The story of. and related puzzles
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1 The story of π and related puzzles Narrator: Niraj Khare Carnegie Mellon University Qatar Being with math is being with the truth and eternity! 1 / 33
2 Time line III (a): series expressions for π Ludolph Van Ceulen using archimedean method with 500 million sides calculated π calculated π to an accuracy of 20 decimal digits by By the time he died in 1610, he accurately found 35 digits! The digits were carved into his tombstone. 2 / 33
3 Time line III (a): series expressions for π Ludolph Van Ceulen using archimedean method with 500 million sides calculated π calculated π to an accuracy of 20 decimal digits by By the time he died in 1610, he accurately found 35 digits! The digits were carved into his tombstone. 2 / 33
4 Ludolph Van Ceulen: Ludolph van Ceulen Dutch-German mathematician Ludolph van Ceulen was a German-Dutch mathematician from Hildesheim. He emigrated to the Netherlands. Wikipedia: Born: January 28, 1540, Hildesheim, Germany Died: December 31, 1610, Leiden, Netherlands Known for: pi Institution: Leiden University Notable student: Willebrord Snellius 3 / 33
5 Time line I: Ancient period The story starts in ancient Egypt and Babylon about 4000 years ago! Around 450 BCE, Anaxagoras proposes squaring the circle from a prison! The puzzle was finally settled in 1882 AD. 4 / 33
6 Time line I: Ancient period Around 450 BCE, Anaxagoras proposes squaring the circle from a prison! The puzzle was finally settled in 1882 AD. Around 250 BC, Archimedes proves that < = < π < = / 33
7 Time line I: Ancient period Around 250 BC, Archimedes proves that < = < π < = / 33
8 A rational number π is irrational An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples:{, 3, 2, 1, 0, 1, 2, 3, }. 5 / 33
9 A rational number π is irrational An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples:{, 3, 2, 1, 0, 1, 2, 3, }. Definition A rational number is a ratio of two integers. 5 / 33
10 A rational number π is irrational An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples:{, 3, 2, 1, 0, 1, 2, 3, }. Definition A rational number is a ratio of two integers. In other words, a number q is a rational if there are integers a and b 0 such that q = a b. Examples: 3.1 = 31 10, 5 / 33
11 A rational number π is irrational An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples:{, 3, 2, 1, 0, 1, 2, 3, }. Definition A rational number is a ratio of two integers. In other words, a number q is a rational if there are integers a and b 0 such that q = a b. Examples: 3.1 = 31 10, 3 = 3 1, 5 / 33
12 A rational number π is irrational An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples:{, 3, 2, 1, 0, 1, 2, 3, }. Definition A rational number is a ratio of two integers. In other words, a number q is a rational if there are integers a and b 0 such that q = a b. Examples: 3.1 = 31 10, 3 = 3 5 1, 5 = 1, 5 / 33
13 A rational number π is irrational An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples:{, 3, 2, 1, 0, 1, 2, 3, }. Definition A rational number is a ratio of two integers. In other words, a number q is a rational if there are integers a and b 0 such that q = a b. Examples: 3.1 = 31 10, 3 = 3 5 1, 5 = 1, / 33
14 A rational number π is irrational An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples:{, 3, 2, 1, 0, 1, 2, 3, }. Definition A rational number is a ratio of two integers. In other words, a number q is a rational if there are integers a and b 0 such that q = a b. Examples: 3.1 = 31 10, 3 = 3 5 1, 5 = 1, / 33
15 is rational! Proof. Let x = x = x = / 33
16 is rational! Proof. Let x = x = x = x x = x = / 33
17 is rational! Proof. Let x = x = x = x x = x = 3124 x = / 33
18 is rational! Proof. Let x = x = x = x x = x = 3124 x = / 33
19 Not everything is rational! Hippasus of Metapontum (/hpss/; Greek:, Hppasos; fl. 3rd century BC), was a Pythagorean philosopher. He was the first one to claim that there were irrational numbers! 7 / 33
