â&#x20AC;&#x2DC;Exploring pi is like exploring the universeâ&#x20AC;&#x2122; - David Chudnovsky

3.141592 WHAT IS PI?

Pi (symbol Ď&#x20AC; ) is a mathematical constant whose value is the ratio of the circumference of a circle to its diameter. This number is the same for every circle. This means that for any circle, the length of the circumference divided by the length of the diameter equals pi.

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6535897 = c d Eg: Draw a circle with a DIAMETER of 1. Then the circumference (the distance all the way around the edge of the circle) will be Pi.

Approximately 22 333 355 7 , 106 , 113

circumference (c) radius (r)

diameter (d)

2 pi in radians form 360 degrees. 02

T H E G R E E K A L P H A B E T

From the Greek alphabet, π (pronounced ‘pee’) is the symbol used for pi. The letters π and p is the sixteenth letter of both the Greek and English alphabet.

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William Jones 1675-1749 The first mathematician to use the Greek letter π to represent the ratio of a circle’s circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706. Jones’ first use of the Greek letter was in the phrase “1/2 Periphery (π)” in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the word periphery.

Leonhard Euler 1707-1783 The Greek letter π was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead. Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly. In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: “for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1”) and the practice was universally adopted in the Western world.

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Chronology Of Pi

Egyptian: G r e a t Pyramid of Giza and M e i d u m Pyramid 3+1/7 = 22/7 = 3.143...*

Ptolemy 377/120 = 3.141666...

Babylonian 25/8 = 3.125* Vitruvius 25/8 = 3.125*

26th

19th

Century BC Century BC

20th

Century BC

Egyptian: R h i n d Papyrus (16/9)^2 = 3.160493...*

20 BC

Zhang Heng √10 = 3.162277...* 730/232 = 3.146551...*

2nd

Century

250 BC 1st

Century

Archimedes 3.1418 (average of the bounds)

L i u X i n 3.1457*

It is clear that the Greeks laid the essential groundwork and even began to build the structure of much of modern mathematics. It should also be emphasized that although some great mathematicians devoted their lives to this work, it nevertheless took three centuries of cumulative effort, each building on the previous work.

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Numbers with an asterisk (*) indicate that not all the decimal places are correct.

Aryabhata 62832/2000 = 3.1416

Zu Chongzhi 355/113 = 3.141592920*

5th

Century

3rd

Century

Fibonacci 3.141818* Al-Khwarizmi 3.1416

9th

Century

7th

Century

Wang Fan 142/45 = 3.155555

L i u H u i 3.14159

12th

13th

Century

Century

15th

Century

Madhava of B h ā s k a r a II S a n g a m a g r a m a 3.14156* 3.14159265359 Brahmagupta 3.1622 (= √10)

J a m s h ī d a l - K ā s h ī 3.14159265358979

Evidently, this required a stable, literate culture over many generations. Geometric results are difficult to transmit in an oral tradition. The level of mathematical analysis attained by Archimedes, Euclid and others is far in advance of anything recorded by the Babylonians or Egyptians.

Jurij Vega 140 decimal places calculated 137 decimal places correct K a t a h i r o T a k e b e 41 decimal places

Yoshisuke Matsunaga 51 decimal places Ludolph van C e u l e n 35 decimal places 20 decimal places

Guilloud&Filliatre I B M 7 0 3 0

250,000 decimal places

Shanks&Wench I B M 7 0 9 0

100,265 decimal places

T o s h i k i y o K a m a t a 24 decimal places

Adriaan van R o o m e n 15 decimal places

Valentinus Otho 6 decimal places

17,526,200 decimal places Chudnovskys

Genuys-IBM704 10,000 decimal places

525,229,270 decimal places

Felton-PEGASUS 7480 decimal places

536,870,898 decimal places

Kanada&Tamura Chudnovskys

Kanada&Tamura

Chudnovskys

John Wrench 1120 decimal places

4,044,000,000 decimal places

D. F. F e r g u s o n 710 decimal places

206,158,430,000 decimal places

18th

Century

G o s p e r SYMBOLICS 3670

R e i t w i e s n e r . e t . a l - E N I A C 1,073,741,799 decimal places 2037 decimal places

John Machin 100 decimal places

16th

250,000 decimal places

J e e n e l & N i c h o l s o n - N O R A C 1,011,196,691 decimal places 3092 decimal places

T h o m a s Fantet de Lagny 127 decimal places calculated 112 decimal places correct

FranĂ§ois ViĂ¨te 9 decimal places

Tamura&Kanada HITACHI M-280H

K ana d a & Ta k a has h i

20th

Century

Century

19th

17th

Century

Century

21st

All records from 1949 onwards were calculated with electronic computers.

