// Numbas version: finer_feedback_settings
{"name": "Complex Numbers; Multiplication 01 (Rect')", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Complex Numbers; Multiplication 01 (Rect')", "tags": [], "metadata": {"description": "Multiplication of complex numbers (in rectangular, Cartesian form)", "licence": "All rights reserved"}, "statement": "<h4>Let $z$ and $w$ be any two complex numbers</h4>\n<h4 style=\"text-align: center;\">$&nbsp; &nbsp;z=a+bi&nbsp; &nbsp;$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; and&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;$&nbsp; &nbsp;w=c+di&nbsp; &nbsp;$&nbsp;</h4>\n<h4 style=\"text-align: left;\">Then&nbsp;</h4>\n<h4 style=\"text-align: center;\">$z w = (a + bi)(c + di)$</h4>\n<h4 style=\"text-align: center;\">$z w = ac-bd+(ad+bc)i$</h4>", "advice": "<h4>We are asked to evaluate the following multiplications of complex numbers.</h4>\n<p>The technique should be very familiar. It is exactly the same as multiplying out two brackets.</p>\n<p>The workings below probably include more steps than really neccesary once you get used to doing it.</p>\n<h4>Given:</h4>\n<p>$\\large z_1=\\var{z1}$&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;&nbsp;$\\large z_2=\\var{z2}$</p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_1z_2=(\\var{z1})(\\var{z2})$&nbsp;</p>\n<p style=\"text-align: center;\">$\\large z_1z_2=(\\var{r1})(\\var{r2})+(\\var{r1})(\\var{i2})i+(\\var{i1})(\\var{r2})i+(\\var{i1})(\\var{i2})i^2$</p>\n<p style=\"text-align: center;\">$\\large z_1z_2=\\simplify{{r1}{r2}}+\\simplify{{r1}{i2}}i+\\simplify{{i1}{r2}}i+\\simplify{{i1}{i2}}i^2$</p>\n<p style=\"text-align: center;\"><strong></strong>$\\large z_1z_2=\\simplify{{r1}{r2}}+\\simplify{{r1}{i2}+{i1}{r2}}i+\\simplify{{i1}{i2}}i^2$</p>\n<p style=\"text-align: left;\">Now here is the extra \"twist\" that comes from doing this with complex numbers. Remember that $i=\\sqrt{-1}$ so $i^2=-1$:</p>\n<p style=\"text-align: center;\">$\\large z_1z_2=\\simplify{{r1}{r2}}+\\simplify{{r1}{i2}+{i1}{r2}}i+\\simplify{{i1}{i2}i^2}$</p>\n<p style=\"text-align: center;\">$\\large z_1z_2=\\simplify{{z1}*{z2}}$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\"></p>\n<hr/>\n<h4>Given:</h4>\n<p>$\\large z_3=\\var{z3}$&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;&nbsp;$\\large z_4=\\var{z4}$</p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_3z_4=(\\var{z3})(\\var{z4})$&nbsp;</p>\n<p style=\"text-align: center;\">$\\large z_3z_4=(\\var{r3})(\\var{r4})+(\\var{r3})(\\var{i4})i+(\\var{i3})(\\var{r4})i+(\\var{i3})(\\var{i4})i^2$</p>\n<p style=\"text-align: center;\">$\\large z_3z_4=\\simplify{{r3}{r4}}+\\simplify{{r3}{i4}}i+\\simplify{{i3}{r4}}i+\\simplify{{i3}{i4}}i^2$</p>\n<p style=\"text-align: center;\"><strong></strong>$\\large z_3z_4=\\simplify{{r3}{r4}}+\\simplify{{r3}{i4}+{i3}{r4}}i+\\simplify{{i3}{i4}}i^2$</p>\n<p style=\"text-align: left;\">Remember that $i=\\sqrt{-1}$ so $i^2=-1$:</p>\n<p style=\"text-align: center;\">$\\large z_3z_4=\\simplify{{r3}{r4}}+\\simplify{{r3}{i4}+{i3}{r4}}i+\\simplify{{i3}{i4}i^2}$</p>\n<p style=\"text-align: center;\">$\\large z_1z_2=\\simplify{{z1}*{z2}}$</p>\n<hr/>\n<h4>Given:</h4>\n<p>$\\large z_5=\\var{z5}$&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;&nbsp;$\\large z_6=\\var{cc1}$</p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_5z_6=(\\var{z5})(\\var{cc1})$</p>\n<p style=\"text-align: center;\">$\\large z_5z_6=(\\var{r5})(\\var{rcc1})+(\\var{r5})(\\var{icc1})i+(\\var{i5})(\\var{rcc1})i+(\\var{i5})(\\var{icc1})i^2$</p>\n<p style=\"text-align: center;\">$\\large z_5z_6=\\simplify{{r5}{rcc1}}+\\simplify{{r5}{icc1}}i+\\simplify{{i5}{rcc1}}i+\\simplify{{i5}{icc1}}i^2$</p>\n<p