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What is pi?

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What is pi?

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  1. “People have calculated billions of digits of pi because of the human desire to do something that’s never been done before. When George Mallory was asked why he wanted to climb Mt. Everest, he replied, ‘Because it’s there’. Well, pi is certainly here. Like the outer planets, it’s built into the fabric of our physical universe and it will always be explored.” - The Story of Pi, Cal.Tech.

  2. What is pi? The ratio of the circumference to the diameter of ANY circle is constant. It is between 3 and . It is close to but NOT EQUAL to 3.14 or . Its digits will NEVER terminate or repeat… (proved in 1766) ...but will ALWAYS continue to fascinate mankind. See “Peel Circle for pi PPT.gsp”

  3. Irrational & Transcendental • IRRATIONAL Cannot be expressed as the quotient of 2 integers This also means it cannot be written as a decimal for it will never terminate or repeat. (speculated early; proved 1767) • TRANSCENDENTAL Cannot be expressed as a root of an algebraic equation with finite terms, rational coefficients - “transcends algebra” (first speculated by Euler 1748, proved by Lindemann 1882)

  4. Our “Pi String” 1998-2002 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 (100) 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 (200) 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 (300) 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 (400) 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 (500) 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 (600) 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 10507922796892589235 (700)4201995611 21290219608640344181 5981362977 47713099605187072113 4999999837 2978049951 0597317328 1609631859 (800) 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 (900) 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989...(1000) Each student adds beads to it on -Day.

  5. Where Can we find pi? IN EVERYTHING CIRCULAR (of course) (See “torus.gsp”)

  6. WHERE ELSE? 1777-1855 German • Area under bell (Gaussian) curveCarl Gauss, “prince of mathematics” • Electricity - formulas for alternating currents and radiation from radio & TV antennas • Probability P (2 integers have no common factors) = P (lattice pt. is visible from origin) = P (needle lands on line) =

  7. Connections to integers (Leibniz) (John Wallis 1655) (Leonard Euler)

  8. can be used to find decimal digits of pi A BIT OF CALCULUS Leibniz - 1st infinite series for pi Newton - converges quicker 1646-1716 1642-1727

  9. Earliest Known Record of Picirca 1650 BC No number has captured the attention and imaginations of people throughout the ages as much as the ratio of a circle’s circumference to its diameter. On the Rhind Papyrus, Egyptian scribe, Ahmes, wrote this ratio as “4 times the square of eight-ninths”

  10. More Attempts to rationalize(all prior to Arabic numerals and decimals) Babylonians, same time as Egyptian Rhind Papyrus, 1650 BC Ptolemy (Alexandria, Egypt) 150 AD Also used by Columbus on his voyage to the New World Archimedes (Syracuse, 287-212 BC) Found pi to be between these two fractions. This average error is only 0.0002! Tsu Ch’ung Chi China, 450 AD Srinivasa Ramanujan (India, 1887-1920) (http://www.science-frontiers.com/sf053/sf053p19.htm) (This is an irrational approximation.)

  11. Archimedes, 250 BC

  12. I have proof! 1767 - Johann Lambert proved irrational 1794 - Adrien-Marie Legendre proved irrational 1840 - Joseph Liouville proved transcendental nos. exist (used limits of continued fractions) 1873 - Charles Hermite provede transcendental 1882 - Ferdinand Lindemann proved transcendental 1728-1777 Swiss French French French German

  13. Interesting digits largest 7-digit sum in the first million digits! • Starting at digit #772 - 9999998 occurs • Starting at digit #509,400 - 112552 occurs • Starting at digit #1,286,368 - 980-7280 occurs • In 1st million, no “123456” but 012345 twice • #357 #358 #359 #360 #361 #362 #363 … 9 0 3 6 0 0 1 A special date - can you guess it? A special telephone number - do you know it? 123456789 first appears at 523,551,502nd digit

  14. Can’t get enough pi digits Circa 1600 - decimal fractions & logarithms invented All by hand - months 1596 … Ludolph van Ceulen (Dutch) calculates 35 digits 1706 … John Machin calculates 100 digits 1874 … William Shanks calculates 707 digits 1947 … Ferguson (using desk calculator) finds 808 digits 1949 … ENIAC computer (DoD & U. of Pen.) finds 2037 digits 1973 … CDC 7600 (Paris) finds 1,000,000 digits (23 hrs) 1989 … 1,000,000,000 digits (USSR Chudnovsky brothers, NY) 1999 … Hitachi SR8000 (Tokyo) 206,158,430,000 digits (37 hrs) But Ferguson finds error in 527th onward used Gauss-Legendre algorithm Why still do this? …to find out more about pi …to test computer architecture & efficiency ... to test software for accuracy and speed

