// Numbas version: finer_feedback_settings {"name": "Complex Arithmetic I & II, Argument and Modulus of Compex Numbers", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "feedback": {"allowrevealanswer": true, "showtotalmark": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "showanswerstate": true, "showactualmark": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "percentPass": 0, "duration": 0, "question_groups": [{"pickingStrategy": "all-ordered", "name": "Group", "pickQuestions": 1, "questions": [{"name": "Complex number arithmetic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}], "functions": {}, "ungrouped_variables": ["z1", "z2", "z3", "z4", "re_z3onz4", "im_z3onz4", "z1plusz2", "z1minusz2", "z1z3", "z1z2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

\\[z_1+z_2=(\\var{z1})+(\\var{z2})=\\simplify{{z1}+{z2}}\\]

\n

\\[z_1-z_2=(\\var{z1})-(\\var{z2})=\\simplify{{z1}-{z2}}\\]

\n

\\[z_1z_3=(\\var{z1})(\\var{z3})=\\var{z1z3}\\]

\n

\\[\\frac{z_3}{z_4}=\\frac{\\var{z3}}{\\var{z4}}=\\frac{(\\var{z3})(\\var{z4bar})}{(\\var{z4})(\\var{z4bar})}=\\simplify{{z3z4bar}/{modz4sq}}\\approx \\var{re_z3onz4}+\\var{im_z3onz4}\\]

", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "

$z_1+z_2=$

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$z_1+z_3=$

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$z_1-z_2=$

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$z_1-z_3=$

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$z_1+z_4=$

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$z_1-z_2=$

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$z_1z_3=$

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$z_1z_2=$

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$z_1z_4=$

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$z_4z_3=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z4*z3}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "

This question will help you to practice adding, subtracting, multiplying and dividing complex numbers in Cartesian/rectangular form. If you wish, you might like to watch this video which explains these operations and how to do problems like these.

\n

\n

Given the complex numbers

\n

\\[z_1=\\var{z1},\\quad z_2=\\var{z2}, \\quad z_3=\\var{z3}, \\quad z_4=\\var{z4},\\]

\n

calculate the following (write your answers in Cartesian/rectangular form).

", "variable_groups": [{"variables": ["z3bar", "z2bar", "z1bar", "z4bar"], "name": "Unnamed group"}, {"variables": ["modz2sq", "modz1sq", "modz3sq", "modz4sq"], "name": "Unnamed group"}, {"variables": ["z3z4bar", "z2z4bar"], "name": "Unnamed group"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"im_z3onz4": {"definition": "im(z3/z4)", "templateType": "anything", "group": "Ungrouped variables", "name": "im_z3onz4", "description": ""}, "z1z3": {"definition": "z1*z3", "templateType": "anything", "group": "Ungrouped variables", "name": "z1z3", "description": ""}, "z1z2": {"definition": "z1*z2", "templateType": "anything", "group": "Ungrouped variables", "name": "z1z2", "description": "