20 Not everything is rational! Hippasus of Metapontum (/hpss/; Greek:, Hppasos; fl. 3rd century BC), was a Pythagorean philosopher. He was the first one to claim that there were irrational numbers! 7 / 33
21 Quotes π is irrational The oldest, shortest words yes and no are those which require the most thought. - Pythagoras A statement or a proposition in mathematics is a sentence that is either true or false 8 / 33
22 Quotes π is irrational A statement or a proposition in mathematics is a sentence that is either true or false but not both. 8 / 33
23 Quotes π is irrational A statement or a proposition in mathematics is a sentence that is either true or false but not both. 8 / 33
24 An engineer or a manager? Tom, Mary and Alice work for Logic is Fun. Two of them are engineers and exactly one of them is a manager. The manager always lies and the engineers always speak the truth. 9 / 33
25 An engineer or a manager? Tom, Mary and Alice work for Logic is Fun. Two of them are engineers and exactly one of them is a manager. The manager always lies and the engineers always speak the truth. Who is the manager? 9 / 33
26 An engineer or a manager? Tom, Mary and Alice work for Logic is Fun. Two of them are engineers and exactly one of them is a manager. The manager always lies and the engineers always speak the truth. Who is the manager? Tom: Mary is an engineer. Alice: Tom is the manager. Mary: Alice is the manager. Alice: Mary is the manager. 9 / 33
27 An engineer or a manager? Tom, Mary and Alice work for Logic is Fun. Two of them are engineers and exactly one of them is a manager. The manager always lies and the engineers always speak the truth. Who is the manager? Tom: Mary is an engineer. Alice: Tom is the manager. Mary: Alice is the manager. Alice: Mary is the manager. 9 / 33
28 A proof by contradiction! To prove: Alice is the manager. Proof. On the contrary assume that Alice is not the manager. 10 / 33
29 A proof by contradiction! To prove: Alice is the manager. Proof. On the contrary assume that Alice is not the manager. Therefore, Alice is an engineer. Thus, she always speak the truth. 10 / 33
30 A proof by contradiction! To prove: Alice is the manager. Proof. On the contrary assume that Alice is not the manager. Therefore, Alice is an engineer. Thus, she always speak the truth. Hence, Tom and Mary are both managers. 10 / 33
31 A proof by contradiction! To prove: Alice is the manager. Proof. On the contrary assume that Alice is not the manager. Therefore, Alice is an engineer. Thus, she always speak the truth. Hence, Tom and Mary are both managers. A contradiction! 10 / 33
32 A proof by contradiction! To prove: Alice is the manager. Proof. On the contrary assume that Alice is not the manager. Therefore, Alice is an engineer. Thus, she always speak the truth. Hence, Tom and Mary are both managers. A contradiction! Therefore, our assumption must be false. So the negation of our assumption is true. 10 / 33
33 A proof by contradiction! To prove: Alice is the manager. Proof. On the contrary assume that Alice is not the manager. Therefore, Alice is an engineer. Thus, she always speak the truth. Hence, Tom and Mary are both managers. A contradiction! Therefore, our assumption must be false. So the negation of our assumption is true. 10 / 33
34 Contrapositive π is irrational If president, then at least 35 years old. 11 / 33
35 Contrapositive π is irrational If president, then at least 35 years old. If not yet 35, then cannot be the president. 11 / 33
36 Contrapositive π is irrational If president, then at least 35 years old. If not yet 35, then cannot be the president. 11 / 33
37 Contrapositive π is irrational If president, then at least 35 years old. If not yet 35, then cannot be the president. If Tom is a parrot, 11 / 33
