SLIDE 1
Algebra, geometry, and Pythagorean triples
Kaloyan Slavov Department of Mathematics ETH Z¨ urich kaloyan.slavov@math.ethz.ch February 9, 2017
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Basics
−2, 1, √ 5, 7
2 / 8
Algebra, geometry, and Pythagorean triples Kaloyan Slavov - - PowerPoint PPT Presentation
Algebra, geometry, and Pythagorean triples Kaloyan Slavov - - PowerPoint PPT Presentation
Algebra, geometry, and Pythagorean triples Kaloyan Slavov Department of Mathematics ETH Z urich kaloyan.slavov@math.ethz.ch February 9, 2017 1 / 8 Basics 2 , 1 , 5 , 7 2 / 8 Basics 1 2 , 1 , 5 , 7 2 7 5 2 / 8
SLIDE 2
SLIDE 3
Basics
−2, 1, √ 5, 7 −2 1 √ 5 7
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SLIDE 4
Basics
algebra geometry −2, 1, √ 5, 7 −2 1 √ 5 7
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SLIDE 5
Basics
algebra geometry −2, 1, √ 5, 7 −2 1 √ 5 7 (2, 1)
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SLIDE 6
Basics
algebra geometry −2, 1, √ 5, 7 −2 1 √ 5 7 (2, 1) x y (2, 1)
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SLIDE 7
Lines
algebra geometry
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SLIDE 8
Lines
algebra geometry x y (2, 1) (−1, 0)
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SLIDE 9
Lines
algebra geometry x y (2, 1) (−1, 0)
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SLIDE 10
Lines
algebra geometry y = x + x y (2, 1) (−1, 0)
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SLIDE 11
Lines
algebra geometry y =
slope
x + x y (2, 1) (−1, 0)
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SLIDE 12
Lines
algebra geometry y = 1 − 0 2 − (−1)
slope
x + x y (2, 1) (−1, 0)
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SLIDE 13
Lines
algebra geometry y = 1 3
slope
x + x y (2, 1) (−1, 0)
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SLIDE 14
Lines
algebra geometry y = 1 3
slope
x + 1 3 x y (2, 1) (−1, 0)
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SLIDE 15
Lines
algebra geometry y = 1 3
slope
x + 1 3 x y (2, 1) (−1, 0) y = 1 3x + 1 3
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SLIDE 16
Circles
algebra geometry
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SLIDE 17
Circles
algebra geometry x y 1
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SLIDE 18
Circles
algebra geometry x y 1 x2 + y2 = 1
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SLIDE 19
Circles
algebra geometry x y 1 (x, y) x y 1 x2 + y2 = 1
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SLIDE 20
Circles
algebra geometry x y 1 (x, y) x y 1 x2 + y2 = 1 Is
- −1
3, 2 √ 2 3
- n this circle?
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SLIDE 21
Circles
algebra geometry x y 1 (x, y) x y 1 x2 + y2 = 1 Is
- −1
3, 2 √ 2 3
- n this circle?
1 9 + 8 9 = 1 = ⇒ yes.
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SLIDE 22
Circles
algebra geometry x y 1 (x, y) x y 1 x2 + y2 = 1 Is
- −1
3, 2 √ 2 3
- n this circle?
1 9 + 8 9 = 1 = ⇒ yes.
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SLIDE 23
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers
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SLIDE 24
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
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SLIDE 25
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2.
