Modulus and argument complex numbers

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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	7.810
ans2	number	9.849
ans3	number	2.236
ans4	number	8.485
b4	integer	-3
b1	integer	6
b2	integer	4
b3	integer	-6
d4	integer	7
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 t
m1	string	The complex number is in the s
z2	number	9 + 4i
m3	string	The complex number is in the f
arg1	number	2.266
z3	number	2 - i
arg2	number	0.418
arg3	number	-0.464
tol	number	0.001
arg4	number	-2.356
m2	string	The complex number is in the f
a1	integer	-5
a3	integer	-6
s8	integer	-1
a4	integer	-2
z4	number	-6 - 6i
z5	number	-2 - 3i
z6	number	1 + 7i
z1	number	-5 + 6i
c2	integer	2
c4	integer	1
f	integer	1
n	integer	5
a2	integer	9
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.

Can an exam override the value of this variable?

Generated value: `number`

7.810

← Depends on:

→ Used by:

Advice
"Part a)" → "Gap 0." - Minimum accepted value
"Part a)" → "Gap 0." - Maximum accepted value

Parts

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

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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Anum Khalid 1 year, 11 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Modulus and argument complex numbers

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Anum Khalid 1 year, 11 months ago

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`