z1z2

"}, "z3z4bar": {"definition": "z3*z4bar", "templateType": "anything", "group": "Unnamed group", "name": "z3z4bar", "description": ""}, "modz2sq": {"definition": "z2*z2bar", "templateType": "anything", "group": "Unnamed group", "name": "modz2sq", "description": ""}, "z1plusz2": {"definition": "z1+z2", "templateType": "anything", "group": "Ungrouped variables", "name": "z1plusz2", "description": ""}, "modz4sq": {"definition": "z4*z4bar", "templateType": "anything", "group": "Unnamed group", "name": "modz4sq", "description": ""}, "z2z4bar": {"definition": "z2*z4bar", "templateType": "anything", "group": "Unnamed group", "name": "z2z4bar", "description": ""}, "z3bar": {"definition": "conj(z3)", "templateType": "anything", "group": "Unnamed group", "name": "z3bar", "description": ""}, "modz1sq": {"definition": "z1*z1bar", "templateType": "anything", "group": "Unnamed group", "name": "modz1sq", "description": ""}, "z4bar": {"definition": "conj(z4)", "templateType": "anything", "group": "Unnamed group", "name": "z4bar", "description": ""}, "modz3sq": {"definition": "z3*z3bar", "templateType": "anything", "group": "Unnamed group", "name": "modz3sq", "description": ""}, "z1bar": {"definition": "conj(z1)", "templateType": "anything", "group": "Unnamed group", "name": "z1bar", "description": ""}, "z2bar": {"definition": "conj(z2)", "templateType": "anything", "group": "Unnamed group", "name": "z2bar", "description": ""}, "z4": {"definition": "random(-10..10)+random(-10..10)i", "templateType": "anything", "group": "Ungrouped variables", "name": "z4", "description": ""}, "re_z3onz4": {"definition": "re(z3/z4)", "templateType": "anything", "group": "Ungrouped variables", "name": "re_z3onz4", "description": ""}, "z1minusz2": {"definition": "z1-z2", "templateType": "anything", "group": "Ungrouped variables", "name": "z1minusz2", "description": ""}, "z1": {"definition": "random(-10..10)+random(-10..10)i", "templateType": "anything", "group": "Ungrouped variables", "name": "z1", "description": ""}, "z2": {"definition": "random(-10..10)+random(-10..10)i", "templateType": "anything", "group": "Ungrouped variables", "name": "z2", "description": ""}, "z3": {"definition": "random(-10..10)+random(-10..10)i", "templateType": "anything", "group": "Ungrouped variables", "name": "z3", "description": ""}}, "metadata": {"description": "

This question provides practice at adding, subtracting, dividing and multiplying complex numbers in rectangular form.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question"}, {"name": "Complex number arithmetic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}], "functions": {}, "ungrouped_variables": ["z1", "z2", "z3", "z4", "re_z3onz4", "im_z3onz4", "z1plusz2", "z1minusz2", "z1z3", "z1z2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

\\[z_1+z_2=(\\var{z1})+(\\var{z2})=\\simplify{{z1}+{z2}}\\]

\n

\\[z_1-z_2=(\\var{z1})-(\\var{z2})=\\simplify{{z1}-{z2}}\\]

\n

\\[z_1z_3=(\\var{z1})(\\var{z3})=\\var{z1z3}\\]

\n

\\[\\frac{z_3}{z_4}=\\frac{\\var{z3}}{\\var{z4}}=\\frac{(\\var{z3})(\\var{z4bar})}{(\\var{z4})(\\var{z4bar})}=\\simplify{{z3z4bar}/{modz4sq}}\\approx \\var{re_z3onz4}+\\var{im_z3onz4}\\]

", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "

$z_1+z_2=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z1+z2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

$z_1+z_3=$

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$z_1-z_2=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z1-z2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

$z_1-z_3=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z1-z3}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

$z_1+z_4=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z1+z4}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

$z_1-z_2=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z1-z2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

$z_1z_3=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z1*z3}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

$z_1z_2=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z1*z2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

$z_1z_4=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z1*z4}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

$z_4z_3=$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{z4*z3}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "

This question will help you to practice adding, subtracting, multiplying and dividing complex numbers in Cartesian/rectangular form. If you wish, you might like to watch this video which explains these operations and how to do problems like these.

\n

\n

Given the complex numbers

\n

\\[z_1=\\var{z1},\\quad z_2=\\var{z2}, \\quad z_3=\\var{z3}, \\quad z_4=\\var{z4},\\]

\n

calculate the following (write your answers in Cartesian/rectangular form).