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SLIDE 26
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2. 32 + 42 = 52
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SLIDE 27
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2. 32 + 42 = 52 62 + 82 = 102
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SLIDE 28
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2. 32 + 42 = 52 62 + 82 = 102 92 + 122 = 152
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SLIDE 29
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2. 32 + 42 = 52 62 + 82 = 102 92 + 122 = 152 . . . 3002 + 4002 = 5002
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SLIDE 30
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2. 32 + 42 = 52, 52 + 122 = 132 62 + 82 = 102 92 + 122 = 152 . . . 3002 + 4002 = 5002
5 / 8
SLIDE 31
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2. 32 + 42 = 52, 52 + 122 = 132, 82 + 152 = 172, 62 + 82 = 102 92 + 122 = 152 . . . 3002 + 4002 = 5002
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SLIDE 32
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2. 32 + 42 = 52, 52 + 122 = 132, 82 + 152 = 172,
- • •
62 + 82 = 102 92 + 122 = 152 . . . 3002 + 4002 = 5002
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SLIDE 33
Pythagorean triples
N = {1, 2, 3, 4, 5, ...} positive integers a b c
Problem
Find examples of a, b, c ∈ N such that a2 + b2 = c2. 32 + 42 = 52, 52 + 122 = 132, 82 + 152 = 172,
- • •
62 + 82 = 102 92 + 122 = 152 . . . 3002 + 4002 = 5002 Non-example: 12 + 12 = ( √ 2)2
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SLIDE 34
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N
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SLIDE 35
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒
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SLIDE 36
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c 2 + b c 2 = 1
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SLIDE 37
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1
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SLIDE 38
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒
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SLIDE 39
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
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SLIDE 40
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational) x y
1
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SLIDE 41
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒
x y
1
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SLIDE 42
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1
x y
1
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SLIDE 43
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1
x y
1
( 3
5, 4 5)
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SLIDE 44
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1 52 + 122 = 132 ⇐ ⇒
x y
1
( 3
5, 4 5)
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SLIDE 45
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1 52 + 122 = 132 ⇐ ⇒ 5 13 2 + 12 13 2 = 1
x y
1
( 3
5, 4 5)
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SLIDE 46
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1 52 + 122 = 132 ⇐ ⇒ 5 13 2 + 12 13 2 = 1
x y
1
( 3
5, 4 5)
( 5
13, 12 13)
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SLIDE 47
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1 52 + 122 = 132 ⇐ ⇒ 5 13 2 + 12 13 2 = 1 152 + 82 = 172 ⇐ ⇒
x y
1
( 3
5, 4 5)
( 5
13, 12 13)
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SLIDE 48
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1 52 + 122 = 132 ⇐ ⇒ 5 13 2 + 12 13 2 = 1 152 + 82 = 172 ⇐ ⇒ 15 17 2 + 8 17 2 = 1
x y
1
( 3
5, 4 5)
( 5
13, 12 13)
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SLIDE 49
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1 52 + 122 = 132 ⇐ ⇒ 5 13 2 + 12 13 2 = 1 152 + 82 = 172 ⇐ ⇒ 15 17 2 + 8 17 2 = 1
x y
1
( 3
5, 4 5)
( 5
13, 12 13)
( 15
17, 8 17)
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SLIDE 50
Geometric interpretation
a2 + b2 = c2 a, b, c ∈ N ⇐ ⇒ a c
- x
2
+ b c
- y
2
= 1 ⇐ ⇒ x2 + y2 = 1 x, y ∈ Q (rational)
32 + 42 = 52 ⇐ ⇒ 3 5 2 + 4 5 2 = 1 52 + 122 = 132 ⇐ ⇒ 5 13 2 + 12 13 2 = 1 152 + 82 = 172 ⇐ ⇒ 15 17 2 + 8 17 2 = 1
Problem
Find examples of x, y ∈ Q such that x2 + y2 = 1. x y
1
( 3
5, 4 5)
( 5
13, 12 13)
( 15
17, 8 17)
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SLIDE 51
Hunting rational points
x y
1
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SLIDE 52
Hunting rational points
x y
1
( 3
5, 4 5)
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SLIDE 53
Hunting rational points
x y
1
( 3
5, 4 5)
( 5
13, 12 13)
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SLIDE 54
Hunting rational points
x y
1
( 3
5, 4 5)
( 5
13, 12 13)
( 15
17, 8 17)
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SLIDE 55
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
( 5
13, 12 13)
( 15
17, 8 17)
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SLIDE 56
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
( 5
13, 12 13)
( 15
17, 8 17)
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SLIDE 57
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
( 5
13, 12 13)
( 15
17, 8 17)
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SLIDE 58
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1
( 5
13, 12 13)
( 15
17, 8 17)
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SLIDE 59
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
( 15
17, 8 17)
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SLIDE 60
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
( 15
17, 8 17)
7 / 8
SLIDE 61
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
( 15
17, 8 17)
7 / 8
SLIDE 62
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
12 13 − 0 5 13 + 1
( 15
17, 8 17)
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SLIDE 63
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
12 13 − 0 5 13 + 1= 2
3
( 15
17, 8 17)
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SLIDE 64
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
12 13 − 0 5 13 + 1= 2
3
( 15
17, 8 17)
7 / 8
SLIDE 65
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
12 13 − 0 5 13 + 1= 2
3
( 15
17, 8 17)
slope m =
7 / 8
SLIDE 66
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
12 13 − 0 5 13 + 1= 2
3
( 15
17, 8 17)
slope m = (rational!)
