Computing data with spreadsheets Example: Computing triangular numbers and their square roots. Recall, we showed 1 ` 2 ` ` n npn ` 1q{2.
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1 Computing data with spreadsheets Example: Computing triangular numbers and their square roots. Recall, we showed 1 ` 2 ` ` n npn ` 1q{2.
2 Computing data with spreadsheets Example: Computing triangular numbers and their square roots. Recall, we showed 1 ` 2 ` ` n npn ` 1q{2. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3)
3 Computing data with spreadsheets Example: Computing triangular numbers and their square roots. Recall, we showed 1 ` 2 ` ` n npn ` 1q{2. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) The result should look like
4 Computing data with spreadsheets Example: Computing triangular numbers and their square roots. Recall, we showed 1 ` 2 ` ` n npn ` 1q{2. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) Next, select cells A3, B3, and C3 and copy.
5 Computing data with spreadsheets Example: Computing triangular numbers and their square roots. Recall, we showed 1 ` 2 ` ` n npn ` 1q{2. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) Next, select cells A3, B3, and C3 and copy. Finally, select cells in as many rows as you want below that, in the columns A, B, and C, and paste.
6 Example: Computing triangular numbers and their square roots. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) Next, select cells A3, B3, and C3 and copy. Finally, select cells in as many rows as you want below that, in the columns A, B, and C, and paste. The spreadsheet will extrapolate for you, and make new formulas relative to their shifted positions.
7 Example: Computing triangular numbers and their square roots. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) Next, select cells A3, B3, and C3 and copy. Finally, select cells in as many rows as you want below that, in the columns A, B, and C, and paste. The spreadsheet will extrapolate for you, and make new formulas relative to their shifted positions. If you don t want it to adjust part of a formula with shifts, add a $ in front of the row number, cell number, or both: A$2, $A2, or $A$2.
8 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b
9 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c b a Geometric proof: Compare the area of the white spaces in b c a b c a b c a b c a versus b c a b c a b c a b c a
10 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions?
11 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions?
12 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc.
13 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z satisfying a 2 ` b 2 c 2?
14 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z satisfying a 2 ` b 2 c 2? Yes! 3 2 `
15 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 `
16 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 `
17 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These are called Pythagorean triples.
18 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These are called Pythagorean triples. Trivial solution: a b c 0.
19 Pythagorean Triples The Pythagorean theorem says the lengths of the sides of a right triangle satisfy the following: a 2 ` b 2 c 2 c a b Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These are called Pythagorean triples. Trivial solution: a b c 0. (Don t forget to look for the simplest solutions!!)
20 Pythagorean Triples Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z ě0 satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These are called Pythagorean triples. Trivial solution: a b c 0. (Don t forget to look for the simplest solutions!!) Infinitely many?
21 Pythagorean Triples Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z ě0 satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These are called Pythagorean triples. Trivial solution: a b c 0. (Don t forget to look for the simplest solutions!!) Infinitely many? Claim: If pa, b, cq is a Pythagorean triple, then so is pna, nb, ncq for any n P Z ě0.
22 Pythagorean Triples Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z ě0 satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These are called Pythagorean triples. Trivial solution: a b c 0. (Don t forget to look for the simplest solutions!!) Infinitely many? Claim: If pa, b, cq is a Pythagorean triple, then so is pna, nb, ncq for any n P Z ě0. So, yes, but this doesn t generate all solutions!
23 Pythagorean Triples Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z ě0 satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These are called Pythagorean triples. Trivial solution: a b c 0. (Don t forget to look for the simplest solutions!!) Infinitely many? Claim: If pa, b, cq is a Pythagorean triple, then so is pna, nb, ncq for any n P Z ě0. So, yes, but this doesn t generate all solutions! We call a Pythagorean triple pa, b, cq primitive if a, b, and c have no common factors, meaning there is no d P Z ą0 such that a, b, and c are all multiples of d.
24 Pythagorean Triples Number theorists ask: Are there integer solutions? If so, are there infinitely many integer solutions? Do they all follow some common pattern, or are the solutions random? etc. Integer solutions: are there a, b, c P Z ě0 satisfying a 2 ` b 2 c 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These are called Pythagorean triples. Trivial solution: a b c 0. (Don t forget to look for the simplest solutions!!) Infinitely many? Claim: If pa, b, cq is a Pythagorean triple, then so is pna, nb, ncq for any n P Z ě0. So, yes, but this doesn t generate all solutions! We call a Pythagorean triple pa, b, cq primitive if a, b, and c have no common factors, meaning there is no d P Z ą0 such that a, b, and c are all multiples of d. Ex: (3,4,5) Non-ex: (6,8,10).
25 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q
26 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generic even numbers are written 2k with k P Z. Generic odd numbers are written 2k ` 1 with k P Z.
27 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generic even numbers are written 2k with k P Z. Generic odd numbers are written 2k ` 1 with k P Z. Square of an even number:
28 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generic even numbers are written 2k with k P Z. Generic odd numbers are written 2k ` 1 with k P Z. Square of an even number: p2kq 2 2 p2k 2 q lomon int
29 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generic even numbers are written 2k with k P Z. Generic odd numbers are written 2k ` 1 with k P Z. Square of an even number: p2kq 2 2 lomon p2k 2 q Even! int
30 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generic even numbers are written 2k with k P Z. Generic odd numbers are written 2k ` 1 with k P Z. Square of an even number: p2kq 2 2 p2k 2 q lomon Square of an odd number: int Even!
