De Moivre's Theorem: $n$th roots of a complex number

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Michael Proudman

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Michael Proudman 7 years, 8 months ago

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Question statement

Find the $\var{n}$ th roots of $\var{z1}$ .

Important: When calculating the roots, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer may not be correct.

For the purposes of this question all arguments of complex numbers are between $0$ and $2\pi$ radians.

Give any introductory information the student needs.

			Name	Type	Generated Value

q1	string	The complex number is in the f
argrn	number	6.085
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
md1	number	1.324
arg1	number	5.093
s2	integer	-1
s1	integer	1
targ1	number	5.0928953575
n	integer	6
a1	integer	2
m1	string	The complex number is in the f
t	integer	1
tol	number	0.001
ans1	number	5.385
gap	number	1.047
adj	number	2 pi
z1	number	2 - 5i
argr1	number	0.849
b1	integer	-5

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Name

Data type

Value

xxxxxxxxxx
 
'The complex number is in the first quadrant.'

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Description

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Generated value: `string`

The complex number is in the first quadrant.

→ Used by:

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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4. Gap Gap 3. Number entry
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5. Gap Gap 4. Number entry
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6. Gap Gap 5. Number entry
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7. Gap Gap 6. Number entry
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Gap-fill

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

Find the modulus and argument of $\var{z1}$ to 3 decimal places.

(i) $|\var{z1}|\;=\;$ Gap 0., to 3 decimal places.

(ii) $\arg(\var{z1})\;=\;$ Gap 1. radians, to 3 decimal places.

Hence find the following $\var{n}$ th roots of $\var{z1}$ i.e. solve for $z$ , $z^\var{n}=\var{z1}$ .

How many roots are there? Gap 2.

All the roots have the same modulus.

Input the modulus here: Gap 3. (to 3 decimal places).

What is the argument of the root with the least argument? Gap 4. radians (to 3 decimal places)

What is the argument of the root with the greatest argument? Gap 5. radians (to 3 decimal places).

If the roots are ordered in terms of their increasing arguments, what is the angle between successive roots? Gap 6. radians (to 3 decimal places).

Advice

Please consult the relevant examples in the lecture notes.

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De Moivre's Theorem: $n$ th roots of a complex number

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Taxonomy: mathcentre

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Michael Proudman 7 years, 8 months ago

Error in variable testing condition

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Generated value: `string`

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Gap-fill

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De Moivre's Theorem: nnth roots of a complex number

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Michael Proudman 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Gap-fill

De Moivre's Theorem: $n$ th roots of a complex number

Generated value: `string`