# Number Pi

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```Index
• Summary
• Keywords
• Contents
• Origin of number &Iuml;€ and of its name.
• Features of number &Iuml;€.
• The approximations of number &Iuml;€ in Ancient Times.
• Mathematics of Ancient Egypt
• Mathematics of Mesopotamia
• Mathematics of Classical Antiquity
• Chinese Mathematics
• Indian Mathematics
• Islamic Mathematics
• Table of approximations
• The approximations of number &Iuml;€ in the Modern Times
• Value of &Iuml;€ with 100 places.
♦ Bibliography
♦ Personal Opinion
Summary
Pi&Acirc; or&Acirc; &Iuml;€&Acirc; is a&Acirc; mathematical constant&Acirc; whose value is the&Acirc; ratio&Acirc; of any&Acirc; circle's
circumference respect to its diameter in&Acirc; Euclidean space; this is the same value as the ratio of a
circle's area to the square of its radius. &Iuml;€ is one of the most important mathematical and physical
constants: many formulae from mathematics,&Acirc; science, and&Acirc; engineering&Acirc; involve &Iuml;€. &Iuml;€ is
an&Acirc; irrational number, which means that its value cannot be expressed exactly as a&Acirc; fraction&Acirc; m/n,
where&Acirc; m&Acirc; and&Acirc; n&Acirc; are integers. Consequently, its&Acirc; decimal representation&Acirc; never ends or repeats.
Keywords
♦ Radius − is any&Acirc; line segment&Acirc; from its center to its&Acirc; perimeter.
♦ Decimal places − they are numbers followings the point.
♦ Circumference&Acirc; − is the distance around a closed&Acirc; curve.
Contents
• Origin of number &Iuml;€ and of its name.
History
The earliest evidenced conscious use of an accurate approximation for the length of a circumference
with respect to its radius is of 3+1/7 in the designs of the&Acirc; Old Kingdom&Acirc; pyramids in Egypt.
The&Acirc; Great Pyramid&Acirc; at Giza, constructed c.2550−2500 BC, was built with a perimeter of
1760&Acirc; cubits&Acirc; and a height of 280 cubits; the ratio 1760/280 &acirc;‰ˆ 2&Iuml;€. Egyptologists such as
Professors Flinders Petrie&Acirc; &Acirc; and I.E.S Edwards&Acirc; have shown that these circular proportions were
deliberately chosen for symbolic reasons by the Old Kingdom scribes and architects.
The name
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The name of the&Acirc; Greek letter &Iuml;€ &Acirc; is&Acirc; pi, and this spelling is commonly used
in&Acirc; typographical&Acirc; contexts when the Greek letter is not available, or its usage could be problematic.
It is not capitalised (&Icirc; ) even at the beginning of a sentence. When referring to this constant, the
symbol &Iuml;€ is always pronounced&Acirc; /'pa&Eacute;&ordf;/, &quot;pie&quot; inEnglish, which is the conventional English
pronunciation of the Greek letter. In Greek, the name of this letter is pronounced&Acirc; [pi].
The&Acirc; constant&Acirc; is named &quot;&Iuml;€&quot; because &quot;&Iuml;€&quot; is the first letter of the&Acirc; Greek&Acirc; words
&Iuml;€&Icirc;&micro;&Iuml;&Icirc;&sup1;&Iuml;†&Icirc;-&Iuml;&Icirc;&micro;&Icirc;&sup1;&Icirc;&plusmn; (periphery) and &Iuml;€&Icirc;&micro;&Iuml;&Icirc;&macr;&Icirc;&frac14;&Icirc;&micro;&Iuml;„&Iuml;&Icirc;&iquest;&Iuml;‚ (perimeter), probably referring to its use in
the formula to find the circumference, or perimeter, of a circle.
• Features of number &Iuml;€.
Definition
Euclid was the first to show that the ratio of a circle to its diameter is a constant quantity.&Acirc; However,
there are various definitions of the number &Iuml;€, but the most common is: &Iuml;€ is the ratio between the
length of a circle to its diameter.
