π is irrational Philosophy of Mathematics Niven and Bourbaki's proof Theπ story of and related puzzles Narrator: Niraj Khare Carnegie Mellon University Qatar Being with math is being with the truth and eternity! Nov, 15, 2017 1 / 33 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 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 1596. By the time he died in 1610, he accurately found 35 digits! The digits were carved into his tombstone. 2 / 33 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 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 1596. By the time he died in 1610, he accurately found 35 digits! The digits were carved into his tombstone. 2 / 33 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof Ludolph Van Ceulen: 3:14159265358979323846264338327950288::: 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 • Around 250 BC, Archimedes proves that 10 223 220 1 3:1408 < 3 71 = 71 < π < 70 = 3 7 ≈ 3:1428. π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 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 • The story starts in ancient Egypt and Babylon about 4000 years ago! π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 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 10 223 220 1 3:1408 < 3 71 = 71 < π < 70 = 3 7 ≈ 3:1428. 4 / 33 • 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. π is irrational Philosophy of Mathematics Niven and Bourbaki's proof Time line I: Ancient period • Around 250 BC, Archimedes proves that 10 223 220 1 3:1408 < 3 71 = 71 < π < 70 = 3 7 ≈ 3:1428. 4 / 33 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 6= 0 such a that q = b . 31 3 −5 Examples: 3:1 = 10 , 3 = 1 , −5 = 1 , 3:127127127 ··· 127 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof A rational number 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; · · · g. 5 / 33 In other words, a number q is a rational if there are integers a and b 6= 0 such a that q = b . 31 3 −5 Examples: 3:1 = 10 , 3 = 1 , −5 = 1 , 3:127127127 ··· 127 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof A rational number 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; · · · g. Definition A rational number is a ratio of two integers. 5 / 33 3 −5 3 = 1 , −5 = 1 , 3:127127127 ··· 127 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof A rational number 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; · · · g. 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 6= 0 such a that q = b . 31 Examples: 3:1 = 10 , 5 / 33 −5 −5 = 1 , 3:127127127 ··· 127 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof A rational number 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; · · · g. 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 6= 0 such a that q = b . 31 3 Examples: 3:1 = 10 , 3 = 1 , 5 / 33 3:127127127 ··· 127 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof A rational number 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; · · · g. 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 6= 0 such a that q = b . 31 3 −5 Examples: 3:1 = 10 , 3 = 1 , −5 = 1 , 5 / 33 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof A rational number 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; · · · g. 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 6= 0 such a that q = b . 31 3 −5 Examples: 3:1 = 10 , 3 = 1 , −5 = 1 , 3:127127127 ··· 127 5 / 33 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof A rational number 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; · · · g. 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 6= 0 such a that q = b . 31 3 −5 Examples: 3:1 = 10 , 3 = 1 , −5 = 1 , 3:127127127 ··· 127 5 / 33 ) 1000x − x = 3127 − 3 ) 999x = 3124 3124 ) x = : 999 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 3:127127127 ··· 127 is rational! Proof. Let x = 3:127127127 ··· 127. 1000x = 3127:127127 ··· 127 x = 3:127127127 ··· 127 6 / 33 3124 ) x = : 999 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 3:127127127 ··· 127 is rational! Proof. Let x = 3:127127127 ··· 127. 1000x = 3127:127127 ··· 127 x = 3:127127127 ··· 127 ) 1000x − x = 3127 − 3 ) 999x = 3124 6 / 33 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 3:127127127 ··· 127 is rational! Proof. Let x = 3:127127127 ··· 127. 1000x = 3127:127127 ··· 127 x = 3:127127127 ··· 127 ) 1000x − x = 3127 − 3 ) 999x = 3124 3124 ) x = : 999 6 / 33 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 3:127127127 ··· 127 is rational! Proof. Let x = 3:127127127 ··· 127. 1000x = 3127:127127 ··· 127 x = 3:127127127 ··· 127 ) 1000x − x = 3127 − 3 ) 999x = 3124 3124 ) x = : 999 6 / 33 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 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 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 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 but not both. π is irrational Philosophy of Mathematics Niven and Bourbaki's proof Quotes • 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 • The oldest, shortest words \yes" and \no" are those which require the most thought. - Pythagoras π is irrational Philosophy of Mathematics Niven and Bourbaki's proof Quotes • A statement or a proposition in mathematics is a sentence that is either true or false but not both. 8 / 33 • The oldest, shortest words \yes" and \no" are those which require the most thought. - Pythagoras π is irrational Philosophy of Mathematics Niven and Bourbaki's proof Quotes • A statement or a proposition in mathematics is a sentence that is either true or false but not both. 8 / 33 Who is the manager? Tom: \Mary is an engineer." Alice: \Tom is the manager." Mary: \Alice is the manager." Alice: \Mary is the manager." π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 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 Tom: \Mary is an engineer." Alice: \Tom is the manager." Mary: \Alice is the manager." Alice: \Mary is the manager." π is irrational Philosophy of Mathematics Niven and Bourbaki's proof 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 π is irrational Philosophy of Mathematics Niven and Bourbaki's proof An engineer or a manager? Tom, Mary and Alice work for `Logic is Fun'.