38 Contrapositive π is irrational If president, then at least 35 years old. If not yet 35, then cannot be the president. If Tom is a parrot, then Tom is a bird. 11 / 33
39 Contrapositive π is irrational If president, then at least 35 years old. If not yet 35, then cannot be the president. If Tom is a parrot, then Tom is a bird. If Tom is not a bird then Tom cannot be a parrot. 11 / 33
40 Contrapositive π is irrational If president, then at least 35 years old. If not yet 35, then cannot be the president. If Tom is a parrot, then Tom is a bird. If Tom is not a bird then Tom cannot be a parrot. 11 / 33
41 History of π is irrational. There are conflicting claims who first guessed that π is not a rational.but was believed by many by 5-th century AD. 12 / 33
42 History of π is irrational. There are conflicting claims who first guessed that π is not a rational.but was believed by many by 5-th century AD. In 1761 to 1776, Lambert and Legendre proved that π is not a ratio of two integers.[cajori, page 246] 12 / 33
43 History of π is irrational. There are conflicting claims who first guessed that π is not a rational.but was believed by many by 5-th century AD. In 1761 to 1776, Lambert and Legendre proved that π is not a ratio of two integers.[cajori, page 246] In 1882, Ferdinand von Lindemann proved transcendence of π (i.e., squaring the circle is impossible). [Berggren, page 407] 12 / 33
44 History of π is irrational. There are conflicting claims who first guessed that π is not a rational.but was believed by many by 5-th century AD. In 1761 to 1776, Lambert and Legendre proved that π is not a ratio of two integers.[cajori, page 246] In 1882, Ferdinand von Lindemann proved transcendence of π (i.e., squaring the circle is impossible). [Berggren, page 407] 12 / 33
45 Lambart s idea: I π is irrational Lemma Let x be a real number. If x is rational, then tan(x) is irrational. 13 / 33
46 Lambart s idea: II π is irrational Theorem π is irrational. Proof. Assume that π = a b π 4 = a 4b. where a and b are integers. Thus, 14 / 33
47 Lambart s idea: II π is irrational Theorem π is irrational. Proof. Assume that π = a b where a and b are integers. Thus, π 4 = a 4b.By previous lemma, tan( π 4 ) must be an irrational. 14 / 33
48 Lambart s idea: II π is irrational Theorem π is irrational. Proof. Assume that π = a b where a and b are integers. Thus, π 4 = a 4b.By previous lemma, tan( π 4 ) must be an irrational. But tan( π 4 ) = 1 = 1 1. A contradiction! 14 / 33
49 Lambart s idea: II π is irrational Theorem π is irrational. Proof. Assume that π = a b where a and b are integers. Thus, π 4 = a 4b.By previous lemma, tan( π 4 ) must be an irrational. But tan( π 4 ) = 1 = 1 1. A contradiction! 14 / 33
50 π is irrational Theorem π is irrational. Irrationality is not limited to numbers! 15 / 33
51 On the contrary assume that π is a rational number. Thus, π = a b for integers a and b 0. Without loss of generality, let a and b be both positive. 16 / 33
52 On the contrary assume that π is a rational number. Thus, π = a b for integers a and b 0. Without loss of generality, let a and b be both positive. 16 / 33
53 On the contrary assume that π is a rational number. Thus, π = a b for integers a and b 0. Without loss of generality, let a and b be both positive. 16 / 33
54 On the contrary assume that π is a rational number. Thus, π = a b for integers a and b 0. Without loss of generality, let a and b be both positive. 16 / 33
55 A bunny found! π is irrational 17 / 33
56 A bunny found! π is irrational For any positive integer n, define f(x) = bn x n (π x) n n! = xn (a bx) n n! 17 / 33
57 A bunny found! π is irrational For any positive integer n, define f(x) = bn x n (π x) n n! = xn (a bx) n n! 17 / 33
58 Some properties of f(x): I For all real x, f(x) = f(π x). For any non-negative integer k, f (k) (π x) = 1 k f (k) (x) where f (k) (x) denotes the k-th derivative of f with respect to x. 18 / 33