", "variable_groups": [{"variables": ["z3bar", "z2bar", "z1bar", "z4bar"], "name": "Unnamed group"}, {"variables": ["modz2sq", "modz1sq", "modz3sq", "modz4sq"], "name": "Unnamed group"}, {"variables": ["z3z4bar", "z2z4bar"], "name": "Unnamed group"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"im_z3onz4": {"definition": "im(z3/z4)", "templateType": "anything", "group": "Ungrouped variables", "name": "im_z3onz4", "description": ""}, "z1z3": {"definition": "z1*z3", "templateType": "anything", "group": "Ungrouped variables", "name": "z1z3", "description": ""}, "z1z2": {"definition": "z1*z2", "templateType": "anything", "group": "Ungrouped variables", "name": "z1z2", "description": "

z1z2

"}, "z3z4bar": {"definition": "z3*z4bar", "templateType": "anything", "group": "Unnamed group", "name": "z3z4bar", "description": ""}, "modz2sq": {"definition": "z2*z2bar", "templateType": "anything", "group": "Unnamed group", "name": "modz2sq", "description": ""}, "z1plusz2": {"definition": "z1+z2", "templateType": "anything", "group": "Ungrouped variables", "name": "z1plusz2", "description": ""}, "modz4sq": {"definition": "z4*z4bar", "templateType": "anything", "group": "Unnamed group", "name": "modz4sq", "description": ""}, "z2z4bar": {"definition": "z2*z4bar", "templateType": "anything", "group": "Unnamed group", "name": "z2z4bar", "description": ""}, "z3bar": {"definition": "conj(z3)", "templateType": "anything", "group": "Unnamed group", "name": "z3bar", "description": ""}, "modz1sq": {"definition": "z1*z1bar", "templateType": "anything", "group": "Unnamed group", "name": "modz1sq", "description": ""}, "z4bar": {"definition": "conj(z4)", "templateType": "anything", "group": "Unnamed group", "name": "z4bar", "description": ""}, "modz3sq": {"definition": "z3*z3bar", "templateType": "anything", "group": "Unnamed group", "name": "modz3sq", "description": ""}, "z1bar": {"definition": "conj(z1)", "templateType": "anything", "group": "Unnamed group", "name": "z1bar", "description": ""}, "z2bar": {"definition": "conj(z2)", "templateType": "anything", "group": "Unnamed group", "name": "z2bar", "description": ""}, "z4": {"definition": "random(-10..10)+random(-10..10)i", "templateType": "anything", "group": "Ungrouped variables", "name": "z4", "description": ""}, "re_z3onz4": {"definition": "re(z3/z4)", "templateType": "anything", "group": "Ungrouped variables", "name": "re_z3onz4", "description": ""}, "z1minusz2": {"definition": "z1-z2", "templateType": "anything", "group": "Ungrouped variables", "name": "z1minusz2", "description": ""}, "z1": {"definition": "random(-10..10)+random(-10..10)i", "templateType": "anything", "group": "Ungrouped variables", "name": "z1", "description": ""}, "z2": {"definition": "random(-10..10)+random(-10..10)i", "templateType": "anything", "group": "Ungrouped variables", "name": "z2", "description": ""}, "z3": {"definition": "random(-10..10)+random(-10..10)i", "templateType": "anything", "group": "Ungrouped variables", "name": "z3", "description": ""}}, "metadata": {"description": "

This question provides practice at adding, subtracting, dividing and multiplying complex numbers in rectangular form.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question"}, {"name": "Arithmetics of complex numbers I", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "d3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "name": "d3"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "b3"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)+s4*random(1..9)*i", "description": "", "name": "b"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s4"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)+d6*i", "description": "", "name": "z2"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)+ s4*random(1..9)*i", "description": "", "name": "a1"}, "e6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..4)", "description": "", "name": "e6"}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s6"}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1..9)+e6*i", "description": "", "name": "z3"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a3"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "c3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)+s2*random(1..9)*i", "description": "", "name": "a"}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)+f6*i", "description": "", "name": "z1"}, "f6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..4)", "description": "", "name": "f6"}, "d6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..4)", "description": "", "name": "d6"}}, "ungrouped_variables": ["a", "b", "f6", "s3", "s2", "s1", "d3", "s6", "s4", "a1", "c3", "a3", "b3", "d6", "e6", "z1", "z2", "z3"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "

$(\\simplify[std]{{a}})(\\simplify[std]{{b}})\\;=\\;$[[0]].

