false, "strings": [".", ")", "("], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 1.0, "answer": "{(a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3)*a}", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "$(\\simplify[std]{{a}})(\\simplify[std]{ {z1}})(\\simplify[std]{ {z3}})\\;=\\;$[[0]].
", "gaps": [{"notallowed": {"message": "Input all numbers as fractions or integers. Also do not include brackets in your answers.
", "showstrings": false, "strings": [".", ")", "("], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 1.0, "answer": "{a*(z1)*(z3)}", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "\nFind the following complex numbers in the form $a+bi\\;$ where $a$ and $b$ are real.
\nInput all numbers as fractions or integers. Also do not include brackets in your answers.
\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(1..9)+s2*random(1..9)*i", "name": "a"}, "f6": {"definition": "s6*random(1..9)", "name": "f6"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "z1": {"definition": "s3*random(1..9)+f6*i", "name": "z1"}, "s6": {"definition": "random(1,-1)", "name": "s6"}, "s5": {"definition": "random(1,-1)", "name": "s5"}, "s4": {"definition": "random(1,-1)", "name": "s4"}, "d6": {"definition": "s4*random(1..9)", "name": "d6"}, "a3": {"definition": "s1*random(1..9)", "name": "a3"}, "b3": {"definition": "s2*random(1..9)", "name": "b3"}, "c3": {"definition": "s3*random(1..9)", "name": "c3"}, "e6": {"definition": "s5*random(3..9)", "name": "e6"}, "z4": {"definition": "s6*s2*random(1..9)+s3*s5*random(1..9)*i", "name": "z4"}, "d3": {"definition": "s4*random(1..9)", "name": "d3"}, "z2": {"definition": "s2*random(1..9)+d6*i", "name": "z2"}, "z3": {"definition": "s6*random(1..9)+e6*i", "name": "z3"}}, "metadata": {"notes": "\n \t\t4/7/2012
\n \t\tAdded tags
\n \t\tSame problem in Complex Numbers_2. Question 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. This is an issue on github.
16/07/2012:
Issue above resolved. Also forbid decimals and brackets. Questions cannot now be answered by simply repeating the expression.
\n \t\tAdded formal cba name mas104220122013CBA1_3 as a tag
\n \t\t", "description": "Multiplication 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": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}