59 Some properties of f(x): I For all real x, f(x) = f(π x). For any non-negative integer k, f (k) (π x) = 1 k f (k) (x) where f (k) (x) denotes the k-th derivative of f with respect to x. For all non-negative integer k, f (k) (0) is an integer. 18 / 33
60 Some properties of f(x): I For all real x, f(x) = f(π x). For any non-negative integer k, f (k) (π x) = 1 k f (k) (x) where f (k) (x) denotes the k-th derivative of f with respect to x. For all non-negative integer k, f (k) (0) is an integer. Therefore, for all non-negative integer k, f (k) (π) is an integer too. 18 / 33
61 Some properties of f(x): I For all real x, f(x) = f(π x). For any non-negative integer k, f (k) (π x) = 1 k f (k) (x) where f (k) (x) denotes the k-th derivative of f with respect to x. For all non-negative integer k, f (k) (0) is an integer. Therefore, for all non-negative integer k, f (k) (π) is an integer too. 18 / 33
62 Some properties of f(x): I For all real x, f(x) = f(π x). For any non-negative integer k, f (k) (π x) = 1 k f (k) (x) where f (k) (x) denotes the k-th derivative of f with respect to x. For all non-negative integer k, f (k) (0) is an integer. Therefore, for all non-negative integer k, f (k) (π) is an integer too. 18 / 33
63 f (k) (0) is an integer: I f(x) = xn (a bx) n n! n (( ) = xn n )a (n i) ( 1) i b i x i n! i i=0 ( (n ) n i) a (n i) ( 1) i b i x n+i = n! i=0 As f(x) is a polynomial of degree 2n, for all k > 2n f (k) (x) = / 33
64 f (k) (0) is an integer: II = f (k) (x) n c a (n i) ( 1) i b i (n + i)(n + i 1) (n + i {k 1})x n+i k i=0 where c = ( n i) and when n + i k. When x = 0, only the term with n + i = k contributes. In particular f (k) (0) = 0 for all k < n. For k = n + i, f (k) (0) = c a(n i) ( 1) i b i (k)(k 1) (k {k 1}) n! = c a(n i) ( 1) i b i (n + i)(n + i 1) (n) (1) n! n! 20 / 33
65 Some properties of f(x): II Define g(x) = f(x) f (2) (x) + f (4) (x) f (6) (x) + + ( 1) k f (2k) (x) + + ( 1) (n 1) f (2(n 1)) (x) + ( 1) n f (2n) (x). Thus, g(0) and g(π) are integers. 21 / 33
66 Some properties of f(x): II Define g(x) = f(x) f (2) (x) + f (4) (x) f (6) (x) + + ( 1) k f (2k) (x) + + ( 1) (n 1) f (2(n 1)) (x) + ( 1) n f (2n) (x). Thus, g(0) and g(π) are integers. Note that g(x) + g (2) (x) = f(x) where g (2) (x) = g (x). 21 / 33
67 Some properties of f(x): II Define g(x) = f(x) f (2) (x) + f (4) (x) f (6) (x) + + ( 1) k f (2k) (x) + + ( 1) (n 1) f (2(n 1)) (x) + ( 1) n f (2n) (x). Thus, g(0) and g(π) are integers. Note that g(x) + g (2) (x) = f(x) where g (2) (x) = g (x). 21 / 33
68 Some properties of f(x): II Define g(x) = f(x) f (2) (x) + f (4) (x) f (6) (x) + + ( 1) k f (2k) (x) + + ( 1) (n 1) f (2(n 1)) (x) + ( 1) n f (2n) (x). Thus, g(0) and g(π) are integers. Note that g(x) + g (2) (x) = f(x) where g (2) (x) = g (x). 21 / 33
69 Anti-derivative of f(x)sin(x) d dx {g (x)sin(x) g(x)cos(x))} = g (x)sin(x) + g (x)cos(x) g (x)cos(x) + g(x)sin(x) = g (x)sin(x) + g(x)sin(x) = {g (x) + g(x)}sin(x) = f(x)sin(x) 22 / 33
70 π 0 π is irrational f(x)sin(x)dx is an integer for all positive integers n. π 0 f(x)sin(x)dx = {g (x)sin(x) g(x)cos(x)} π 0 = {g (π)sin(π) g(π)cos(π)} {g (0)sin(0) g(0)cos(0)} 23 / 33
71 π 0 π is irrational f(x)sin(x)dx is an integer for all positive integers n. π 0 f(x)sin(x)dx = {g (x)sin(x) g(x)cos(x)} π 0 = {g (π)sin(π) g(π)cos(π)} {g (0)sin(0) g(0)cos(0)} = g(π) + g(0) 23 / 33
72 π 0 π is irrational f(x)sin(x)dx is an integer for all positive integers n. π 0 f(x)sin(x)dx = {g (x)sin(x) g(x)cos(x)} π 0 = {g (π)sin(π) g(π)cos(π)} {g (0)sin(0) g(0)cos(0)} = g(π) + g(0) Hence, π 0 f(x)sin(x)dx is an integer for all positive integers n. 23 / 33
73 π 0 π is irrational f(x)sin(x)dx is an integer for all positive integers n. π 0 f(x)sin(x)dx = {g (x)sin(x) g(x)cos(x)} π 0 = {g (π)sin(π) g(π)cos(π)} {g (0)sin(0) g(0)cos(0)} = g(π) + g(0) Hence, π 0 f(x)sin(x)dx is an integer for all positive integers n. 23 / 33
