Denis's copy of Divide two elements of $\mathbb{Z}[i]$

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Denis Flynn

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Denis Flynn 8 years, 3 months ago

Created this as a copy of Divide two elements of

$\mathbb{Z}[i]$ .

Name	Status	Author	Last Modified
Divide two elements of $\mathbb{Z}[i]$	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Denis's copy of Divide two elements of $\mathbb{Z}[i]$	draft	Denis Flynn	26/01/2017 12:08

Question statement

In the Gaussian integer ring $\mathbb{Z}[i]$ , find the remainder $r=a+bi$ , where $a \gt 0,\;b \gt 0$ , on dividing $\simplify[all,!collectNumbers]{{a}+{b}*i}$ by $\simplify[all,!collectNumbers]{{c}+{d}*i}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	116
c	integer	21
p1	integer	3
d	integer	-9
m	integer	5
n	integer	2
p	integer	4
b	integer	29
t	integer	4
v	integer	3

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Name

Data type

Value

xxxxxxxxxx
 
c*t-d*v+m

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Description

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Generated value: `integer`

116

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1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Remainder $r=\;$ Gap 0.

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

$z=\;$ Gap 1.

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

Advice

We have:

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

on taking the nearest integer values.

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

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

So the remainder is $r=\simplify{{m}+{n}*i}$ .

Note that:

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

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Denis's copy of Divide two elements of $\mathbb{Z}[i]$

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Taxonomy: mathcentre

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Denis Flynn 8 years, 3 months ago

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Generated value: `integer`

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Denis's copy of Divide two elements of Z[i]\mathbb{Z}[i]

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

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Denis Flynn 8 years, 3 months ago

Error in variable testing condition

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Generated value: integer

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Denis's copy of Divide two elements of $\mathbb{Z}[i]$

Generated value: `integer`