“People have calculated billions of digits of pi

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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


 41 






... 
(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
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“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
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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.
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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 ...
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