Irrational and transcendental number
&Iuml;€ is an irrational number, which means it can not be expressed as a fraction of two integers, as
demonstrated by Johann Heinrich Lambert in 1761 (or 1767).&Acirc; It is also a transcendental number,
which means it is not the root of any polynomial with integer coefficients.&Acirc; In the nineteenth century
German mathematician Ferdinand Lindemann proved this fact and permanently closing the problem
of squaring the circle indicating that there isn't solution.
• The approximations of number &Iuml;€ in
The Ancient Times.
◊ Mathematics of Ancient Egypt
The approximate value of &Iuml;€ in ancient Egypt was wrote by scribe Ahmes in 1800 a.&Acirc; C., in the
Rhind papyrus, where he used an approximate value of &Iuml;€ by saying that: the area of a circle is similar
to a square whose side equals the diameter of the circle decreased in 1 / 9, that's&Acirc; equal to 8 / 9 of the
diameter.&Acirc; In modern notation:
◊ Mathematics of Mesopotamia
Some mathematicians of of Mesopotamia used in the calculation of segments, values of &Iuml;€ equal to 3,
reaching in some cases approximate values, such as 3 + 1 / 8.
◊ Mathematics of Classical Antiquity
The Greek mathematician Archimedes (III century BC) was able to determine the value of &Iuml;€ between
the interval by 3 10/71, as the minimum value, and 3 1 / 7, the maximum value.&Acirc; With this
approximation of Archimedes gets a value with an error of between 0.024% and 0.040% on the actual
value.&Acirc; The method used by Arqu&Atilde;-medes was simple and consisted circumscribe and inscribe
regular polygons of n−sided circles and calculate the perimeter of these polygons. Archimedes started
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with hexagons circumscribed and inscribed, and was doubling the number of sides to reach 96−sided
polygons.
◊ Chinese Mathematics
In 263, the mathematician Liu Hui was the first to suggest that 3.14 was a good approximation using a
polygon of 968 or 1926 sides. Later &Acirc; he estimated the value of &Iuml;€ as: 3,14159 using a polygon of
3,072 sides.
In finals of V century, the Chinese mathematician and astronomer Zu Chongzhi calculated the value
of &Iuml;€ as: 3.1415926 and which he called &quot;default&quot; and 3.1415927 &quot;excess value&quot; and gave two
rational approximations of &Iuml;€: 22 /&Acirc; 7 and 355/113, both well−known, the second one being so good
and precise that wassn't equaled until more than nine centuries later, in the XV century.
◊ Indian Mathematics
Indian mathematician, Aryabhata estimated the value of &Iuml;€ as 3.1416 using a regular polygon of 384
sides inscribed.&Acirc; Around 1400 Madhava get an accurate approximation to 11 digits (3.14159265359),
being the first to use series to estimate.
◊ Islamic Mathematics
In the ninth century Al−jwarizmi in his &quot;Algebra&quot; (Hisab al yabr ua al muqabala) notes that the
practical man used 22 / 7 as the value of &Iuml;€, the geometer uses 3, and the astronomer 3.1416. In the
fifteenth century, the Persian mathematician al−Kashi Ghiyath was able to calculate the approximate
value of &Iuml;€ with nine digits, using a sexagesimal numerical basis, equivalent to a 16 digit decimal
approximation: 2&Iuml;€ = 6.2831853071795865.
◊ Table of approximations
Mathematician or
Year
Culture
document
~1900&Acirc; b.&Acirc; C. Papyrus of Ahmes
Egyptian
~1600&Acirc; b.&Acirc; C. Tablet of Susa
Babylonian
The Bible&Acirc; (Reyes I,
Jewish
~600&Acirc; b.&Acirc; C.
7,23)
~500&Acirc; b.&Acirc; C. Bandhayana
Indian
~250&Acirc; b. C.
