Complex numbers in real-imaginary form

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Complex numbers in real-imaginary form	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Utilice las potencias $i$ para resolver lo siguiente...	Ready to use	Luis Hernandez	08/11/2020 01:25

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Question statement

Write the following complex numbers in real-imaginary ( $x+iy$ ) form.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	4
n	integer	7002
moda4	integer	0
modn4	integer	2

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..5)

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Description

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

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Generated value: `integer`

→ Used by:

moda4

Advice
"Part b)" - prompt
"Part b)" → "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

Gap-fill

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

$\frac{1+i}{1-i}=$ Unnamed gap

Advice

a)

To write $z=\frac{1+i}{1-i}$ in real-imaginary form, multiply $z$ by the complex conjugate of the denominator, to obtain

$z=\frac{(1+i)(1+i)}{(1-i)(1+i)}=\frac{1+2i-1}{1+1}=i.$

b)

To express $z=\left(\frac{1+i}{1-i}\right)^\var{a}$ in real-imaginary form, use part a), so that we only need to write $i^\var{a}$ in real-imaginary form.

The result of raising $i$ to an arbitrary integer power $n$ can be determined by using the fact that $i^2=-1$ , $i^3=-i$ , and $i^4=i$ .

Divide the power by $4$ , and calculate the remainder (i.e. calculate $n\mod 4$ ), which will be either $0$ , $1$ , $2$ , or $3$ . If the remainder is $0$ , then $i^n=1$ ; if the remainder is $1$ , then $i^n=i$ ; if the remainder is $2$ , then $i^n=-1$ ; if the remainder is $3$ , then $i^n=-i$ .

Here, $n \bmod 4 =\var{moda4}$ , so $i^\var{a}=\simplify{i^{a}}$ .

c)

Using the method from part b), $n \bmod 4=\var{n} \bmod 4=\var{modn4}$ , so $i^{\var{n}}=i^{\var{modn4}}=\simplify{i^{modn4}}$ .

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This question is used in the following exam:

Complex variables by Newcastle University Mathematics and Statistics in Content created by Newcastle University.

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Taxonomy: mathcentre

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Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

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Complex numbers in real-imaginary form

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: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`