Gaussian integers

(1)

Gaussian integers

Season 3

Episode 03 Time frame 1 period

Prerequisites :

^Basi^algebra^skills.

Objectives :

•

^Disover ômplex^numbers ând ^some ôf ^their ^properties ^through ^Gaussianîntegers.

•

^Link ^geometry^and ^algebra.

Materials :

•

^Mathing ârds ^with â ^pair ôf ^Gaussianîntegers ôr â^pair ôf ^pointsônâ ^lattie.

•

^Lattie ^with âxesând ^the ûnits

1

^and

i

(one for eah student).

•

^Beamer ând ^lesson âbout ^Gaussian întegers.

1 – Matching game with Gaussian integers and points. 15 mins

Students are handed out ards with either a pair of Gaussian integers of apair of points

on alattie. Thay mingleto math the ards by pairs.

2 – Experimental study of the operations in

^Z

[ i ] ^{25 mins}

The teaher introdues the notations of the set of Gaussian integers Z

[ i ]

^and ^the ^two

operations,

+

^and

×

^.

Working inteamsof two,three offour, studentshave toexperiment onoperationsinthe

set of Gaussianintegers. The aimis tond the graphialanalogous to

•

^addition^;

•

multipliation by a real number;

•

multipliation by

i

;

•

multipliation by a number

ib

.

Answers are then given with abeamer.?

(2)

Document Lesson

Definition 1

A Gaussian integer is a number of the form a + bi where a and b are two integers and i is an imaginary number such that i ² = − 1. The set of all Gaussian integers is usually noted

^Z

[ i ].

Every Gaussianinteger isuniquely assoiated toa pointin asquare lattie.This proess

denes akind of oordinatesystem onthe lattie, with unit

1

ôn^the ^horizontalâxisând

i

on the vertial one.

bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc

b b b b

b b

0 i

1 2 − 3 i

−4 + 4 i

3 + 2 i

The set of Gaussian integers is an extension of the set of standard integers. Addition

and multipliationin the set Z

[ i ]

^follow^the ^same ^rules âs âdditionând multipliation of integers,with theaddedpropertiesthat

i ² = − 1

ând ^that^the ^real^parts ândîmaginary

parts annotbe mixed up inaddition.Wededue easilythe following properties.

Proposition 1

Let a + bi and c + di be two Gaussian integers. Their sum is equal to the Gaussian integer

( a + c ) + ( b + d ) i while their product is equal to

( ac − bd ) + ( ad + bc ) i.

(3)

Operations on Gaussian integers have simple geometrial ounterparts. Addition by a

Gaussian integer

a + bi

is a translation by the vetor dened by the lattiepoints

0

^and

a + bi

. The example belowshows addition by

2 + i

.

b b b

b b

0 i

1 2 + i

− 4 − 3 i

−2 − 2 i

1 − 4 i

3 − 3 i

−5 + 2 i

− 3 + 3 i

Multipliation by a Gaussian integer

a + bi

is the omposition of a rotation and an ho- mothetyaround the lattiepoint

0

^(the ôrigin),^whose ângleând ^ratio âre ^dened ^by ^the

lattie points

0

^and

a + bi

.Here are a fewexamples.

•

Multipliation by

i

is a rotationof angle

π

2

âround ^the ôrigin, âs ^the ^distane ^between

0

^and

i

isequal to

1

^.

•

Multipliation by

2

îs ân ^homothety ôf ^ratio

2

âround ^the ôrigin, âs ^the ângle ^dened

by the point

2

^is ^equal^to

0

^.

•

Multipliation by

2 i

is the omposition of an homothety of ratio

2

ând â ^rotation ôf

angle

π

2

^, ^both ^around ^the ^origin.

b b b b

b

b b b

0 3 + i

−1 + 3 i

× i

−2 + i

− 4 + 2 i

× 2

− 2 − 2 i

4 − 4 i

× 2 i

(4)

Document 1

^Lattie ^for experiments

bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc

Gaussian integers