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Copyright Audrey Weeks 2005 www.calculusinmotion.com “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. Our Story of Pi Begins 1650BC Formal Geometry Begins 600BC 300BC Thales Euclid Pythagoras Decimal Fractions Invented Logarithms Invented Calculus Discovered 1100 1600 Today Algebra Invented Computers & Arabic Numerals (1,2,3...) Invented Calculators (World's 1st Novel Written) (general public not even aware of the date) 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com What is pi? circumference diameter The ratio of the circumference to the diameter of ANY circle is constant. It is between 3 and 3 71 . It is close to but NOT EQUAL to 3.14 or 22. 7 Its digits will NEVER terminate or repeat… (proved in 1766) ...but will ALWAYS continue to fascinate mankind. See “Peel Circle for Pi.gsp” (runs in GSP4) 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com Irrational & Transcendental 227 3.14 • 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 Unlike 3 which solves x 2 3 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) 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Our “Pi String” Each student adds 3.1415926535 5820974944 8214808651 4811174502 4428810975 4564856692 7245870066 7892590360 3305727036 0744623799 9833673362 6094370277 0005681271 1468440901 4201995611 5187072113 5024459455 7101000313 5982534904 1857780532 8979323846 5923078164 3282306647 8410270193 6659334461 3460348610 0631558817 0113305305 5759591953 6274956735 4406566430 0539217176 4526356082 2249534301 2129021960 4999999837 3469083026 7838752886 2875546873 1712268066 Copyright Audrey Weeks 2005 www.calculusinmotion.com beads to it on 2643383279 0628620899 0938446095 8521105559 2847564823 4543266482 4881520920 4882046652 0921861173 1885752724 8602139494 2931767523 7785771342 4654958537 8640344181 2978049951 4252230825 5875332083 1159562863 1300192787 5028841971 8628034825 5058223172 6446229489 3786783165 1339360726 9628292540 1384146951 8193261179 8912279381 6395224737 8467481846 7577896091 1050792279 5981362977 0597317328 3344685035 8142061717 8823537875 6611195909 -Day. 6939937510 3421170679 5359408128 5493038196 2712019091 0249141273 9171536436 9415116094 3105118548 8301194912 1907021798 7669405132 7363717872 6892589235 4771309960 1609631859 2619311881 7669147303 9375195778 2164201989... (100) (200) (300) (400) (500) (600) (700) (800) (900) (1000) 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com Where Can we find pi? IN EVERYTHING CIRCULAR (of course) h r SA 21 dh r 2 C d h A r2 V 31 r 2h r SA dh 2 r 2 V r 2h SA 4 r V 43 r 3 2 SA 4 r 2a V 2 2 r 2a (See “torus.gsp”) 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com WHERE ELSE? • Area under bell (Gaussian) curve Carl Gauss, “prince of mathematics” y 1 y=e -2 -x 2 A= 1777-1855 German -1 1 x 2 • Electricity - formulas for alternating currents and radiation from radio, TV, microwave antennas -1 1 2 (frequency)(capaci tance) inductive reac tance 2 (frequence)(induc tance) capacitive reac tance ElectroMagnetic Radiation antenna (wattage)(gain) 4 (dis tance)2 3.1415926535897932384626433832795028841971693993751058209749445923078 ... WHERE ELSE? Copyright Audrey Weeks 2005 www.calculusinmotion.com • Probability 6 P (2 integers have no common factors) = 2 P (lattice pt. is visible from origin) = 62 P (needle lands on line) = 2 • Rivers dist. between || lines = length of needle ("Buffon's Needle Problem", 1777) actual length (as it meanders) 3.14 direct length (beg. to end, straight) (this is an average) Calculated by Hans-Henrik Stolum, Cambridge University (from “Fermat’s Enigma” by Simon Singh) 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com Connections to integers 1 1 1 1 1 1 1 41 ... (Leibniz) 3 5 7 9 11 13 15 2 6 1 1 1 1 1 1 1 1 ... 4 9 16 25 36 49 64 1 2 4 4 6 6 8 8 10 10 2 2 ... 3 3 5 5 7 7 9 9 11 (John Wallis 1655) 1 5 7 11 13 17 19 23 29 3 2 ... (Leonard Euler) 2 6 6 10 14 18 18 22 30 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Earliest Known Record of Pi circa 1650 BC Copyright Audrey Weeks 2005 www.calculusinmotion.com 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” 8 2 4 256 approx. 3.1604938... 81 9 less than 1% error ! 