what is pi?

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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.

1650BC 600BC 300BC 1100 1600 2001Algebra Invented

Decimal Fractions InventedLogarithms InventedCalculus Discovered

Our Story of Pi Begins

ThalesPythagoras

Euclid

Formal Geometry Begins

Arabic Numerals (1,2,3...) Invented(World's 1st Novel Written)

(general public not even aware of the date)

Computers & Calculators Invented

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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 .

circumferencediameter

227

Its digits will NEVER terminate or repeat…

(proved in 1766)

...but will ALWAYS continue to fascinate

mankind.

See “Peel Circle for pi PPT.gsp”

173

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Irrational & Transcendental

• IRRATIONAL

Cannot be expressed as the quotient of 2 integersThis also means it cannot be written as a decimal for it will never terminate or repeat. (speculated early; proved 1767)

• TRANSCENDENTALCannot 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)

227 3.14

2Unlike 3 which solves x 3

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Copyright Audrey Weeks 2003Our “Pi String” 1998-2002Each student adds beads to it on -Day.

3.1415926535 8979323846 2643383279 5028841971 69399375105820974944 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 1050792279 6892589235 (700) 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 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)

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Where Can we find pi?IN EVERYTHING CIRCULAR

(of course)

2

C d

A r

2

2

SA dh 2 r

V r h

2

2

12

13

SA dh r

V r h

2

343

SA 4 r

V r

2

2 2

SA 4 r a

V 2 r a

r

h

r

h

(See “torus.gsp”)

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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) =

26

1

dist. between II lines = 2(needle length)

1777-1855German

-2 -1 1 2x

1

-1

y

A=y = e-x2

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Connections to integers

2

1 1 1 1 1 1 14 1 ...

3 5 7 9 11 13 15

1 1 1 1 1 1 1 16 ...

1 4 9 16 25 36 49 64

2 2 4 4 6 6 8 8 10 102 ...

1 3 3 5 5 7 7 9 9 11

3 5 7 11 13 17 19 23 292 ...

2 6 6 10 14 18 18 22 30

(Leonard Euler)

(John Wallis 1655)

(Leibniz)

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Copyright Audrey Weeks 2003A BIT OF CALCULUS

Leibniz - 1st infinite series for pi

Newton - converges quicker1642-1727

1646-1716

3 5 7

x

0 2Using area, found arctan x

x x xManipulating int egrand, arctan x x ...

3 5 7

Let x 1, arctan (1

11 1

4 1 ...3 54 7

) ,

dx1 x

can be used to find decimal digits of pi

3 5 7

2

3 5 7

dxFound arcsin x

1 x

1 x 1 3 x 1 3 5 xBinomial Thm. to int egrand arcsin x x ...

2 3 2 4 5 2 4 6 7

1 1Let x , a

1 1 1 3 1 3 56 .rcsin ,

2 2..

2 2 3 26 2 4 5 2 2 4 6 7 2

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Earliest Known Record of Picirca 1650 BC

On the Rhind Papyrus,

Egyptian scribe, Ahmes,

wrote this ratio as

“4 times the square of eight-ninths”2

84

925681

approx. 3.160493

less than 1% error

8...

!

No number has captured the attention and imaginationsof people throughout the ages as much as the ratio of a circle’s circumference to its diameter.

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More Attempts to rationalize (all prior to Arabic numerals and decimals)

227 3.142857

22371 3.1408450704...

355113 3.141592920354 ...

Archimedes (Syracuse, 287-212 BC)Found pi to be between these two fractions.

This average error is only 0.0002!

377120 3.1416

3.125258

Babylonians, same time as Egyptian Rhind Papyrus, 1650 BC

Tsu Ch’ung Chi China, 450 AD

Ptolemy (Alexandria, Egypt) 150 ADAlso used by Columbus on his voyage to the New World

Srinivasa Ramanujan (India, 1887-1920)(http://www.science-frontiers.com/sf053/sf053p19.htm)

42143

3.14159265258...22

1 11 12 2

1 4311

16539 ...

4 21431 1222 2

97 If 16,539 replaced by , 97(This is an irrational approximation.)

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Archimedes, 250 BC

Area CircleArea Square

Circumference of CircleDiameter

but also ...

10 17713 3

0

1

2

3

4

5

6

3 4 5 6Inner polygon perimeter / 2rOuter polygon perimeter / 2r

He began with a regular hexagonand kept doubling sides to a 96-gon!

Later, the Chinese continued this doubling to over 3000 sides to get 3.14159.

r

r

Area Circle = 12.1 cm2

Area Square = 3.9 cm2

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I have proof!1767 - Johann Lambert proved irrational

1794 - Adrien-Marie Legendre proved irrational1840 - Joseph Liouville proved transcendental nos. exist

(used limits of continued fractions)

1873 - Charles Hermite proved e transcendental

1882 - Ferdinand Lindemann proved transcendental

1728-1777

4 4

First, he proved -

i.e.,

Q.E.D.

If x is rational, (x 0), then tan x cannot be rational.

If tan x is rational, then x must be irrational or 0.

Since tan = 1, must be irrational.

2

Swiss

French

French

French

German

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Interesting 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

largest 7-digit sum in the first million digits!

A special date - can you guess it?

A special telephone number - do you know it?

123456789 first appears at 523,551,502nd digit

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Copyright Audrey Weeks 2003Can’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 1600 - decimal fractions & logarithms invented

But Ferguson finds error in 527th onward

All by hand - months

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Copyright Audrey Weeks 2003-TV STAR TREK (1 min.)

STARGATE (4 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, 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!

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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.

18(rounded to nearest th)circumference of head

ie 1

Swiss 1707-1783

German 1879-1955

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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!

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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”

CADAEIBF

“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

30 cubits

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Pi SongTo the tune of

“O Christmas Tree”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

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pi scentThe 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 http://www.givenchy.com/givenchy/givenchy.html

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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 GeisterHimmlisch wie du und göttlich!Noch reiner in AeonenWird das uns strahlenWie im lichten Morgenrot! (30)

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)

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)

Sir, I send a rhyme excellingIn sacred truth and rigid spelling.Numerical sprites elucidateFor 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)

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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

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Indiana legislature, 1897Author 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

“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]

3.23, 2 2.041

163

9

“Fools Rush In”

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Interesting web sitesJoy 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

what is pi?

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