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SLIDE 67
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
12 13 − 0 5 13 + 1= 2
3
( 15
17, 8 17)
slope m = 1 4 (rational!)
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SLIDE 68
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
12 13 − 0 5 13 + 1= 2
3
( 15
17, 8 17)
slope m = 1 4 (rational!) slope m = 1 10 (try!)
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SLIDE 69
Hunting rational points
x y
1
(−1, 0)
( 3
5, 4 5)
slope m =
4 5 − 0 3 5 + 1 = 1
2
( 5
13, 12 13)
slope m =
12 13 − 0 5 13 + 1= 2
3
( 15
17, 8 17)
slope m = 1 4 (rational!) slope m = 1 10 (try!)
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SLIDE 70
The computation
x y
1
(−1, 0) slope m =
1 10
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SLIDE 71
The computation
x y
1
(−1, 0) slope m =
1 10
y =x +
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SLIDE 72
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x +
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SLIDE 73
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10
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SLIDE 74
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
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SLIDE 75
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1
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SLIDE 76
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1
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SLIDE 77
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒
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SLIDE 78
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0
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SLIDE 79
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 x1 = −1
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SLIDE 80
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 x1 = −1 x1x2 = − 99 101 Vieta’s formula
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SLIDE 81
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 x1 = −1 x1x2 = − 99 101 Vieta’s formula = ⇒ x2 = 99 101
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SLIDE 82
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 99
101,
- x1 = −1
x1x2 = − 99 101 Vieta’s formula = ⇒ x2 = 99 101
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SLIDE 83
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 99
101,
- x1 = −1
x1x2 = − 99 101 Vieta’s formula = ⇒ x2 = 99 101, y2 = 20 101
8 / 8
SLIDE 84
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 99
101, 20 101
- x1 = −1
x1x2 = − 99 101 Vieta’s formula = ⇒ x2 = 99 101, y2 = 20 101
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SLIDE 85
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 99
101, 20 101
- x1 = −1
x1x2 = − 99 101 Vieta’s formula = ⇒ x2 = 99 101, y2 = 20 101 Check: 99 101 2 + 20 101 2 = 1
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SLIDE 86
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 99
101, 20 101
- x1 = −1
x1x2 = − 99 101 Vieta’s formula = ⇒ x2 = 99 101, y2 = 20 101 Check: 99 101 2 + 20 101 2 = 1 ⇐ ⇒
8 / 8
SLIDE 87
The computation
x y
1
(−1, 0) slope m =
1 10
y = 1 10 x + 1 10 (x, 1
10x + 1 10)
x2 + y2 = 1 Solve: x2 + 1 10x + 1 10 2 = 1 ⇐ ⇒ 101x2 + 2x − 99 = 0 99
101, 20 101
- x1 = −1
x1x2 = − 99 101 Vieta’s formula = ⇒ x2 = 99 101, y2 = 20 101 Check: 99 101 2 + 20 101 2 = 1 ⇐ ⇒ 992 + 202 = 1012
8 / 8