// Numbas version: exam_results_page_options
{"name": "Paul's copy of Arithmetics of complex numbers I", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Paul's copy of Arithmetics of complex numbers I", "tags": ["addition of complex numbers", "checked2015", "complex numbers", "multiplication of complex numbers", "product of complex numbers"], "metadata": {"description": "<p>Elementary examples of multiplication and addition of complex numbers. Four parts.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Express the following in the form $x+iy\\;$ where $x$ and $y$ are real numbers.</p>\n<p>Write your answers using only integers or fractions, not decimals. &nbsp;To input, e.g., $2+3i$, type 2+3i putting the i last.</p>\n<p>To input, e.g., $\\frac{3}{10}+\\frac{4}{5}i$, type 3/10+4/5i or (3/10)+(4/5)i. &nbsp;There should be no brackets in your answer except to indicate fractions as here.</p>", "advice": "<p>a) &nbsp;</p>\n<p>Multiplying two complex numbers and extracting their real and imaginary parts, we find:</p>\n<p>\\[\\begin{eqnarray*}\\simplify[]{Re((x_1 + iy_1)(x_2 + iy_2))} &amp;=&amp; x_1 x_2 -y_1y_2 \\\\ \\simplify[]{Im((x_1 + iy_1)(x_2 + iy_2))} &amp;=&amp; x_1 y_2 +y_1x_2 \\end{eqnarray*} \\]</p>\n<p>So we have:<br/>\\[\\begin{eqnarray*}\\simplify[]{Re({a}*{b})} &amp;=&amp; \\simplify[]{{Re(a)}*{Re(b)} - {Im( a)}*{Im(b)} = {Re(a*b)}}\\\\ \\simplify[]{Im({a}*{b})} &amp;=&amp; \\simplify[]{{Re(a)}*{Im(b)} + {Im( a)}*{Re(b)} = {Im(a*b)}} \\end{eqnarray*} \\]<br/>Hence the solution is :</p>\n<p>\\[(\\simplify[std]{{a}})(\\simplify[std]{{b}})=\\var{a*b}\\]<br/>b)</p>\n<p>This is calculated in a similar way once the expression is written as:</p>\n<p>$(\\simplify[std]{{a1}})^2=&nbsp;(\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})$ then we find:</p>\n<p>\\[\\begin{eqnarray*}(\\simplify[std]{{a1}})^2&amp;=&amp;&nbsp;(\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})\\\\ &amp;=&amp; \\simplify[]{({Re(a1)}*{Re(a1)} - {Im(a1)}*{Im(a1)})+ ({Re(a1)}*{Im(a1)} + {Im(a1)}*{Re(a1)})i}\\\\ &amp;=&amp; \\simplify[std]{{a1^2}} \\end{eqnarray*} \\]<br/>c) <br/>We know that $i^2=-1$ which gives $i^3=i^2i=-i$.</p>\n<p>Hence:<br/>\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}&amp;=&amp;\\simplify[std]{{a3} + {b3} * i -{c3} -({d3} * i)}\\\\ &amp;=&amp;\\simplify[std]{ {a3} -{c3} + ({b3} -{d3}) * i}\\\\ &amp;=&amp;\\simplify[std]{{a3 -c3} + {b3 -d3} * i} \\end{eqnarray*} \\]<br/>d) <br/>This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses, <br/>and then to multiply the result of that calculation by the third set of parentheses.</p>\n<p>So we obtain:<br/>\\[ \\begin{eqnarray*} (\\var{z1})(\\var{z2})(\\var{z3})&amp;=&amp;((\\var{z1})(\\var{z2}))(\\var{z3})\\\\ &amp;=&amp;(\\var{z1*z2})(\\var{z3})\\\\ &amp;=&amp;\\var{z1*z2*z3} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "d3": {"name": "d3", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", 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variables", "definition": "s3*random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "s6": {"name": "s6", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "s6*random(1..9)+e6*i", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": 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Also do not include brackets in your answers.</span> </span></span></p>"}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$(\\simplify[std]{{a1}})^2\\;=\\;$[[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a1^2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", ")", "("], "showStrings": false, "partialCredit": 0, "message": "<p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>"}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}\\;=\\;$[[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", ")", "("], "showStrings": false, "partialCredit": 0, "message": "<p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>"}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$(\\simplify[std]{{z1}}) (\\simplify[std]{{z2}}) (\\simplify[std]{{z3}})\\;=\\;$[[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{z1*z2*z3}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", ")", "("], "showStrings": false, "partialCredit": 0, "message": "\n            <p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>\n            <p>&nbsp;</p>\n            "}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Paul McFadden", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18299/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Paul McFadden", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18299/"}]}