\n

 

\n

 

", "scripts": {}, "gaps": [{"answer": "{a*b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "

Do not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.

", "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}, {"prompt": "

$(\\simplify[std]{{a1}})^2\\;=\\;$[[0]].

", "scripts": {}, "gaps": [{"answer": "({a1^2})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "

Do not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.

", "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}, {"prompt": "

$\\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}\\;=\\;$[[0]].

", "scripts": {}, "gaps": [{"answer": "{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "

Do not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.

", "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}, {"prompt": "

$(\\simplify[std]{{z1}}) (\\simplify[std]{{z2}}) (\\simplify[std]{{z3}})\\;=\\;$[[0]].

", "scripts": {}, "gaps": [{"answer": "{z1*z2*z3}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "\n

Do not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.

\n

 

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Express the following in the form $a+bi\\;$ where $a$ and $b$ are real.

\n

Do 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:

\n

Added tags.

\n

4/07/2012:

\n

Added tags.

\n

16/07/2012:

\n

Added 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.

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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*} \\]

\n

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 :

\n

\\[(\\simplify[std]{{a}})(\\simplify[std]{{b}})=\\var{a*b}\\]
b)

\n

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 $i^3=i^2i=-i$.

\n

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.

\n

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*} \\]

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Multiplication and addition of complex numbers. Four parts.

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$\\var{e6*i}(\\simplify[std]{{a}})\\;=\\;$[[0]].

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$(\\simplify[std]{{a}})(\\simplify[std]{{z4}})\\;=\\;$[[0]].

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$\\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}\\;=\\;$[[0]].

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$(\\simplify[std]{{a}})(\\simplify[std]{ {z1}})(\\simplify[std]{ {z3}})\\;=\\;$[[0]].

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Input all numbers as fractions or integers. Also do not include brackets in your answers.

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Find the following complex numbers in the form $a+bi\\;$ where $a$ and $b$ are real.

\n

Input all numbers as fractions or integers. Also do not include brackets in your answers.

", "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

a)

\n

The solution is given by:

\n


$\\simplify[std]{{e6*i}}(\\simplify[std]{{a}})=\\simplify[std]{{a*e6*i}}$

\n


b)

\n

$\\simplify[std]{{a}*{z4}={a*z4}}$

\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)

\n

This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses, 

\n

and then to multiply the result of that calculation by the third set of parentheses.

\n

So 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*} \\]

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Inverse and division of complex numbers.  Four parts.

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$\\displaystyle \\simplify[std]{{c1}/{z1}} = $ [[0]]

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$\\displaystyle \\simplify[std]{{c2}/{z2}}\\;=\\;$[[0]]

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$\\displaystyle \\simplify[std]{{z1}/{z3}}\\;=\\;$[[0]].

\n

Do not include brackets in your answer.

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$\\displaystyle \\simplify[std]{{z3}/{z2}}\\;=\\;$[[0]].

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Express the following in the form $a+bi$.

\n

Input $a$ and $b$ as fractions or integers and not as decimals.

", "tags": ["checked2015", "complex numbers", "conjugate of a complex number", "division of complex numbers", "inverse of complex numbers", "multiplication of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus", "!collectLikeFractions"]}, "preamble": {"css": "", "js": ""}, "type": "question", "advice": "\n \n \n

Division 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*} \\]

\n \n "}, {"name": "Arithmetics of complex numbers IV", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"rz3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a3=re(z1),a3+random(1,-1),a3)", "description": "", "name": "rz3"}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..9)", "description": "", "name": "c1"}, "z1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..9)+s1*random(1..9)*i", "description": "", "name": "z1"}, "a3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..9)", "description": "", "name": "a3"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "z2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "re(z1)+s2*random(1,2)+s4*random(1..9)*i", "description": "", "name": "z2"}, "z3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "rz3+s1*random(1..9)*i", "description": "", "name": "z3"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "description": "", "name": "c2"}}, "ungrouped_variables": ["s3", "s2", "s1", "s4", "a3", "rz3", "c2", "c1", "z1", "z2", "z3"], "functions": {}, "parts": [{"prompt": "\n

\\[\\displaystyle z=\\simplify[!collectNumbers]{({z3}*{z2})/{z1}}\\]

\n

$z=\\;\\;$[[0]].

