Arithmetics of complex numbers I

Error

There was an error loading the page.

Contributors

Newcastle University Mathematics and Statistics

History

○

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Created this.

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

There are 37 other versions that do you not have access to.

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

Automatically regenerate variables?

Name

Data type

Value

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

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `number`

-6 - 4i

← Depends on:

→ Used by:

Advice
"Part a)" - prompt
"Part a)" → "Unnamed gap" - Correct answer

Parts

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

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:

$\begin{eqnarray*} (\var{z1})(\var{z2})(\var{z3})&=&((\var{z1})(\var{z2}))(\var{z3})\\ &=&(\var{z1*z2})(\var{z3})\\ &=&\var{z1*z2*z3} \end{eqnarray*}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is used in the following exams:

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `number`

Parts

Gap-fill

Loading...

Error

Arithmetics of complex numbers I

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`