Paul's copy of Arithmetics of complex numbers I

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Paul McFadden

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Paul McFadden 2 years, 5 months ago

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Created this as a copy of Arithmetics of complex numbers I.

Name	Status	Author	Last Modified
Arithmetics of complex numbers I	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Complex Numbers Ex Sheet 1	draft	Violeta CIT	17/10/2017 19:49
Complex Numbers Ex Sheet 1	draft	Joshua Harnwell	02/11/2017 08:36
Exprese lo siguiente en la forma $a + bi \;$ donde $a$ y $b$ son reales.	Ready to use	Luis Hernandez	08/11/2020 01:10
Maria's copy of Arithmetics of complex numbers I	draft	Maria Aneiros	25/05/2019 08:20
Rechnen mit komplexen Zahlen I	Ready to use	Ulrich Görtz	08/02/2021 19:57
Multiplication of complex numbers (i)	Ready to use	Shaheen Charlwood	27/10/2022 18:53
Paul's copy of Arithmetics of complex numbers I	Ready to use	Paul McFadden	08/11/2024 14:56

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Question statement

Express the following in the form $x+iy\;$ where $x$ and $y$ are real numbers.

Write your answers using only integers or fractions, not decimals. To input, e.g., $2+3i$ , type 2+3i putting the i last.

To input, e.g., $\frac{3}{10}+\frac{4}{5}i$ , type 3/10+4/5i or (3/10)+(4/5)i. There should be no brackets in your answer except to indicate fractions as here.

Give any introductory information the student needs.

			Name	Type	Generated Value

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

Automatically regenerate variables?

Name

Data type

Value

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

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `number`

-1 + 7i

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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

Multiplying two complex numbers and extracting their real and imaginary parts, we find:

$\begin{eqnarray*}\simplify[]{Re((x_1 + iy_1)(x_2 + iy_2))} &=& x_1 x_2 -y_1y_2 \\ \simplify[]{Im((x_1 + iy_1)(x_2 + iy_2))} &=& x_1 y_2 +y_1x_2 \end{eqnarray*}$

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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Paul's copy of Arithmetics of complex numbers I

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Paul McFadden 2 years, 5 months ago

Paul McFadden 2 years, 5 months ago

Paul McFadden 2 years, 5 months ago

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Generated value: `number`

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Paul's copy of Arithmetics of complex numbers I

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Paul McFadden 2 years, 5 months ago

Paul McFadden 2 years, 5 months ago

Paul McFadden 2 years, 5 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`