Presentation on theme: "3.1415926535897932384626433832795028841971693993751058209749445923078... Copyright Audrey Weeks 2003 “People have calculated billions of digits of pi."— Presentation transcript:
Copyright Audrey Weeks 2003 “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.
Copyright Audrey Weeks 2003 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”
Copyright Audrey Weeks 2003 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)
Copyright Audrey Weeks 2003 Our “Pi String” Each student adds beads to it on -Day (100) (200) (300) (400) (500) (600) (700) (800) (900) (1000)
Copyright Audrey Weeks 2003 Where Can we find pi? IN EVERYTHING CIRCULAR (of course) (See “torus.gsp”)
Copyright Audrey Weeks 2003 WHERE ELSE? Area under bell (Gaussian) curve Carl 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) = German
Copyright Audrey Weeks 2003 A BIT OF CALCULUS Leibniz - 1st infinite series for pi Newton - converges quicker can be used to find decimal digits of pi
Copyright Audrey Weeks 2003 Earliest Known Record of Pi circa 1650 BC On the Rhind Papyrus, Egyptian scribe, Ahmes, wrote this ratio as “4 times the square of eight-ninths” 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.
Copyright Audrey Weeks 2003 More Attempts to rationalize (all prior to Arabic numerals and decimals) Archimedes (Syracuse, BC) Found pi to be between these two fractions. This average error is only ! Babylonians, same time as Egyptian Rhind Papyrus, 1650 BC Tsu Ch’ung Chi China, 450 AD Ptolemy (Alexandria, Egypt) 150 AD Also used by Columbus on his voyage to the New World Srinivasa Ramanujan (India, ) (http://www.science-frontiers.com/sf053/sf053p19.htm) (This is an irrational approximation.)
Copyright Audrey Weeks 2003 Archimedes, 250 BC
Copyright Audrey Weeks 2003 I have proof! Johann Lambert proved irrational Adrien-Marie Legendre proved irrational Joseph Liouville proved transcendental nos. exist (used limits of continued fractions) Charles Hermite proved e transcendental Ferdinand Lindemann proved transcendental Swiss French German
Copyright Audrey Weeks 2003 Interesting digits Starting at digit # occurs Starting at digit #509, occurs Starting at digit #1,286, occurs In 1st million, no “123456” but twice #357 #358 #359 #360 #361 #362 #363 … largest 7-digit sum in the first million digits! A special date - can you guess it? A special telephone number - do you know it? first appears at 523,551,502nd digit
Copyright Audrey Weeks 2003 Can’t get enough pi digits 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) Why still do this?…to find out more about pi …to test computer architecture & efficiency... to test software for accuracy and speed used Gauss-Legendre algorithm Circa decimal fractions & logarithms invented But Ferguson finds error in 527th onward All by hand - months
Copyright Audrey Weeks TV STAR TREK (1 min.) STARGATE (4 min.) From the original series, 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. 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!
Copyright Audrey Weeks 2003 More misc. pi facts 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 German
Copyright Audrey Weeks 2003 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, , , 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 ) 9 hours... 42,000 digits!
Copyright Audrey Weeks 2003 Pi In print 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 “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
Copyright Audrey Weeks 2003 Pi Song To the tune of “O Christmas Tree” Oh, number Pi Oh, number Pi Your digits are unending, Oh, number Pi Oh, number Pi No pattern are you sending. You're three point one four one five nine, And even more if we had time, Oh, number Pi Oh, number Pi For circle lengths unbending.
Copyright Audrey Weeks 2003 pi scent 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. Cologne by Givenchy This was their 1999 advertisement at
Copyright Audrey Weeks 2003 Pi mnemonics 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. 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) 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 Newhouse’s geometry classroom studies carefully out at the Port Charlotte High School. (21) Sir, I send a rhyme excelling In sacred truth and rigid spelling. Numerical sprites elucidate For me the lexicon's dull weight. (21) A mnemonic is a verse to assist memory Sol y Luna y Mundo proclaman al Eterno Autor del Cosmo. (11) Count the letters in each word! May I have a large container of coffee? … (8)
Copyright Audrey Weeks 2003 CAN YOU FIND 402 digits of PI ? 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 “Circle Digits” By Michael Keith
Copyright Audrey Weeks 2003 Indiana legislature, 1897 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 Pf. Waldo (Purdue, checking school grant) overhears; coaches Senate Feb Senate votes to postpone further consideration of this bill “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 90 o 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] “Fools Rush In”
Copyright Audrey Weeks 2003 Interesting web sites Joy of Piwww.joyofpi.com Friends of Pi Clubhttp://www.astro.univie.ac.at/~wasi/PI/pi_club.html Search Digits in Pihttp://www.angio.net/pi/piquery The Pi Trivia Gamehttp://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 Languageshttp://www.cecm.sfu.ca/pi/yapPing.html Listen to Pi on Polyphonhttp://home.t-online.de/home/HAEL.YGGS/polyphon.htm Pi Day Songshttp://www.winternet.com/~mchristi/piday.html At the Exploratoriumhttp://www.exploratorium.edu/learning_studio/pi