style=\"text-align: center;\">$\\large z_5z_6=\\simplify{{r5}{rcc1}}+\\simplify{{r5}{icc1}+{i5}{rcc1}}i+\\simplify{{i5}{icc1}}i^2$</p>\n<p style=\"text-align: left;\">Remember that $i=\\sqrt{-1}$ so $i^2=-1$:</p>\n<p style=\"text-align: center;\">$\\large z_5z_6=\\simplify{{r5}{rcc1}}+\\simplify{{r5}{icc1}+{i5}{rcc1}}i+\\simplify{{i5}{icc1}i^2}$</p>\n<p style=\"text-align: center;\">$\\large z_5z_6=\\simplify{{z5}*{cc1}}$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\"></p>\n<hr/>\n<h4>Given:</h4>\n<p>$\\large z_1=\\var{z1}$&nbsp; &nbsp; and&nbsp; &nbsp; &nbsp; $\\large z_1^*=\\var{cc1}$</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_1z_1^*=(\\var{z1})(\\var{cc1})$</p>\n<p style=\"text-align: center;\">$\\large z_1z_1^*=(\\var{r1})(\\var{rcc1})+(\\var{r1})(\\var{icc1})i+(\\var{i1})(\\var{rcc1})i+(\\var{i1})(\\var{icc1})i^2$</p>\n<p style=\"text-align: center;\">$\\large z_1z_1^*=\\simplify{{r1}{rcc1}}+\\simplify{{r1}{icc1}}i+\\simplify{{i1}{rcc1}}i+\\simplify{{i1}{icc1}}i^2$</p>\n<p style=\"text-align: center;\">$\\large z_1z_1^*=\\simplify{{r1}{rcc1}}+\\simplify{{r1}{icc1}+{i1}{rcc1}}i+\\simplify{{i1}{icc1}}i^2$</p>\n<p style=\"text-align: left;\">Remember that $i=\\sqrt{-1}$ so $i^2=-1$:</p>\n<p style=\"text-align: center;\">$\\large z_1z_1^*=\\simplify{{r1}{rcc1}}+\\simplify{{r1}{icc1}+{i1}{rcc1}}i+\\simplify{{i1}{icc1}i^2}$</p>\n<p style=\"text-align: center;\">$\\large z_1z_1^*=\\simplify{{z1}*{cc1}}$&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\"></p>\n<hr/>\n<h4>Given:</h4>\n<p>$\\large z_2=\\var{z1}$ and $\\large z_2^*=\\var{cc2}$</p>\n<p></p>\n<p></p>\n<p style=\"text-align: center;\">Evaluate:<br/>$\\large z_2z_2^*=(\\var{z2})(\\var{cc2})$</p>\n<p style=\"text-align: center;\">$\\large z_2z_2^*=(\\var{r2})(\\var{rcc2})+(\\var{r2})(\\var{icc2})i+(\\var{i2})(\\var{rcc2})i+(\\var{i2})(\\var{icc2})i^2$</p>\n<p style=\"text-align: center;\">$\\large z_2z_2^*=\\simplify{{r2}{rcc2}}+\\simplify{{r2}{icc2}}i+\\simplify{{i2}{rcc2}}i+\\simplify{{i2}{icc2}}i^2$</p>\n<p style=\"text-align: center;\">$\\large z_2z_2^*=\\simplify{{r2}{rcc2}}+\\simplify{{r2}{icc2}+{i2}{rcc2}}i+\\simplify{{i2}{icc2}}i^2$</p>\n<p style=\"text-align: left;\">Remember that $i=\\sqrt{-1}$ so $i^2=-1$:</p>\n<p style=\"text-align: center;\">$\\large z_2z_2^*=\\simplify{{r2}{rcc2}}+\\simplify{{r2}{icc2}+{i2}{rcc2}}i+\\simplify{{i2}{icc2}i^2}$</p>\n<p style=\"text-align: center;\">$\\large z_2z_2^*=\\simplify{{z2}*{cc2}}$</p>", "rulesets": {}, "extensions": [], "variables": {"r1": {"name": "r1", "group": "CN1", "definition": "random(2..25)", "description": "", "templateType": "anything"}, "i1": {"name": "i1", "group": "CN1", "definition": "random(2..25 except r1)", "description": "", "templateType": "anything"}, "r2": {"name": "r2", "group": "CN2", "definition": "random(2..25 except r1)", "description": "", "templateType": "anything"}, "i2": {"name": "i2", "group": "CN2", "definition": "random(2..25 except r1 except r2)", "description": "", "templateType": "anything"}, "r3": {"name": "r3", "group": "CN3", "definition": "random(-25..25 except 0 except r1 except r2)", "description": "", "templateType": "anything"}, "i3": {"name": "i3", "group": "CN3", "definition": "random(2..25 except r3 except r1 except r2)", "description": "", "templateType": "anything"}, "r4": {"name": "r4", "group": "CN4", "definition": "random(-10..10)", "description": "", "templateType": "anything"}, "i4": {"name": "i4", "group": "CN4", "definition": "random(-10..10 except 0 except r4)", "description": "", "templateType": "anything"}, "r5": {"name": "r5", "group": "CN5", "definition": "random(-10..10 except