// Numbas version: finer_feedback_settings
{"name": "Exprese lo siguiente en la forma $ a + bi \\; $ donde $ a $ y $ b $ son reales.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Exprese lo siguiente en la forma $ a + bi \\; $ donde $ a $ y $ b $ son reales.", "tags": [], "metadata": {"description": "<p>Ejemplos elementales de multiplicaci&oacute;n y suma de n&uacute;meros complejos.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Exprese lo siguiente en la forma $ a + bi \\; $ donde $ a $ y $ b $ son reales.</p>\n<p>No incluya decimales en sus respuestas, solo fracciones o enteros. Tampoco incluya corchetes en sus respuestas. recuerde que $i^2=-1$</p>", "advice": "<p>a)<br/>La f&oacute;rmula para multiplicar n&uacute;meros complejos es<br/>\\[\\begin{eqnarray*}\\simplify[]{Re((a + ib)(c + id))} &amp;=&amp; ac -bd \\\\ \\simplify[]{Im((a + ib)(c + id))} &amp;=&amp; ad +bc \\end{eqnarray*} \\]</p>\n<p>Entonces tenemos:<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>Esto se calcula de manera similar una vez que la expresi&oacute;n se escribe como:</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/>Sabemos que $ i^2 = -1 $ lo que da $ i^3 = i^2 i = -i$.</p>\n<p>Por lo tanto:<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)</p>\n<p>Esto se puede calcular usando la f&oacute;rmula dos veces, primero para multiplicar los dos primeros conjuntos de par&eacute;ntesis,<br/>y luego multiplicar el resultado de ese c&aacute;lculo por el tercer conjunto de par&eacute;ntesis.</p>\n<p>As&iacute; obtenemos:</p>\n<p>\\[ \\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": [], "variables": {"d3": {"name": "d3", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything"}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "s6*random(1..9)+e6*i", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "s2*random(1..9)+d6*i", "description": "", "templateType": "anything"}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "s3*random(1..9)+f6*i", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "f6": {"name": "f6", "group": "Ungrouped variables", "definition": "s4*random(1..4)", "description": "", "templateType": "anything"}, "s6": {"name": "s6", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "s1*random(1..9)+ s4*random(1..9)*i", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "e6": {"name": "e6", "group": "Ungrouped variables", "definition": "s3*random(1..4)", "description": "", "templateType": "anything"}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "s1*random(1..9)+s2*random(1..9)*i", "description": "", "templateType": "anything"}, "d6": {"name": "d6", "group": "Ungrouped variables", "definition": "s1*random(1..4)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s3*random(1..9)+s4*random(1..9)*i", "description": "", "templateType": "anything"}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "f6", "s3", "s2", "s1", "d3", "s6", "s4", "a1", "c3", "a3", "b3", "d6", "e6", "z1", "z2", "z3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", 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"vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": [".", "(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p><span class=\"rwRO\" style=\"color: #3f52b8; font-family: Tahoma, 'Sans Serif', Arial; font-size: 11px; background-color: #fdf2c4;\"><span class=\"sq\" style=\"color: #9a9a9a;\"><a style=\"font-family: Verdana, Arial, Helvetica, sans-serif; font-size: medium; background-color: #ffffff;\" href=\"../../32/complex-numbers/\">D</a><span style=\"color: #000000; font-family: Verdana, Arial, Helvetica, sans-serif; font-size: medium; background-color: #ffffff;\">o not include decimals in your answers, only fractions or integers. 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\\;=\\;$[[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a1^2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": [".", ")", "("], "showStrings": false, "partialCredit": 0, "message": "<p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>"}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}\\;=\\;$[[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": [".", ")", "("], "showStrings": false, "partialCredit": 0, "message": "<p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>"}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$(\\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, "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", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}