Divide two elements of $\mathbb{Z}[\sqrt{2}]$

Error

There was an error loading the page.

Contributors

Newcastle University Mathematics and Statistics

History

○

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Created this.

There is only one version of this question that you have access to.

Question statement

In the quadratic number ring $\mathbb{Z}[\sqrt{2}]$ , find the remainder $r=a+b\sqrt{2}$ , where $a \gt 0,\;b \gt 0$ , on dividing $\simplify[all,!collectNumbers]{{a}+{b}*sqrt(2)}$ by $\simplify[all,!collectNumbers]{{c}+{d}*sqrt(2)}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	18
c	integer	35
b	integer	30
d	integer	-10
m	integer	3
n	integer	5
p	integer	5
t	integer	1
v	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
c*t+2*d*v+m

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: `integer`

← Depends on:

→ Used by:

Statement
Advice
"Unnamed part" - prompt

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

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

Remainder $r=\;$ Gap 0.

Find $z=c+d\sqrt{2}$ such that $\simplify[all,!collectNumbers]{{a}+{b}*sqrt(2)=({c}+{d}*sqrt(2))*z+r}$

$z=\;$ Gap 1.

Please input your answers in the form a+b*sqrt(2) where a and b are integers. Do not include decimal numbers.

Advice

We have:

$\begin{align}\simplify[all,!collectNumbers]{({a} + {b} * sqrt(2)) / ({c} + {d} * sqrt(2))}& = \simplify[all,!collectNumbers]{(({a} + {b} * sqrt(2)) * ({c} + { -d} * sqrt(2))) / (({c} + {d} * sqrt(2)) * ({c} + { -d} * sqrt(2)))}\\ & = \simplify{{a * c - 2*b * d} / {c ^ 2 -2*d ^ 2} + ({b * c -(a * d)} / {c ^ 2 - 2*d ^ 2}) * sqrt(2)}\\&=\simplify[all,!collectNumbers]{{(a * c - 2*b * d) / (c ^ 2 - 2*d ^ 2)} + {(b * c -(a * d)) / (c ^ 2 - 2*d ^ 2)} * sqrt(2)}\\ & \approx \simplify[all,!collectNumbers]{{t} + {v}* sqrt(2)}\end{align}$

on taking the nearest integer values.

Taking $z= \simplify[all,!collectNumbers]{{t} + {v}* sqrt(2)}$ and on calculating we find:

$\simplify[all,!collectNumbers]{{a} + {b} * sqrt(2) = ({c} + {d} * sqrt(2)) * ({t} + {v} * sqrt(2)) + {m} + {n} * sqrt(2)}$

So the remainder is $r=\simplify[all,!collectNumbers]{{m}+{n}*sqrt(2)}$ .

Note that:

$N(r) = \simplify[all,!collectNumbers]{Abs({m ^ 2 -(2 * n ^ 2)}) = {Abs(m ^ 2 -(2 * n ^ 2))}} \lt\simplify[all,!collectNumbers]{ N({c} + {d} * Sqrt(2)) = Abs({c ^ 2 -(2 * d ^ 2)}) = {Abs(c ^ 2 -(2 * d ^ 2))}}.$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is used in the following exams:

Loading...

Error

Divide two elements of $\mathbb{Z}[\sqrt{2}]$

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

Loading...

Error

Divide two elements of Z[√2]\mathbb{Z}[\sqrt{2}]

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

Divide two elements of $\mathbb{Z}[\sqrt{2}]$

Generated value: `integer`