Century

Willebrord Snell 35 decimal places

W i l l i a m Rutherford 152 decimal places

K

Christoph Grienberger 38 decimal places

Zacharias Dase 200 decimal places

T a k a h a s h i

Isaac Newton 16 decimal places Takakazu Seki 11 decimal places 16 decimal places Abraham Sharp 71 decimal places

n

a

d

a

2,576,980,377,524 decimal places

Thomas Clausen 248 decimal places

Fabrice Bellard

2,699,999,990,000 decimal places

L e h m a n n 261 decimal places

Shigeru Kondo

5,000,000,000,000 decimal places 10,000,000,000,050 decimal places

W i l l i a m Rutherford 440 decimal places William Shanks 707 decimal places calculated 527 decimal places correct

a

1,241,100,000,000 decimal places

R i c h t e r 500 decimal places

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ONE HUNDRED AND NINETY ONE DIGITS OF PI

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THE FIRST MILLION DECIMAL PLACES OF PI CONSISTS OF

•99,959 ZEROS• •99,758 ONES•

•100,026 TWOS•

•100,229 THREES•

•100,230 FOURS•

•100,359 FIVES• •99,458 SIXES•

•99,800 SEVENS•

•99,985 EIGHTS•

•100,106 NINES• 08

D O N

Y O U

E

E

D

H E L P

REMEMBERING

P

I

U S E

?

A

MNUEMONIC! 09

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 digits 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 imbecilic 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 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 ensured. An able title suddenly existed-â&#x20AC;?Circle Digits.â&#x20AC;? 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 is incontestably complete. soon I will locate publisher. There a narrative will I trust immediately appear, producing fame. The end.

CIRCLE DIGITS : A SELF-REFERENTIAL STORY By Michael Keith

This is a mnemonic for the first 402 decimal places of pi. Count the number of letters in each word of the story to obtain the successive decimals of pi.

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A R C H I M E D E S O F S Y R A C U S E Archimedes (287 BC - 212 BC ) was a Greek mathematician, philosopher and inventor who wrote important works on geometry, arithmetic and mechanics. Archimedes estimated the value of pi by drawing two polygons around the circle’s center. One outside the circle (circumscribed) so its perimeter was greater than the circle’s and one inside the circle (inscribed) so its perimeter was less than the circle’s. Since π is the ratio of any circle’s circumference and its diameter, it will be greater than the circumference of any inscribed regular polygon and less than that of any circumscribed regular polygon. The more sides the polygons have, the closer it will resemble an actual circle thus the better the estimate of pi. Archimedes used a polygon with 96 sides to show that the value of pi was between 3 10/70 and 3 10/71. After Syracuse was captured, Archimedes was killed by a Roman soldier. It is said that he was so absorbed in his calculations he told his killer not to disturb him.

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‘Do not disturb my circles’ - Archimedes

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INSCRIBED REGULAR POLYGONS n n

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= number = 4, 5,

of 6,

sides 7, 8

n

=

9,

10,

11,

12,

13

The more sides a polygon has, the closer it will resemble an actual circle.

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P O L Y G O N A P P R O X I M A T I O N E R A

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Mathematician

Number of sides on polygon

Pi

96

3.141818

Fibonacci Italy, 1200 AD

805,306,368

3.14159265358979

1,073,741,824

3.1415926535897932

al-Kashi Samarkand, 1430

A Van Roomen Italy, 1593

32,212,254,720

3.14159265358979323846

4.611686e+18

3.14159265358979323846 26433832795029

Ludolph van Ceulen Leiden, 1596 16

T H G R E A P Y R A M I D O F G I Z It appears that the value of pi was built into the Great Pyramid of Giza, hundreds of years before the Greeks allegedly discovered it. How was this value built into the great pyramid? The vertical height of the pyramid holds the same relationship to the perimeter of its base (distance around the pyramid) as the radius of a circle bears to its circumference. If we equate the height of the pyramid to the radius of a circle, than the distance around the pyramid is equal to the circumference of that circle. The ratio of the perimeter to height of 1760/280 cubits equates to 2Ď&#x20AC; to an accuracy of better than 0.05% (corresponding to the wellknown approximation of Ď&#x20AC; as 22/7). Some Egyptologists consider this to have been the result of deliberate design proportion.

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E T S A

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decimal places of pi is all that is needed to calculate the cicumference of the earth, with an error of no more than one quarter of an inch in 25,000 miles would result.

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39 digits of Pi is all that is needed to compute the circumference of the entire universe to the accuracy of less than the diameter of a hydrogen atom.

S Q U A R I N G T H E C I R C L E

The term refers to the famous problem of finding an area with straight-line boundaries equal in area to a circle of given diameter.