r4)", "description": "", "templateType": "anything"}, "i5": {"name": "i5", "group": "CN5", "definition": "random(-10..10 except r5)", "description": "", "templateType": "anything"}, "z1": {"name": "z1", "group": "CN1", "definition": "r1+i(i1)", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "group": "CN2", "definition": "r2+i(i2)", "description": "", "templateType": "anything"}, "z3": {"name": "z3", "group": "CN3", "definition": "r3+i(i3)", "description": "", "templateType": "anything"}, "z4": {"name": "z4", "group": "CN4", "definition": "r4+i(i4)", "description": "", "templateType": "anything"}, "z5": {"name": "z5", "group": "CN5", "definition": "r5+i(i5)", "description": "", "templateType": "anything"}, "cc1": {"name": "cc1", "group": "CN1", "definition": "conj(z1)", "description": "", "templateType": "anything"}, "cc2": {"name": "cc2", "group": "CN2", "definition": "conj(z2)", "description": "", "templateType": "anything"}, "cc3": {"name": "cc3", "group": "CN3", "definition": "conj(z3)", "description": "", "templateType": "anything"}, "cc4": {"name": "cc4", "group": "CN4", "definition": "conj(z4)", "description": "", "templateType": "anything"}, "cc5": {"name": "cc5", "group": "CN5", "definition": "conj(z5)", "description": "", "templateType": "anything"}, "rcc1": {"name": "rcc1", "group": "CN1", "definition": "re(cc1)", "description": "", "templateType": "anything"}, "icc1": {"name": "icc1", "group": "CN1", "definition": "im(cc1)", "description": "", "templateType": "anything"}, "rcc2": {"name": "rcc2", "group": "CN2", "definition": "re(cc2)", "description": "", "templateType": "anything"}, "icc2": {"name": "icc2", "group": "CN2", "definition": "im(cc2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "CN1", "variables": ["r1", "i1", "z1", "cc1", "rcc1", "icc1"]}, {"name": "CN2", "variables": ["r2", "i2", "z2", "cc2", "rcc2", "icc2"]}, {"name": "CN3", "variables": ["r3", "i3", "z3", "cc3"]}, {"name": "CN4", "variables": ["r4", "i4", "z4", "cc4"]}, {"name": "CN5", "variables": ["r5", "i5", "z5", "cc5"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<h4>If:</h4>\n<p>$\\large z_1=\\var{z1}$&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;&nbsp;$\\large z_2=\\var{z2}$</p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_1z_2=$ [[0]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\"></p>\n<hr/>\n<h4>If:</h4>\n<p>$\\large z_3=\\var{z3}$ &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp; $\\large z_4=\\var{z4}$</p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_3z_4=$ [[1]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\"></p>\n<hr/>\n<h4>If:</h4>\n<p>$\\large z_5=\\var{z5}$,&nbsp; &nbsp; &nbsp; $\\large z_6=\\var{cc1}$</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_5z_6=$ [[2]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<hr/>\n<h4>If:</h4>\n<p>$\\large z_1=\\var{z1}$&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;&nbsp;$\\large z_1^*=\\var{cc1}$</p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_1z_1^*=$ [[3]]</p>\n<hr/>\n<h4>If:</h4>\n<p>$\\large z_2=\\var{z2}$&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;&nbsp;$\\large z_2^*=\\var{cc2}$</p>\n<p style=\"text-align: left;\"></p>\n<h4 style=\"text-align: left;\">Evaluate:</h4>\n<p style=\"text-align: center;\">$\\large z_1z_1^*=$ [[4]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{z1}*{z2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{z3}*{z4}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{z5}*{cc1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{z1}*{cc1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{z2}*{cc2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}], "type": "question", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}