Complex Numbers_1

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Bill Foster

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Bill Foster 55 years, 2 months ago

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Question statement

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

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

Give any introductory information the student needs.

			Name	Type	Generated Value

a	number	3 - 5i
b	number	7 - 2i
f6	integer	-3
s3	integer	1
s2	integer	-1
s1	integer	1
z1	number	1 - 3i
s6	integer	-1
s4	integer	-1
a1	number	2 - 9i
d6	integer	4
a3	integer	4
b3	integer	-4
c3	integer	9
e6	integer	4
d3	integer	-8
z2	number	-7 + 4i
z3	number	-6 + 4i

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xxxxxxxxxx
 
s1*random(1..9)+s2*random(1..9)*i

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Description

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Generated value: `number`

3 - 5i

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
1. Gap Mathematical expression
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Part c) Gap-fill
1. Gap Mathematical expression
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Part d) Gap-fill
1. Gap Mathematical expression
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

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

Advice

a)
The formula for multiplying complex numbers is

$\begin{eqnarray*}\simplify[]{Re((a + ib)(c + id))} &=& ac -bd \\ \simplify[]{Im((a + ib)(c + id))} &=& ad +bc \end{eqnarray*}$

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 :

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

This is calculated in a similar way once the expression is written as:

$(\simplify[std]{{a1}})^2= (\simplify[std]{{a1}}) (\simplify[std]{{a1}})$ then we find:

$\begin{eqnarray*}(\simplify[std]{{a1}})^2&=& (\simplify[std]{{a1}}) (\simplify[std]{{a1}})\\ &=& \simplify[]{({Re(a1)}*{Re(a1)} - {Im(a1)}*{Im(a1)})+ ({Re(a1)}*{Im(a1)} + {Im(a1)}*{Re(a1)})i}\\ &=& \simplify[std]{{a1^2}} \end{eqnarray*}$
c)
We know that

$i^2=-1$ which gives

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

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.

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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This question is used in the following exams:

mathcentre: Basic complex numbers by Bill Foster in Bill's workspace.
Test eksamen Matematikk 1 by Frode Naesje in Frode's workspace.
Complex Arithmetic by Hayley Bishop in Hayley's workspace.
Maths Support: Complex arithmetic by Christian Lawson-Perfect in Maths Support Wiki.
Complex Numbers by Gareth Woods in Gareth's workspace.
Test exam by Inna Namestnikova in Inna's workspace.
Complex numbers test 1 by Tom Button in Tom's workspace.
Complex numbers: 1 by Aneela Arshad in Aneela's workspace.
CN questions collection by Tatiana Tyukina in Tatiana's workspace.
Michael's copy of Maths Support: Complex arithmetic by Michael Proudman in .Complex Numbers.

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Taxonomy: mathcentre

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Taxonomy: Context

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Generated value: `number`

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Complex Numbers_1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`