Archimedes&Acirc; from
Siracusa
~150
263
263
~300
Claudius Ptolemy
Liu Hui
Wang Fan
Chang Hong
~500
Zu Chongzhi
~500
~600
~800
1220
Aryabhata
Brahmagupta
Al−Juarismi
Fibonacci
28/34&Acirc; ~ 3,1605
25/8 = 3,125
Error
(parts per
million)
6016 ppm
5282 ppm
3
45070 ppm
3,09
beetwen 3 10/71 y 3 1/7
16422 ppm
Approximation
&lt;402 ppm
Greek
he used 211875/67441 ~
3,14163
Greek−Egyptian 377/120 = 3,141666...
Chinese
3,14159
Chinese
157/50 = 3,14
Chinese
101/2&Acirc; ~ 3,1623
entre 3,1415926 y
3,1415929
Chinese
emple&Atilde;&sup3; 355/113 ~
3,1415929
Indian
3,1416
Indian
101/2&Acirc; ~ 3,1623
Persian
3,1416
Italian
3,141818
13,45 ppm
23,56 ppm
0,84 ppm
507 ppm
6584 ppm
&lt;0,078 ppm
0,085 ppm
2,34 ppm
6584 ppm
2,34 ppm
72,73 ppm
3
1400
Indian
1424
Al−Kashi
Persian
3,14159265359
2&Iuml;€ =
6,2831853071795865
0,085 ppm
0,1 ppm
• The approximations of number &Iuml;€ in
The Modern Times
When the first computers were designed, there began to appear programs for calculating the number
&Iuml;€ with figures as much as possible.&Acirc; In 1949, ENIAC was able to break all records, earning 2037
decimal places in 70 hours,&Acirc; a few years later (1954) a NORAC arrived to 3092 figures.&Acirc; During the
decade of the 1960s IBM were breaking records, until a 7030 IBM was able to reach in 1966 to
250,000 decimal places (8 h 23 min).&Acirc; During this time computers were being tested new algorithms
for generating sets of numbers from &Iuml;€.
In 2009 they found more than two and a half billion decimals using a supercomputer T2K Tsukuba
System, formed by 640 high−performance computers, which together get processing speeds of 95
teraflops.&Acirc; To do this it took 73 hours and 36 minutes.
Year Discoverer
1949
1954
1959
1967
1973
1981
1982
1986
1986
1987
1988
1989
1989
1991
1994
1995
1997
1999
1999
2002
2004
2009
Computer used
G.W. Reitwiesner and others ENIAC
NORAC
Guilloud
IBM 704
CDC 6600
Guillord y Bouyer
CDC 7600
FACOM M−200
Guilloud
Bailey
CRAY−2
HITAC S−810/20
NEC SX−2
others
Hitachi S−820
CRAY−2 y
Brothers Chudnovsky
IBM−3090/VF
Brothers Chudnovsky
IBM 3090
Brothers Chudnovsky
Brothers Chudnovsky
HITAC S−3800/480
Hitachi SR2201
Hitachi SR8000
Hitachi SR8000
Hitachi SR8000/MP
Hitachi
Daisuke Takahashi
T2K Tsukuba System
Number of decimal
places
2.037
3.092
16.167
500.000
1.001.250
2.000.036
2.000.050
29.360.111
67.108.839
134.217.700
201.326.000
480.000.000
1.011.196.691
2.260.000.000
4.044.000.000
6.442.450.000
51.539.600.000
68.719.470.000
206.158.430.000
1.241.100.000.000
1.351.100.000.000
2.576.980.370.000
5. Value of &Iuml;€ with 100 places.
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This is the value of &Iuml;€ with its first 100 places:
3, 1415926535 8979323846 2643383279 5028841971 6939937510
5820974944 5923078164 0628620899 8628034825 3421170679
Bibliography
Internet, especially Wikipedia but not at all, I used other websites too.
Personal Opinion
I think this work is very interesting because I could to learn things about very specific number, the
number &Iuml;€.
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