  15. -TV From the original series, 1967 - episode #36 “Wolf in the Fold”. The main computer of the Starship Enterprise is possessed by an evil alien entity. Kirk, Spock and the gang have a plan to send the entity into deep space but must first find a way to keep the computer “busy” so it doesn’t detect their plan. STAR TREK (1 min.) STARGATE (4 min.) Courtesy of Randy Coombs (3/00) The main characters are trying to uncover a secret hidden by a mysterious puzzle. The legend is that the ancient Norse god, Thor, created the puzzle so that when mankind developed enough to solve the puzzle, we would be ready for the secret behind it!

  16. More misc. pi facts German 1879-1955 • Albert Einstein born 3 / 14 / 1879 (Pi-Day) • Symbol introduced by Leonard Euler, 1737 • Euler (using DeMoivre’s work) • Hat size = Although used first by William Jones in 1706 (short for “periphery”), he did not have the weight to make it popular. Once the renowned Euler (“Oiler”) picked it up (previously using “p” or “c”) it became the standard. Swiss 1707-1783

  17. More misc. pi facts • To find the circumference of a circle the size of the known universe, accurate to within the radius of one proton, how many decimal places of pi would be needed? only 39! • Consider the following series of integers, each using one more digit of pi: 3, 31, 314, 3141, 31415, 314159, 3141592, etc. Out of the first 1000 numbers in this series, only 4 are prime! • The world record for pi-recitation (from memory) is held by Hiroyuki Gotu, age 21. (Seattle Times 2-26-1995) 9 hours ... 42,000 digits!

  18. Pi In print “And he made a molten sea, 10 cubits from brim to brim, round in compass ... and a line of 30 cubits did compass it round about.” (cubit = dist. from elbow to tip of fingers) Large brass casting in Solomon’s Temple • Bible - I Kings vii.23 (Solomon’s Temple) • Jules Verne - “20,000 Leagues Under the Sea” • A 1970 advertisement - “The Nautilus was stationary, floating near a mountain which formed a sort of quay”(lake) … “imprisoned by a circle of walls, measuring 2 miles in diameter and 6 in circumference” CADAEIB F

  19. Pi Song Oh, number PiOh, number PiYour digits are unending,Oh, number PiOh, number PiNo pattern are you sending.You're three point one four one five nine,And even more if we had time,Oh, number PiOh, number PiFor circle lengths unbending. http://www.winternet.com/~mchristi/piday.html To the tune of “O Christmas Tree”

  20. Cologne by Givenchy pi scent This was their 1999 advertisement at http://www.givenchy.com/givenchy/givenchy.html The Inspiration The answer lay in the quest itself. From the exploration of new territories to the conquest of space, men have always endeavored to push back the frontiers of the known world and reveal the mysteries of the unknown. Man’s essential character lies in his strength and determination in pushing back his limits. The Name Resonant with history and mystery, is a link between past, present and future. Pi is the universal number, the transcendental number, the ruling number. Since Archimedes’ discovery of , more than 2000 years ago, has been the object of a ceaseless quest. This letter of the Greek alphabet is used in mathematics to express the constant ratio of the circumference of a circle to its diameter. Today man is still seeking to establish ’s unlimited decimals. The Bottle Designed by Serge Mansau for Givenchy, the bottle is a study in purity. Its two sculpted backs, with their irregular density, modulate the amber tones of the fragrance. The bottle’s broad, full base gives it a masculine foundation and allure. To complete this construction, an innovative closing system crowns the bottle. The curved shape of the cap, in bronze-colored metal, symbolically evokes the name.

  21. A mnemonic is a verse to assist memory Count the letters in each word! Pi mnemonics May I have a large container of coffee? … (8) How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard … (24) Que j’aime à faire apprendre un nombre utile aux sages! Immortel Archimède, artisite ingénieur, (31) Qui de ton jugement peut priser la valeur? Pour moi, ton problème eut de pareils avantages. Sir, I send a rhyme excelling In sacred truth and rigid spelling. Numerical sprites elucidate For me the lexicon's dull weight. (21) Dir, o Held, o alter Philosoph, du Riesengenie! Wie viele Tausendre bewundern Geister Himmlisch wie du und göttlich! Noch reiner in Aeonen Wird das uns strahlen Wie im lichten Morgenrot! (30) Sol y Luna y Mundo proclaman al Eterno Autor del Cosmo. (11) Wie? O! Dies (24) Mach ernstlich so vielen viele Müh’! Lernt immerhin, Jünglinge, leichte Verselein, Wie so zum Beispiel dies dürfte zu merken sein! Yes. I know a great geometric pi number which Mrs Weeks’ geometry classroom studies carefully out at the Campbell Hall School. (21)