\n ", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{re(conj(z1)*z3*z2)}/{abs(z1)^2}+{im(conj(z1)*z3*z2)}/{abs(z1)^2}*i", "mustmatchpattern": {"message": "Your answer is not in the form $a+bi$.", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "nameToCompare": ""}, "vsetRangePoints": 5, "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "answerSimplification": "std", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"prompt": "\n

\\[\\displaystyle z=\\simplify[!collectNumbers]{({z2}*{z1})}(\\var{z3})^{-1}\\]

\n

$z=\\;\\;$[[0]].

\n ", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{re(conj(z3)*z1*z2)}/{abs(z3)^2}+{im(conj(z3)*z1*z2)}/{abs(z3)^2}*i", "mustmatchpattern": {"message": "Your answer is not in the form $a+bi$.", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "nameToCompare": ""}, "vsetRangePoints": 5, "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "answerSimplification": "std", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Express the following complex numbers $z$ in the form $a+bi$.

\n

Input $a$ and $b$ as fractions and not as decimals.

", "tags": ["algebra of complex numbers", "checked2015", "complex arithmetic", "complex numbers", "division of complex numbers", "inverse of complex numbers", "multiplication of complex numbers", "product of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus", "!collectlikefractions"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Composite multiplication and division of complex numbers. Two parts.

"}, "advice": "\n

a)
\\[\\begin{eqnarray*}z=\\simplify[!collectNumbers]{({z3}*{z2})/{z1}} &=&\\simplify[!collectNumbers]{({z3}*{z2}*{conj(z1)})/({z1}*{conj(z1)})}\\\\ &=&\\simplify[!collectNumbers]{({z3*z2}*{conj(z1)})/({abs(z1)^2})}\\\\ &=&\\simplify[!collectNumbers]{{z3*z2*conj(z1)}/{abs(z1)^2}}\\\\ &=& \\simplify[std]{{re(z3*z2*conj(z1))}/{abs(z1)^2}+{im(z3*z2*conj(z1))}/{abs(z1)^2}*i} \\end{eqnarray*} \\]

\n

b)
\\[\\begin{eqnarray*}z= \\simplify[!collectNumbers]{({z2}*{z1})}(\\var{z3})^{-1} &=& \\simplify[!collectNumbers]{({z2}*{z1})/{z3}}\\\\ &=&\\simplify[!collectNumbers]{({z2}*{z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\ &=&\\simplify[!collectNumbers]{({z2*z1}*{conj(z3)})/({abs(z3)^2})}\\\\ &=&\\simplify[!collectNumbers]{{z2*z1*conj(z3)}/{abs(z3)^2}}\\\\ &=& \\simplify[std]{{re(z2*z1*conj(z3))}/{abs(z3)^2}+{im(z2*z1*conj(z3))}/{abs(z3)^2}*i} \\end{eqnarray*} \\]

\n "}, {"name": "The modulus of complex numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "name": "tol", "description": ""}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..4)", "name": "b3", "description": ""}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s5", "description": ""}, "c4": {"templateType": "anything", "group": "Ungrouped variables", 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{"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "name": "n", "description": ""}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c2+d2*i", "name": "z3", "description": ""}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*i", "name": "z1", "description": ""}, "z4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3+b3*i", "name": "z4", "description": ""}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z2*z3),3)", "name": "ans2", "description": ""}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z4)^n,3)", "name": "ans3", "description": ""}, "d4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "name": "d4", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "f", "description": ""}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2+b2*i", "name": "z2", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(3..9)", "name": "b1", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "name": "a2", "description": ""}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z1),3)", "name": "ans1", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "name": "b2", "description": ""}, "s8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s8", "description": ""}, "ans4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z5/z6),3)", "name": "ans4", "description": ""}, "z6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c4+d4*i", "name": "z6", "description": ""}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..4)", "name": "a3", "description": ""}, "a4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s8*random(1..9)", "name": "a4", "description": ""}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s3", "description": ""}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s6", "description": ""}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s7", "description": ""}}, "ungrouped_variables": ["ans1", "ans2", "ans3", "ans4", "b4", "b1", "b2", "b3", "d4", "d2", "s8", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "z2", "tol", "c4", "a1", "a3", "a2", "a4", "z4", "z5", "z6", "z1", "c2", "z3", "f", "n"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans1+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "

$|\\var{z1}|=\\;\\;$[[0]]

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$|(\\var{z2})(\\var{z3})|=\\;\\;$[[0]]

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$|(\\var{z4})^{\\var{n}}|=\\;\\;$[[0]]

", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans4+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "

Let \\[z=\\frac{\\var{z5}}{\\var{z6}}\\]
$|z|=\\;\\;$[[0]]

", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Find the modulus of each of the following complex numbers, leaving your answer in decimal form to 3 decimal places:

", "tags": ["checked2015", "complex number", "complex numbers", "division of complex numbers", "mas1602", "MAS1602", "modulus of a complex number", "modulus of complex numbers", "modulus of the division of complex numbers", "modulus of the power of complex numbers", "modulus of the product of complex numbers", "multiplication of complex numbers", "multiply complex numbers", "product of complex numbers", "properties of the modulus of complex numbers", "rationalise the denominator", "rationalising the denominator"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

15/07/2015:

\n

Added tags.

\n

5/07/2012:

\n

Added tags.

\n

Question appears to be working correctly.  

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Finding the modulus of four complex numbers; includes finding the modulus of a product, a power and a quotient.

"}, "advice": "

Recall that $|a+bi|=\\sqrt{a^2+b^2}$ and that:

\n

1. $ |z^n| = |z|^n$

\n

2. $ |z_1z_2|=|z_1|\\;|z_2|$

\n

3. $ |z_1/z_2|=|z_1|/|z_2|$

\n

a) \\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.

\n

b) \\[ \\begin{eqnarray*} |(\\var{z2})(\\var{z3})|&=&|\\var{z2}|\\;|\\var{z3}|\\\\ &=& \\var{abs(z2)}\\times \\var{abs(z3)}\\\\ &=&\\var{abs(z2*z3)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.

\n

c) \\[ \\begin{eqnarray*} |(\\var{z4})^{\\var{n}}|&=&|\\var{z4}|^{\\var{n}}\\\\ &=& \\var{abs(z4)}^{\\var{n}}\\\\ &=& \\var{abs(z4)^n}\\\\ &=&\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.

\n

d) \\[ \\begin{eqnarray*} \\left|\\frac{\\var{z5}}{\\var{z6}}\\right|&=&\\frac{|\\var{z5}|}{|\\var{z6}|}\\\\ &=& \\frac{\\var{abs(z5)}}{\\var{abs(z6)}}\\\\ &=& \\var{abs(z5/z6)}\\\\ &=&\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.

"}, {"name": "Modulus and argument complex numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s8*random(1..9)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=3,-1,1)", "description": "", "name": "s5"}, "arg4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z4),3)", "description": "", "name": "arg4"}, "b4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b4"}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=2,-1,t=3,1,-1)", "description": "", "name": "s7"}, "arg2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z2),3)", "description": "", "name": "arg2"}, "s8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=4,-1,t=3,1,-1)", "description": "", "name": "s8"}, "arg3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z3),3)", "description": "", "name": "arg3"}, "m4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q1,t=2,q3,t=3,q2,q4)", "description": "", "name": "m4"}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c2+d2*i", "description": "", "name": "z3"}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*i", "description": "", "name": "z1"}, "z4": {"templateType": "anything", "group": "Ungrouped variables", 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{"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "f"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2+b2*i", "description": "", "name": "z2"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(3..9)", "description": "", "name": "b1"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "description": "", "name": "m1"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the third quadrant.'", "description": "", "name": "q3"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "name": "s4"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z2),3)", "description": "", "name": "ans2"}, "q4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the fourth quadrant.'", "description": "", "name": "q4"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "a3"}, "a4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s8*random(1..9)", "description": "", "name": "a4"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=2,-1,t=3,-1,1)", "description": "", "name": "s3"}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "name": "s6"}, "arg1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z1),3)", "description": "", "name": "arg1"}}, "ungrouped_variables": ["ans1", "ans2", "ans3", "ans4", "b4", "b1", "b2", "b3", "d4", "d2", "q1", "q3", "q2", "q4", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "m4", "m1", "z2", "m3", "arg1", "z3", "arg2", "arg3", "tol", "arg4", "m2", "a1", "a3", "s8", "a4", "z4", "z5", "z6", "z1", "c2", "c4", "f", "n", "a2", "t"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "