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com More Attempts to rationalize (all prior to Arabic numerals and decimals) 25 3.125 8 377 3.1416 120 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 223 3.1408450704... Archimedes (Syracuse, 287-212 BC) 71 22 Found pi to be between these two fractions. 3.142857 This average error is only 0.0002! 7 355 3.141592920354 ... Tsu Ch’ung Chi China, 450 AD 113 4 2143 Srinivasa Ramanujan (India, 1887-1920) 3.14159265258... (http://www.science-frontiers.com/sf053/sf053p19.htm) 22 4 97 2 1 2 3 1 1 1 1 1 16539... If 16,539 replaced by , 97 2 1 1 (This is an irrational approximation.) 2 1 4 2143 22 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Archimedes, 250 BC Copyright Audrey Weeks 2005 www.calculusinmotion.com 1 3 10 3 7 71 Circumference of Circle Diameter but also ... Area Circle = 12.1 cm 2 Area Square = 3.9 cm 2 r Area Circle Area Square r 6 5 4 3 2 He began with a regular hexagon and kept doubling sides to a 96-gon! Later, the Chinese continued this doubling to over 3000 sides to get 3.14159. 1 0 3 4 5 6 Inner polygon perimeter / d Outer polygon perimeter / d 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com I have proof! 1767 - Johann Lambert proved irrational First, he proved If x is rational, (x 0), then tan x cannot be rational. i.e., If tan x is rational, then x must be irrational or 0. Since tan 4 = 1, 4 must be irrational. Q.E.D. 1728-1777 Swiss 1794 - Adrien-Marie Legendre proved 2 irrational French 1840 - Joseph Liouville proved transcendental nos. exist (used limits of continued fractions) French 1873 - Charles Hermite proved e transcendental transcendental French 1882 - Ferdinand Lindemann proved German 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Interesting digits Copyright Audrey Weeks 2005 www.calculusinmotion.com • Starting at digit #772 - 9999998 occurs largest 7-digit sum in the first million digits! • Starting at digit #509,400 - 112552 occurs A special date - can you guess it? • Starting at digit #1,286,368 - 980-7280 occurs A special telephone number - do you know it? • In 1st million, no “123456” but 012345 twice 123456789 first appears at 523,551,502nd digit • #357 #358 #359 #360 #361 #362 #363 …9 0 3 6 0 0 1 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Can’t get enough pi digits Copyright Audrey Weeks 2005 www.calculusinmotion.com Circa 1600 - decimal fractions & logarithms invented 1596 … Ludolph van Ceulen (Dutch) calculates 35 digits 1706 … John Machin calculates 100 digits All by hand - months But Ferguson finds 1874 … William Shanks calculates 707 digits error in 527th onward 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) used Gauss-Legendre algorithm 2002 … Hitachi (Tokyo U.) 1,240,000,000,000 digits (400 hrs) 2 trillion calcs. / sec; 5 years to design program; Prof. Kanada + 9 others at Info. Tech. Cntr. Why still do this? …to find out more about pi …to test computer architecture & efficiency ... to test software for accuracy and speed 3.1415926535897932384626433832795028841971693993751058209749445923078 ... -TV Copyright Audrey Weeks 2005 www.calculusinmotion.com STAR TREK (1 min.) 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 and Spock 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. STARGATE (4 min.) Courtesy of Randy Coombs - season 2, 1998, episode #28 or #206 “Thor’s Chariot” 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 worthy of the secret behind it! 3.1415926535897932384626433832795028841971693993751058209749445923078 ... More misc. pi facts Copyright Audrey Weeks 2005 www.calculusinmotion.com • Albert Einstein, Waclaw Sierpinski born 3/14/1879, 3/14/1882 (Pi-Day) German 1879-1955 • Symbol Polish 1882-1969 introduced by Leonard Euler, 1737 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 • ei 1 Euler (using DeMoivre’s work) • Hat size = circumference of head (rounded to nearest 1 th) 8 3.1415926535897932384626433832795028841971693993751058209749445923078 ... More misc. pi facts Copyright Audrey Weeks 2005 www.calculusinmotion.com • 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! 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com Pi In print 30 cubits • Bible - I Kings vii.23 (Solomon’s Temple) “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 10 cubits • Jules Verne - “20,000 Leagues Under the Sea” “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” • A 1970 advertisement CADAEIB F 3.1415926535897932384626433832795028841971693993751058209749445923078 ... “Sliding” Pi In canadian SUBWAY Artist’s Plaque Copyright Audrey Weeks 2005 www.calculusinmotion.com photos and information courtesy of Larry Ottman: http://home.gwu.edu/~ottmanl/ottmanpresent/frame0001.html INSPIRED TILEWORK FOR THE DOWNSVIEW SUBWAY STATION NEAR TORONTO Artist’s Directions The rectangles overlap each other by the digit of pi being represented. A darker color shows the layering. The more rectangles that overlap, the darker the color. 