  22. CAN YOU FIND 402 digits of PI ? “Circle Digits” By Michael Keith For a time I stood pondering on circle sizes. The large computer mainframe quietly processed all of its assembly code. Inside my entire hope lay for figuring out an elusive expansion value: pi. Decimals expected soon. I nervously entered a format procedure. The mainframe processed the request. Error. I, again entering it, carefully retyped. This iteration gave zero error printouts in all - success. Intently I waited. Soon, roused by thoughts within me, appeared narrative mnemonics relating digit to verbiage! The idea appeared to exist but only in abbreviated fashion - little phrases typically. Pressing on I then resolved, deciding firmly about a sum of decimals to use - likely around four hundred, presuming the computer code soon halted! Pondering these ideas, words appealed to me. But a problem of zeros did exist. Pondering more, solution subsequently appeared. Zero suggests a punctuation element. Very novel! My thoughts were culminated. No, periods, I concluded. All residual marks of punctuation - zeros. First digit expansion answer then came before me. On examining some problems unhappily arose. That imbecillic bug! The printout I possessed showed four nine as foremost decimals. Manifestly troubling. Totally every number looked wrong. Repairing the bug took much effort. A pi mnemonic with letters truly seemed good. Counting of all the letters probably should suffice. Reaching for a record would be be helpful. Consequently, I continued, expecting a good final answer from computer. First number slowly displayed on the flat screen - 3. Good. Trailing digits apparently were right also. Now my memory scheme must probably be implementable. The technique was chosen, elegant in scheme; by self reference a tale mnemonically helpful was assured. An able title suddenly existed - “Circle Digits”. Taking pen I began. Words emanated uneasily. I desired more synonyms. Speedily I found my (alongside me) Thesaurus. Rogets is probably an essential in doing this, instantly I decided. I wrote and erased more. The Rogets clearly assisted immensely. My story proceeded (how lovely!) faultlessly. The end, above all, would soon joyfully overtake. So, this memory helper story I incontestably complete. Soon I will locate publisher. There a narrative will I trust immediately appear, producing fame. THE END. 360 words - ignore periods other punctuation = 0 words > 9 letters = 2 digits word for no. = digit

  23. “Fools Rush In” Indiana legislature, 1897 “A bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana, free of cost by paying any royalties whatever on the same, provided it is accepted and adopted.” “...It has been found that the circular area is to the quadrant of the circumference,as the area of an equilateral rectangle is to the square on one side. The diameter employed as the linear unit according to the present rule in computing the circle’s area is entirely wrong…” (This makes no sense … if meant to be “eq. tri”, then here!) …“Furthermore, it has revealed the ratio of the chord and arc of 90o as 7:8, and the ratio of the diagonal and one side of a square as 10:7, and the ratio of the diameter and circumference is 5/4:4(so now ) “In further proof of the value of the author’s proposed contribution to education … and State of Indiana” … (claims the Dr. solved other classic unsolvable problems). [sq. circle] (These ancient problems have been proven to be unsolvable.) [trisect angle] Author of Bill - Edwin J. Goodman, M.D. of Indiana - Introduced Jan. 18, 1897 Preamble: Body: Feb. 5 - House votes 67 to 0 in favor; bill forwarded to the Senate Feb. 10 - Pf. Waldo (Purdue, checking school grant) overhears; coaches Senate Feb. 12 - Senate votes to postpone further consideration of this bill

  24. Interesting web sites Joy of Pi www.joyofpi.com Friends of Pi Club http://www.astro.univie.ac.at/~wasi/PI/pi_club.html Search Digits in Pi http://www.angio.net/pi/piquery The Pi Trivia Game http://eveander.com/trivia/ The Pi Project ? http://rene.ma.utexas.edu/users/tyilk/PiProj/Index.htm Brief History of Pi ? http://rene.ma.utexas.edu/users/tyilk/PiProj/PiHistory.htm Recite Digits in Languages http://www.cecm.sfu.ca/pi/yapPing.html Listen to Pi on Polyphon http://home.t-online.de/home/HAEL.YGGS/polyphon.htm Pi Day Songs http://www.winternet.com/~mchristi/piday.html At the Exploratorium http://www.exploratorium.edu/learning_studio/pi

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