// Numbas version: exam_results_page_options {"showQuestionGroupNames": false, "feedback": {"feedbackmessages": [], "allowrevealanswer": true, "showanswerstate": true, "showtotalmark": true, "advicethreshold": 0, "showactualmark": true, "intro": ""}, "showstudentname": true, "navigation": {"preventleave": true, "showfrontpage": true, "showresultspage": "oncompletion", "onleave": {"message": "", "action": "none"}, "browse": true, "allowregen": true, "reverse": true}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "Group", "pickQuestions": 1, "questions": [{"name": "Complex number arithmetic: add and subtract", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Given the complex numbers

\\[a=\\var{z1}\\\\b=\\var{z2}\\\\c=\\var{z3}\\\\d=\\var{z4}\\]

calculate the following. Enter your answers in the form

\\[x+yi\\]

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$a+b=$

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No brackets are required or allowed.

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$a+c=$

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No brackets are required or allowed.

$a-b=$

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No brackets are required or allowed.

$a-c=$

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No brackets are required or allowed.

$a+d=$

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No brackets are required or allowed.

$\\simplify{{n}*b+{m}*d}=$

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

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Given the complex numbers

\\[a=\\var{a},\\quad b=\\var{b},\\quad c=\\var{c},\\quad d=\\var{d},\\]

calculate the following. Enter your answers in the form

\\[x+iy\\]

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$ab=$ [[0]]

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Expand the brackets $(\\var{a})(\\var{b})$.

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Simplify using the fact $i^2=-1$.

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No brackets are required or allowed.

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$ac=$ [[0]]

No brackets are required or allowed.

$\\frac{a}{b}=$ [[0]]

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What is the complex conjugate of $b$?

$\\overline{b}=$

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Multiply the numerator ($\\var{a}$) by $\\overline{b}$.

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Multiply the denominator ($\\var{b}$) by $\\overline{b}$.

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Now divide your two answers.

\\[\\frac{a}{b}=\\frac{a\\overline{b}}{b\\overline{b}}\\]

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No brackets are required or allowed.

$\\frac{a}{c}=$ [[0]]

No brackets are required or allowed.

$ad=$ [[0]]

No brackets are required or allowed.

$\\frac{b}{d}=$ [[0]]

No brackets are required or allowed.

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Solve the quadratic equation:

\\[\\simplify{x^2-2*{re(a)}*x+{abs(a)^2}}=0\\]

You may use the following formula: if $ax^2+bx+c=0$ then

\\[x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}\\]

Enter your answers in the form $x+yi$.

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$x_1=\\;$ [[0]] (solution with positive imaginary part)

$x_2=\\;$ [[1]] (solution with negative imaginary part)

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This question is about converting complex numbers from polar to rectangular form and from rectangular to polar form. The argument $\\theta$ should be in the range

\\[-180<\\theta\\leq180\\]

Give your answers correct to at least 1 decimal place.

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Write the number $\\var{r1}\\angle\\var{theta1}^\\circ$ in rectangular form.

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Write the number $\\var{r2}\\angle\\var{theta2}^\\circ$ in rectangular form.

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Write the number $\\simplify{{x3}+i*{y3}}$ in polar form.

[[0]]$\\angle\\;$[[1]]$^\\circ$

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Write the number $\\simplify{{x4}+i*{y4}}$ in polar form.

[[0]]$\\angle\\;$[[1]]$^\\circ$

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Quiz covering basic arithmetic with complex numbers, solving a quadratic with complex solutions and converting to/from polar and rectangular forms.

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