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com Pi Songs To the tune of Oh, number Pi “O Christmas Tree” 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. http://www.winternet.com/~mchristi/piday.html 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com Pi Songs Pi is here, can’t ignore it. To the tune of A ratio, let’s explore it. “Winter Wonderland” Distance around to Lyrics modified by Distance straight through Audrey Weeks Thinkin’ in a winter numberland. Pi’s a number that is transcendental. This was proved in eighteen-eighty-two. Its never-ending digits aren’t sequential, But you can find as many as you choose. Later on, we’ll conspire, As we work with numbers higher. So much to explore, Can’t wait to know more. Thinkin’ in a winter numberland. Inspired by Hampton Schools’ “Winter Numberland” event 2003. 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com Pi Songs Circles in the snow, To the tune of “Jingle Bells” ‘Round and ‘round we fly. Lyrics modified by How far did we go? Audrey Weeks Diameter times pi! Pi r squared finds out, Area that’s plowed. Oh what fun it is to shout Our formulas out loud! (Refrain ) Refrain: Oh…Pi day songs All day long. Oh, what fun it is, To sing a jolly pi day song In a great math class like this. 3.1415926535897932384626433832795028841971693993751058209749445923078 ... pi scent Copyright Audrey Weeks 2005 www.calculusinmotion.com Cologne by Givenchy 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. 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Count the letters in each word! Pi mnemonics Copyright Audrey Weeks 2005 www.calculusinmotion.com A mnemonic is a verse to assist memory 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) Sir, I send a rhyme excelling Qui de ton jugement peut priser la valeur? In sacred truth and rigid spelling. Pour moi, ton problème eut de pareils avantages. Numerical sprites elucidate For me the lexicon's dull weight. (21) Dir, o Held, o alter Philosoph, du Riesengenie! Sol y Luna y Mundo proclaman Wie viele Tausendre bewundern Geister al Eterno Autor del Cosmo. (11) Himmlisch wie du und göttlich! Wie? O! Dies (24) Noch reiner in Aeonen Macht ernstlich so vielen viele Müh’! Wird das uns strahlen Lernt immerhin, Jünglinge, leichte Verselein, Wie im lichten Morgenrot! (30) 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) More at: http://www.geocities.com/CapeCanaveral/Lab/3550/pimnem.htm 3.1415926535897932384626433832795028841971693993751058209749445923078 ... CAN YOU FIND 402 digits of PI ? Copyright Audrey Weeks 2005 www.calculusinmotion.com “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 360 words - ignore periods other punctuation = 0 I trust immediately appear, producing fame. words > 9 letters = 2 digits word for no. = digit THE END. 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Indiana legislature, 1897 Copyright Audrey Weeks 2005 www.calculusinmotion.com “Fools Rush In” Author of Bill - Edwin J. Goodman, M.D. of Indiana - Introduced Jan. 18, 1897 Preamble: “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 Body: 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 163 9 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 3.23, 2 2.041) “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] 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 Petr Beckmann, A History of Pi (New York: St. Martin's Press, 1971). pp. 174-177 3.1415926535897932384626433832795028841971693993751058209749445923078 ... Copyright Audrey Weeks 2005 www.calculusinmotion.com 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/ (200 million digits in 2005!) Recite Digits in Languages http://www.cecm.sfu.ca/pi/yapPing.html Listen to Pi on Polyphon http://www.jvshly.de/piworld/pi_poly.htm Pi Day Songs http://www.winternet.com/~mchristi/piday.html At the Exploratorium http://www.exploratorium.edu/learning_studio/pi 3.1415926535897932384626433832795028841971693993751058209749445923078 ...