$(\\simplify[std]{{a}})(\\simplify[std]{{b}})\\;=\\;$[[0]].
\n\n
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Do not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.
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", "scripts": {}, "gaps": [{"answer": "{z1*z2*z3}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "\nDo not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.
\n\n ", "showStrings": false, "partialCredit": 0, "strings": [".", ")", "("]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "
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.
", "tags": ["addition of complex numbers", "checked2015", "complex numbers", "mas1602", "MAS1602", "multiplication of complex numbers", "product of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/07/2015:
\nAdded tags.
\n4/07/2012:
\nAdded tags.
\n16/07/2012:
\nAdded 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.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Elementary examples of multiplication and addition of complex numbers. Four parts.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "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*} \\]