Maria's copy of Modulus and argument complex numbers

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Maria Aneiros 5 years, 11 months ago

Created this as a copy of Modulus and argument complex numbers.

Name	Status	Author	Last Modified
Modulus and argument complex numbers	Ready to use	Newcastle University Mathematics and Statistics	07/04/2023 08:05
Blathnaid's copy of Modulus and argument complex numbers	draft	Blathnaid Sheridan	25/11/2019 14:17
Maria's copy of Modulus and argument complex numbers	draft	Maria Aneiros	25/05/2019 08:20
Modulus and argument calculations	Ready to use	Shaheen Charlwood	27/10/2022 18:54

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Question statement

Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\pi$ and $\pi$ .

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	7.211
ans2	number	3.162
ans3	number	8.062
ans4	number	10.817
b4	integer	-3
b1	integer	4
b2	integer	1
b3	integer	-9
d4	integer	5
d2	integer	-4
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 t
m1	string	The complex number is in the s
z2	number	3 + i
m3	string	The complex number is in the f
arg1	number	2.554
z3	number	7 - 4i
arg2	number	0.322
arg3	number	-0.519
tol	number	0.001
arg4	number	-2.159
m2	string	The complex number is in the f
a1	integer	-6
a3	integer	-6
s8	integer	-1
a4	integer	-8
z4	number	-6 - 9i
z5	number	-8 - 3i
z6	number	2 + 5i
z1	number	-6 + 4i
c2	integer	7
c4	integer	2
f	integer	2
n	integer	4
a2	integer	3
t	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.

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Generated value: `number`

7.211

← 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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This question is used in the following exam:

Maths 1A - Homework Week 7 (Complex Numbers) by Maria Aneiros in Maria's workspace.

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Maria's copy of Modulus and argument complex numbers

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Maria Aneiros 5 years, 11 months ago

Error in variable testing condition

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Generated value: `number`

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Gap-fill

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Maria's copy of Modulus and argument complex numbers

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Maria Aneiros 5 years, 11 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`