Episode 03 – Gaussian integers European section – Season 3

(1)

Episode 03 – Gaussian integers

European section – Season 3

Episode 03 – Gaussian integers

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Gaussian integers on a lattice

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Gaussian integers on a lattice

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b

0 Episode 03 – Gaussian integers

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Gaussian integers on a lattice

b

0 b

1

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Gaussian integers on a lattice

b

0 b

1 b i

Episode 03 – Gaussian integers

(6)

Gaussian integers on a lattice

b

0 b

1 b i

b

2 3i

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Gaussian integers on a lattice

b

0 b

1 b i

b

2 − 3i

b

−4 + 4i

Episode 03 – Gaussian integers

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Gaussian integers on a lattice

b

0 b

1 b i

b

2 3i

b

−4 + 4i

b

3 + 2i

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Operations in the set of Gaussian integers

Definition (Gaussian integers)

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 ].

Episode 03 – Gaussian integers

(10)

Operations in the set of Gaussian integers

Definition (Gaussian integers)

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 ].

Golden rule

Addition and multiplication in the set Z [i ] follow the same rules as

addition and multiplication of integers but “real parts” and “imaginary

parts” cannot be mixed up in addition.

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Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Episode 03 – Gaussian integers

(12)

Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Question 2

What is the graphical effect of multiplying a Gaussian integer by

another ?

(13)

Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Question 2

What is the graphical effect of multiplying a Gaussian integer by another ?

Study the problem for :

Episode 03 – Gaussian integers

(14)

Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Question 2

What is the graphical effect of multiplying a Gaussian integer by another ?

Study the problem for :

multiplication by a real number b ;

(15)

Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Question 2

What is the graphical effect of multiplying a Gaussian integer by another ?

Study the problem for :

multiplication by a real number b ; multiplication by the imaginary number i ;

Episode 03 – Gaussian integers

(16)

Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Question 2

What is the graphical effect of multiplying a Gaussian integer by another ?

Study the problem for :

multiplication by a real number b ;

multiplication by the imaginary number i ;

multiplication by the imaginary number bi.

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Addition and translation

b b b

0 i

1 Episode 03 – Gaussian integers

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Addition and translation

b b b

0 i

1 b

2 + i

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Addition and translation

b b b

0 i

1 b

2 + i

Episode 03 – Gaussian integers

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Addition and translation

b b b

0 i

1 b

2 + i

b

4 3i

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Addition and translation

b b b

0 i

1 b

2 + i

b

− 4 − 3i

b

−2 − 2i

Episode 03 – Gaussian integers

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Addition and translation

b b b

0 i

1 b

2 + i

b

4 3i

b

−2 − 2i

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Addition and translation

b b b

0 i

1 b

2 + i

b

− 4 − 3i

b

−2 − 2i

b

1 − 4i

Episode 03 – Gaussian integers

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Addition and translation

b b b

0 i

1 b

2 + i

b

4 3i

b

−2 − 2i

b

3 − 3i

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Addition and translation

b b b

0 i

1 b

2 + i

b

− 4 − 3i

b

−2 − 2i

b

1 − 4i

b

3 − 3i

Episode 03 – Gaussian integers

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Addition and translation

b b b

0 i

1 b

2 + i

b

4 3i

b

−2 − 2i

b

3 − 3i

b

−5 + 2i

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Addition and translation

b b b

0 i

1 b

2 + i

b

− 4 − 3i

b

−2 − 2i

b

1 − 4i

b

3 − 3i

b

−5 + 2i

b − 3 + 3i

Episode 03 – Gaussian integers

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Addition and translation

b b b

0 i

1 b

2 + i

b

4 3i

b

−2 − 2i

b

3 − 3i

b

−5 + 2i

b − 3 + 3i

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Multiplication, homothety and rotation

b b b

b

− 2 + i

Episode 03 – Gaussian integers

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Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

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Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

Episode 03 – Gaussian integers

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Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

0

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Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

0 b

3 + i

Episode 03 – Gaussian integers

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Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

0 b

3 + i

b − 1 + 3i

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Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

0 b

3 + i

b − 1 + 3i

Episode 03 – Gaussian integers

(36)

Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

0 b

3 + i

b − 1 + 3i

b

−2 − 2i

(37)

Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

0 b

3 + i

b − 1 + 3i

b

−2 − 2i

b

4 − 4i

Episode 03 – Gaussian integers

(38)

Multiplication, homothety and rotation

b b b

b

− 2 + i

b

−4 + 2i

0 b

3 + i

b − 1 + 3i

b

−2 − 2i

Episode 03 – Gaussian integers European section – Season 3

Episode 03 – Gaussian integers

European section – Season 3

Episode 03 – Gaussian integers

Gaussian integers on a lattice

Gaussian integers on a lattice

b

0

Episode 03 – Gaussian integers

Gaussian integers on a lattice

b

0

b

1

Gaussian integers on a lattice

b

0

b

1

b i

Episode 03 – Gaussian integers

Gaussian integers on a lattice

b

0

b

1

b i

b

2 3i

Gaussian integers on a lattice

b

0

b

1

b i

b

2 − 3i

b

−4 + 4i

Episode 03 – Gaussian integers

Gaussian integers on a lattice

b

0

b

1

b i

b

2 3i

b

−4 + 4i

b

3 + 2i

Operations in the set of Gaussian integers

Definition (Gaussian integers)

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 ].

Episode 03 – Gaussian integers

Operations in the set of Gaussian integers

Definition (Gaussian integers)

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 ].

Golden rule

Addition and multiplication in the set Z [i ] follow the same rules as

addition and multiplication of integers but “real parts” and “imaginary

parts” cannot be mixed up in addition.

Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Episode 03 – Gaussian integers

Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Question 2

What is the graphical effect of multiplying a Gaussian integer by

another ?

Addition and multiplication of points ?

Question 1

What is the graphical effect of adding a Gaussian integer to another ?

Question 2

What is the graphical effect of multiplying a Gaussian integer by another ?

Study the problem for :

Episode 03 – Gaussian integers

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 ].

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 ].