Complex Numbers: Modulus, Argument

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John Sheekey

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John Sheekey 7 years, 1 month ago

Created this as a copy of Complex Numbers: Modulus, Argument.

Name	Status	Author	Last Modified
Complex Numbers: Modulus, Argument	Ready to use	Bill Foster	01/06/2016 09:47
David's copy of Complex Numbers: Modulus, Argument	Ready to use	David Rickard	01/06/2016 09:47
Complex Numbers: Modulus, Argument	Ready to use	Christoph Walti	01/06/2016 09:47
Graham's copy of Complex Numbers: Modulus, Argument	Ready to use	Graham Wynn	01/06/2016 09:47
Graham's copy of Graham's copy of Complex Numbers: Modulus, Argument	Ready to use	Graham Wynn	01/06/2016 09:47
Sabri's copy of Graham's copy of Graham's copy of Complex Numbers: Modulus, Argument	Ready to use	Sabri Al-Safi	01/06/2016 09:47
Complex Numbers: Modulus, Argument	draft	John Sheekey	06/02/2018 17:03
Complex Numbers: Modulus, Argument	draft	Harry Flynn	24/05/2018 12:55
Complex Numbers: Modulus, Argument	draft	Harry Flynn	24/05/2018 12:55
Complex Numbers: Modulus, Argument	draft	Harry Flynn	24/05/2018 13:01
Complex Numbers: Modulus, Argument	draft	Harry Flynn	24/05/2018 13:04
Complex Numbers: Modulus, Argument	draft	Harry Flynn	25/05/2018 12:01
Gemma's copy of Complex Numbers: Modulus, Argument	draft	Gemma Crowe	04/06/2018 10:40

There are 10 other versions that do you not have access to.

Question statement

Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\pi$ and $\pi$ . If your answer is outside this range, simply add or subtract $2\pi$ to your answer.

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

Input all answers to 3 decimal places.

Give any introductory information the student needs.

			Name	Type	Generated Value

ans1	number	5.000
ans2	number	8.544
ans3	number	2.236
ans4	number	3.162
b4	integer	-3
b1	integer	-4
b2	integer	-8
b3	integer	3
d4	integer	-1
d2	integer	1
q1	string	The complex number is in the f
q3	string	The complex number is in the t
q2	string	The complex number is in the s
q4	string	The complex number is in the f
s3	integer	-1
s2	integer	-1
s1	integer	-1
s7	integer	1
s6	integer	1
s5	integer	-1
s4	integer	1
m4	string	The complex number is in the s
m1	string	The complex number is in the t
m3	string	The complex number is in the f
arg1	number	-2.214
z3	number	2 + i
arg2	number	-1.212
arg3	number	0.464
tol	number	0.001
arg4	number	1.893
m2	string	The complex number is in the f
a1	integer	-3
a3	integer	-1
s8	integer	1
a4	integer	8
z4	number	-1 + 3i
z5	number	8 - 3i
z6	number	6 - i
z1	number	-3 - 4i
z2	number	3 - 8i
c4	integer	6
f	integer	6
n	integer	5
a2	integer	3
t	integer	3
c2	integer	2

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
precround(abs(z1),3)

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`

5.000

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Part b) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Part c) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Part d) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Gap-fill

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

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

Input both answers to 3 decimal places.

Advice

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

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

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

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.

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

Complex number in the Second Quadrant.

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.

Complex number in the Third Quadrant.

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.

a)

Modulus

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

Argument

{m1}

Hence we see that:

$\begin{eqnarray*} \arg(\var{z1}) &=& \var{arg(z1)}\\ &=& \var{arg1}\; \mbox{radians} \end{eqnarray*}$ to 3 decimal places.

b)

Modulus

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

Argument

{m2}

Hence we see that:

$\begin{eqnarray*} \arg(\var{z2}) &=& \var{arg(z2)}\\ &=& \var{arg2}\; \mbox{radians} \end{eqnarray*}$ to 3 decimal places.

c)

Modulus

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

Argument

{m3}

Hence we see that:

$\begin{eqnarray*} \arg(\var{z3}) &=& \var{arg(z3)}\\ &=& \var{arg3}\; \mbox{radians} \end{eqnarray*}$ to 3 decimal places.

d)

Modulus

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

Argument

{m4}

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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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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History

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Checkpoint description

John Sheekey 7 years, 1 month ago

Error in variable testing condition

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Warning

Generated value: `number`

Parts

Gap-fill

Complex number in the Second Quadrant.

Complex number in the Third Quadrant.

a)

Modulus

Argument

b)

Modulus

Argument

c)

Modulus

Argument

d)

Modulus

Argument

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Complex Numbers: Modulus, Argument

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

John Sheekey 7 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Complex number in the Second Quadrant.

Complex number in the Third Quadrant.

a)

Modulus

Argument

b)

Modulus

Argument

c)

Modulus

Argument

d)

Modulus

Argument

Generated value: `number`