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The â&#x20AC;&#x153;squaring the circleâ&#x20AC;? method of understanding pi has fascinated mathematicians because traditionally the circle represents the infinite, immeasurable, and even spiritual world while the square represents the manifest, measurable, and comprehensive world. Archimedes proved that the area of a circle is equal to that of a right-angled triangle having the two shorter sides equal to the radius of the circle and its circumference respectively. The idea of his proof is as follows. Consider first a square inscribed in the circle.The square is made up of four triangles. The area of the square isnâ&#x20AC;&#x2122;t a very good approximation to that of the circle, but we can improve it by replacing the square by a regular octagon, with all its points on the circle. Now, this octagon can by divided into eight triangles, following the same procedure as for the square. The process is repeated: the octagon is replaced by a regular 16-sided polygon, with all its points on the circle.

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I R R A T I O N A L N U M B E R S The number pi is a irrational number. An irrational number is any real number that cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer. It was not until the 18th century that Johann Heinrich Lambert proved that Ď&#x20AC; is irrational. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Pi cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate Ď&#x20AC;); consequently, its decimal representation never ends and never repeats.

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LAMBERT’S CONTINUED FRACTION EXPANSION Lambert’s ‘Mémoires de l’Académie royale des sciences de Berlin’ (1768).

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THE PERFECT CIRCLE A perfect circle is only perfect in imagination, because it is impossible to calculate the exact ratio of a circle. Mathematicians and scientists simply decide that they are close enough, virtually perfect. The perfect circle is possible through formulated thought, but it is impossible in practice due to Piâ&#x20AC;&#x2122;s irrationality, and technically not existing because of it. Some would argue that all circles and polygons; that it is more correct to say that a circle has an infinite number of corners than to view a circle as being cornerless.

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â&#x20AC;&#x153;The circle is the first perfect shape. The equidistant arrangement of the outer points from the centre, defining and ideal, are impossible to achieve by human hand. The space speaks of potential- the tension between what is achieved and what could be achieved. From the circle, we derive ideals and focus, both the halo of saints and the cross-haired target in gun sights.â&#x20AC;? -Data

Flow

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PI PATTERNS Is the frequency of the digits found in π truly random?

There are no recurring patterns since pi is an irrational number. In the first 4 billion digits of pi there does not appear to be any repeating patterns longer than 10 digits. Pi is considered as a “pseudo-random” number, since the number is, after all, always the same. As is often the case in Mathematics, in order to understand the randomness π, one must also have a good concept of infinity. Since there are only 10 digits in our number system and many different ways in which we use numbers in our lives (birth, marriage and death dates, ID numbers like social security or driver’s licenses, telephone numbers, credit card numbers, addresses, etc.) it’s actually pretty easy to come across many similarities which may appear to be significant if you believe that they must be! And with an infinite stream of random digits to pick and choose from, you should be able to find ALL of them within the digits of π.

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“ There’s a beauty to Pi that keeps us looking at it... The digits of Pi are extrememly random. They really have no pattern, and in mathematics that’s really the same as saying they have every pattern” -Peter Borwein

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E Q U A T I O N S

MATHEMATICAL E Q U A T I O N S O F P I Here are various equations and formulas of pi from the geometry, calculus and modern era to show the complexity of such a basic number.

Archimedes 287 - 212 BC

Fibonacci 1180 - 1250

Lord Brounker 1620 - 1684

Euler 1707 - 1783

Salamin/Brent 1976

J and P.Borwein 1984

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WHY CALCULATE PI TO SO MANY DECIMAL PLACES? The ubiquitous nature of Ď&#x20AC; makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published and the number is celebrated on Pi Day.

Several people have endeavored to memorize the value of Ď&#x20AC; with increasing precision, leading to records of over 67,000 digits.

But why calculate pi to a trillion places when apparently only 39 decimal places is enough to calculate the circumference of the visible Universe? Answers include testing out algorithms for computing such numbers and to demonstrate computer power. Also to see if there would be some pattern.

In this age of high-tech precision instruments, where we assure ourselves that perfection is attainable, pi is an ever-present, sometimes grating reminder that there are puzzles that can be solved and there are mysteries that, perhaps, can not.

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N

I

V

E

Pi shows up everywhere. In mathematics, pi appears in many fundamental equations that have nothing to do with circles. In science, pi is inextricable from measuring everything from ocean waves to economic statistics. Pi helps describe how the universe works and how organic things grow. These things are beyong our rational grasp but mathematics sets itself to coming to grips with approximating to better approximations to these numbers.

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R

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F N

U

M

B

E

The real essence of pi is circles, it is rotations, it is spinning things round; anything in the universe that spins round has rotational symmetry. The laws of phyics are symmetric; space is symmetric and one of the important symmetries of space is that you can rotate things. Whenever rotations comesup, there are circles thus you will find pi.

R

S

â&#x20AC;&#x153;There are those who seek knowledge for the sake of knowledge...â&#x20AC;? -Bernard of Clairvaux

B Y B E V E R L E Y C H A N

The Circular Constant
The Circular Constant

A book about pi, based on BBC Radio 4's programme 'In Our Time'

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