Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop The Wallis Product, a Connection to Pi and Probability, and Maybe the Gamma Function Perhaps? Aba Mbirika Assistant Professor of Mathematics May 8th, 2017 A Math Presentation for the Riemann Seminar Class Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Questions I may or may not address (in no particular order) Why is π so freaking cool? How many digits of π can you recite? What does π have to do with combinatorics? Coin flips? Drunken walks? Are Ralph Waldo Emerson and Henry David Thoreau transcendentalists? And is π transcendental? Does this talk have ANYTHING at all to do with the gamma 3 1 function and more precisely the value Γ 2 being 2 factorial? Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Introducing my co-speaker, Sophie Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Venn diagram of the Real Numbers Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Sophie suggests Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop The 16th and 17th Centuries Fran¸coisVi`eta(1579, France) p p p q p p 2 2 2 + 2 2 + 2 + 2 = · · ··· π 2 2 2 John Wallis (1650, England) 1 π 2 · 2 4 · 4 6 · 6 8 · 8 Y 2n · 2n = · · · ··· = 2 1 · 3 3 · 5 5 · 7 7 · 9 (2n − 1)(2n + 1) n=1 Equation above comes up BIG TIME later in the talk! Lord Brouncker (1650, England) 4 12 = 1 + π 32 2 + 52 2 + . 2 + .. Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop James Gregory (1668, Scotland) π 1 1 1 = 1 − + − ··· 4 3 5 7 Does anyone see a Calc II way to prove this? Maybe use the Taylor series expansion for arctan(x) and a judicious choice for x. Abraham Sharp (1699, England) π r1 1 1 1 1 = 1 − + − + + ··· 6 3 3 × 3 32 × 5 33 × 7 34 × 9 Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop The 18th Century Georges-Louis Leclerc, Comte de Buffon (1760, France) A plane is ruled with parallel lines 1 inch apart. A needle of length 1 inch is dropped randomly on the plane. What is the probability that it will be lying across one of the lines? 2 ≈ 63:66% π Johann Lambert (1761, Germany) proves that π is irrational. In this same paper, he conjectures that π is transcendental. Remember this for a few slides from now when I speak of Lindemann. Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Leonhard Euler (1748, Switzerland) publishes Introductio in analysin infinitorum. Math historians say that it was this publication that catapulted the symbol π into popular use. QUESTION: What is the coolest of all of Euler's formula? ANSWER: eiπ + 1 = 0 Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop How cool do these folks think this equation is? Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop How cool do these folks think this equation is? Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop How cool do these folks think this equation is? Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Can this guy wash his shirt anymore? Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop The 19th Century to the present William Rutherford (1841, England) π 1 1 1 = 4 arctan − arctan + arctan 4 5 70 99 He uses this to estimate π to 208 places (152 correct). But even 152 decimal places broke the world record at the time. The modern quest for EVER-more digits begins. Ferdinand Lindemann (1882, Germany) proves that π is transcendental. Srinivasa Ramanujan (1913, India) gives many remarkable approximations of π: 1 2 192 4 9 + 22 =3 :1415926525826461252 ::: 355 :0003 113 1 − 3533 =3 :1415926535897943 :::. Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Sophie wonders Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Recall from a previous slide, John Wallis (1655) arrived at his celebrated formula 1 π 2 · 2 4 · 4 6 · 6 8 · 8 Y 2n · 2n = · · · ··· = 2 1 · 3 3 · 5 5 · 7 7 · 9 (2n − 1)(2n + 1) n=1 Many textbook proofs of this formula rely the family fIng of definite integral Z π 2 n In := (sin x) dx 0 by repeated partial integration. (Hint: Let u = sinn−1 x and n−1 dv = sin x dx. And derive the recursion formula In = n In−2 π with initial values I0 = 2 and I1 = 1.) This is cool problem for Calculus II. I highly suggest you write this down and give it a go later if you want. Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Overview for the rest of the talk Introduce two equivalent combinatorial problems Coin flips problem, and Drunken walks problem. Give a geometric proof of Wallis' product formula. Use this to derive the solution to the combinatorial problems. And of course, we will connect ALL of this to π and perhaps 3 the gamma function or more precisely the value of Γ 2 1 which is 2 factorial. Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Flipping Coins Suppose we flip a fair coin 2n times. What are the chances that we get an equal amount of heads and tails? Label the head and tail sides of the coin as H and T respectively. We can view the 2n coin flips as sequences of H's and T 's. Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop The sequences for n = 1 (i.e., flipping a coin 2 × 1 = 2 times) HH TT HT TH So there are exactly 2 ways to get exactly one head and one tail. So there is a .5 probability that this event occurs when we flip a coin 2 times. Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop The sequences for n = 2 (i.e., flipping a coin 2 × 2 = 4 times): HHHH THHH HHHT THHT HHTH THTH HHTT THTT HTHH TTHH HTHT TTHT HTTH TTTH HTTT TTTT QUESTION: What is the probability that you get an equal number of heads and tails when you flip a coin 4 times? ANSWER: Well this is silly question since I have highlighted certain sequences above in bold red! Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Flipping Coins QUESTION: How many sequences for a given n are there (i.e., flipping a coin 2 × n = 2n times)? Why? ANSWER: You're absolutely right! There are exactly 22n possible sequences! Suppose we flip a fair coin 2n times. What are the chances that we get an equal amount of heads and tails? We can then rephrase the original question: Out of the 22n possible sequences, how many sequences contain exactly n H's and n T 's? Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Example Consider the case for n = 5. Then the number of ways of getting exactly five H's is the number of ways of choosing 5 of the 10 sequence slots in which to place them. Hence there are 10 10! = = 252 5 5! 5! ways. Now since there are exactly 210 = 1024 possible sequences of H's and T 's, the probability of getting exactly five H's and five 252 T 's is 1024 = :24609375. This number is close to p1 which is approximately :252313. 5π For n = 20, the probability is .125371 while p 1 is approximately 20π .126157. QUESTION: What is your conjecture???? Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Goal: As n gets large, the probability of getting exactly n heads and n tails out of 2n coin flips approaches the number p1 . nπ Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Drunken Walks The coin flip problem can be recast in the following drunken walk model: Why pi? A brief history of π Coin toss and drunken walk problem A simple geometric proof Poop Drunken Walks Here is the drunken walk model: Consider a walk where we go forward each time, however at each step forward we veer 45 degrees left or right. Let H be denoted by a step forward that veers left and T to be one that veers right. If after 2n steps we return to the original horizontal starting level, then we must have taken exactly n left-steps and n right-steps.