$|\\var{z1}|=\\;\\;$[[0]], $\\arg(\\var{z1})=\\;\\;$[[1]] radians

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Input both answers to 3 decimal places.

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$|\\var{z2}|=\\;\\;$[[0]], $\\arg(\\var{z2})=\\;\\;$[[1]] radians

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Input both answers to 3 decimal places.

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$|\\var{z3}|=\\;\\;$[[0]], $\\arg(\\var{z3})=\\;\\;$[[1]] radians

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Input both answers to 3 decimal places.

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$|\\var{z4}|=\\;\\;$[[0]], $\\arg(\\var{z4})=\\;\\;$[[1]] radians

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Input both answers to 3 decimal places.

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Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\\pi$ and $\\pi$.

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When calculating the argument pay particular attention to the quadrant in which the complex number lies.

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Input all answers to 3 decimal places.

", "tags": ["arctan", "arg", "argument", "argument of complex numbers", "checked2015", "complex number", "complex numbers", "mas1602", "MAS1602", "mod", "modulus", "modulus argument form", "modulus of complex numbers", "quadrants and complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

15/7/2015:

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Added tags.

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5/07/2012:

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Added tags.

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Changed some of the grammar in the advice section.

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Question appears to be working correctly.

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The presentation in IE on using Test Run is not good.

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9/07/2012:

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Display in Advice set out properly.

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13/07/2009:

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Set new tolerance variable tol=0.001 for all numeric input.

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Finding the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi < \\theta \\le \\pi$.

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You have to be careful with using a standard calculator when you are finding the argument of a complex number.

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If $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.

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Using a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi < \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants – you get the correct value.

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However, The calculator gives the wrong value for complex numbers in the other quadrants.

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Complex number in the Second Quadrant.

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Since $\\arctan(b/a)$ does not distinguish between the second and fourth quadrants and the calculator gives the argument for the fourth quadrant you have to add $\\pi$ onto the calculator value.

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Complex number in the Third Quadrant.

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Since $\\arctan(b/a)$ does not distinguish between the first and third quadrants and the calculator gives the argument for the first quadrant you have to take away $\\pi$ from the calculator value.

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a)Modulus.

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\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.

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Argument.

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{m1}

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Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.

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b)Modulus.

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\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.

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Argument.

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{m2}

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Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.

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c)Modulus.

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\\[ \\begin{eqnarray*} |\\var{z3}|&=&\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &=& \\var{abs(z3)}\\\\ &=&\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.

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Argument.

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{m3}

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Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &=& \\var{arg(z3)}\\\\ &=& \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.

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d)Modulus.

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\\[ \\begin{eqnarray*} |\\var{z4}|&=&\\sqrt{(\\var{a3})^2+(\\var{b3})^2}\\\\ &=& \\var{abs(z4)}\\\\ &=&\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.

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Argument.

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{m4}

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Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z4}) &=& \\var{arg(z4)}\\\\ &=& \\var{arg4}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.

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Questions about complex arithmetic; argument and modulus of complex numbers; 

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Denis's copy of Nick's copy of Henrik Skov's copy of Complex numbers

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