74 π 0 π is irrational f(x)sin(x)dx is not an integer for LARGE n. = π 0 π = bn n! 0 π f(x)sin(x)dx b n x n (π x) n sin(x)dx n! 0 x n (π x) n sin(x)dx 24 / 33
75 π 0 π is irrational f(x)sin(x)dx is not an integer for LARGE n. π 0 < bn x n (π x) n sin(x)dx n! 0 25 / 33
76 π 0 π is irrational f(x)sin(x)dx is not an integer for LARGE n. π 0 < bn x n (π x) n sin(x)dx n! 0 25 / 33
77 War between exponent and factorial! 0 < bn ( π ) 2n (π) n! 2 y = 6n n! 26 / 33
78 War between exponent and factorial! 0 < bn ( π ) 2n (π) n! 2 y = 6n n! 26 / 33
79 War between exponent and factorial! 0 < bn ( π ) 2n (π) n! 2 y = 6n n! 26 / 33
80 π 0 π is irrational f(x)sin(x)dx is not an integer for LARGE n: III As n, b n ( π ) 2n lim (π) = 0. n n! 2 27 / 33
81 π 0 π is irrational f(x)sin(x)dx is not an integer for LARGE n: III As n, b n ( π ) 2n lim (π) = 0. n n! 2 π 0 < bn x n (π x) n sin(x)dx n! 0 27 / 33
82 π 0 π is irrational f(x)sin(x)dx is not an integer for LARGE n: III As n, b n ( π ) 2n lim (π) = 0. n n! 2 π 0 < bn x n (π x) n sin(x)dx n! 0 < bn ( π ) 2n (π) n! 2 27 / 33
83 π 0 π is irrational f(x)sin(x)dx is not an integer for LARGE n: III As n, b n ( π ) 2n lim (π) = 0. n n! 2 π 0 < bn x n (π x) n sin(x)dx n! 0 < bn ( π ) 2n (π) n! 2 < / 33
84 π 0 π is irrational f(x)sin(x)dx is not an integer for LARGE n: III As n, b n ( π ) 2n lim (π) = 0. n n! 2 π 0 < bn x n (π x) n sin(x)dx n! 0 < bn ( π ) 2n (π) n! 2 < / 33
85 Bunny or a pigeon? For large integer n, bn n! π 0 xn (π x) n sin(x)dx is an integer. 0 < bn n! π 0 x n (π x) n sin(x)dx < / 33
86 Formal and informal references: Informal References Documentaries: Math and rise of civilizations: math-rise-of-civilization-science-docs-documentaries-2/ BBC: Story of Mathematics: http: // Websites: The history of pi by David Wilson Papers2000/wilson.html Archimedes Approximation of Pi archimedes/archimedes.html Euclid s Elements Wekipedia: Ludolph Van Ceulen s biography 29 / 33
87 Formal and informal references: Formal References(I) George E. Andrews, Peter Paule. Some questions concerning computer-generated proofs of a binomial double-sum identity, J. Symbolic Comput. 16(1993), P. Backman, The history of Pi, The Golem Press. Boulder Colorado, J.L.Berggren, J., Borwein, P. Borwein, Pi: A Source Book, Springer, D. Blatner, The joy of Pi, Walker Publishing Company, Inc Newyork, F. Cajori, A history of Mathematics, MacMillan and Co. London, Sir T. Heath,A History of Greek Mathematics: From Thales 30 / 33
88 Formal and informal references: Formal References (II) Bourbaki, N Fonctions d une variable relle, chap. IIIIII, Actualits Scientifiques et Industrielles (in French), 1074, Hermann, pp , Jeffreys, Harold, Scientific Inference (3rd ed.), Cambridge University Press, p. 268, Niven, Ivan, A simple proof that is irrational (PDF), Bulletin of the American Mathematical Society, 53 (6), p. 509, / 33
89 Acknowledgments π is irrational I would specially like to thank: i) Prof. Marion Oliver for his support, interest in history of mathematics and encouragement for the concept of Explore Math. ii) Prof. H. Demirkoparan and Prof. Z. Yilma for their help and suggestions. iii) Kara, Angela, Catalina and Geetha for promoting the event and taking care of logistics. iv) Ghost of Ludolph Van Ceulen for haunting me and pushing me to explore more about Pi :). 32 / 33
90 Something to carry home! Q: What will a logician choose: a half of an egg or eternal bliss? 33 / 33
91 Something to carry home! Q: What will a logician choose: a half of an egg or eternal bliss? A: A half of an egg! Because nothing is better than eternal bliss, and a half of an egg is better than nothing. 33 / 33
92 Something to carry home! Q: What will a logician choose: a half of an egg or eternal bliss? A: A half of an egg! Because nothing is better than eternal bliss, and a half of an egg is better than nothing. 33 / 33
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