31 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generic even numbers are written 2k with k P Z. Generic odd numbers are written 2k ` 1 with k P Z. Square of an even number: p2kq 2 2 p2k 2 q lomon Square of an odd number: p2k ` 1q 2 4k 2 ` 4k ` 1 int Even!
32 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generic even numbers are written 2k with k P Z. Generic odd numbers are written 2k ` 1 with k P Z. Square of an even number: p2kq 2 2 p2k 2 q lomon int Even! Square of an odd number: p2k ` 1q 2 4k 2 ` 4k ` 1 2 p2k looooomooooon 2 ` 2kq `1 integer
33 Primitive Pythagorean triples (PPTs) More examples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generic even numbers are written 2k with k P Z. Generic odd numbers are written 2k ` 1 with k P Z. Square of an even number: p2kq 2 2 p2k 2 q lomon int Even! Square of an odd number: p2k ` 1q 2 4k 2 ` 4k ` 1 2 p2k looooomooooon 2 ` 2kq `1 integer Odd! Similarly, you can show that sum of evens and odds follows the pattern ` even odd even even odd odd odd even
34 Primitive Pythagorean triples (PPTs) You try: For solutions a, b, c P ZZ to a 2 ` b 2 c 2, complete the following table. a b c even even odd odd even odd even odd Which could possibly be primitive?
35 Primitive Pythagorean triples (PPTs) You try: For solutions a, b, c P ZZ to a 2 ` b 2 c 2, complete the following table. a b c even even odd odd even odd even odd Which could possibly be primitive? Without loss of generality (WLOG), assume a is odd and b is even. (WLOG here means that sincea and b are interchangeable, we don t need to also consider the cases where the reverse is true)
36 Looking for more primitive Pythagorean triples (PPTs) Without loss of generality (WLOG), assume a is odd and b is even, so c is odd. (WLOG here means that sincea and b are interchangeable, we don t need to also consider the cases where the reverse is true) If a 2 ` b 2 c 2, then a 2 c 2 b 2
37 Looking for more primitive Pythagorean triples (PPTs) Without loss of generality (WLOG), assume a is odd and b is even, so c is odd. (WLOG here means that sincea and b are interchangeable, we don t need to also consider the cases where the reverse is true) If a 2 ` b 2 c 2, then a 2 c 2 b 2 pc bqpc ` bq.
38 Looking for more primitive Pythagorean triples (PPTs) Without loss of generality (WLOG), assume a is odd and b is even, so c is odd. (WLOG here means that sincea and b are interchangeable, we don t need to also consider the cases where the reverse is true) If a 2 ` b 2 c 2, then a 2 c 2 b 2 pc bqpc ` bq. Claims: Assume a, b, c ą Both c b and c ` b are positive odd integers.
39 Looking for more primitive Pythagorean triples (PPTs) Without loss of generality (WLOG), assume a is odd and b is even, so c is odd. (WLOG here means that sincea and b are interchangeable, we don t need to also consider the cases where the reverse is true) If a 2 ` b 2 c 2, then a 2 c 2 b 2 pc bqpc ` bq. Claims: Assume a, b, c ą Both c b and c ` b are positive odd integers. 2. There are no divisors common to c b and c ` b.
40 Looking for more primitive Pythagorean triples (PPTs) Without loss of generality (WLOG), assume a is odd and b is even, so c is odd. (WLOG here means that sincea and b are interchangeable, we don t need to also consider the cases where the reverse is true) If a 2 ` b 2 c 2, then a 2 c 2 b 2 pc bqpc ` bq. Claims: Assume a, b, c ą Both c b and c ` b are positive odd integers. 2. There are no divisors common to c b and c ` b. 3. Both c b and c ` b are perfect squares. Later we will talk about the uniqueness of prime factorizations. For today, assume that if n ab for positive integers n, a, and b, then there is a one-to-one correspondence between tprime divisors of n, counting multiplicityu and tprime divs of a, counting multu\tprime divs of b, counting multu.
41 Looking for more primitive Pythagorean triples (PPTs) Let pa, b, cq be a PPT with a, b, c ą 0. We ve shown that c b and c ` b are positive perfect squares with no common divisors.
42 Looking for more primitive Pythagorean triples (PPTs) Let pa, b, cq be a PPT with a, b, c ą 0. We ve shown that c b and c ` b are positive perfect squares with no common divisors. Define s, t P Z ą0 by c ` b s 2 and c b t 2.
43 Looking for more primitive Pythagorean triples (PPTs) Let pa, b, cq be a PPT with a, b, c ą 0. We ve shown that c b and c ` b are positive perfect squares with no common divisors. Define s, t P Z ą0 by c ` b s 2 and c b t 2. Theorem Primitive Pythagorean triples are classified by positive integers a, b, and c such that a st, b s2 t 2, and c s2 ` t 2, 2 2 for odd integers s ą t ě 1 with no common factors.
44 Looking for more primitive Pythagorean triples (PPTs) Let pa, b, cq be a PPT with a, b, c ą 0. We ve shown that c b and c ` b are positive perfect squares with no common divisors. Define s, t P Z ą0 by c ` b s 2 and c b t 2. Theorem Primitive Pythagorean triples are classified by positive integers a, b, and c such that a st, b s2 t 2, and c s2 ` t 2, 2 2 for odd integers s ą t ě 1 with no common factors. You try: Exercise 5, parts (c), (d), and/or (e).
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