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The Early History of Pi
Egypt
In 1856 a Scotsman by the name of Henry Rhind purchased a
papyrus in Egypt
...
C
...

Problems of algebra, linear systems, mensuration, and mathematical amusements were assembled
...

In one problem we find the statement:

...

From this statement, the “student” was expected to compute the area of any
circle
...

Effectively, using the fact that the true area of a circle of diameter 9 is
2

/

π

= (4 5)2 ≈ 63 62

r

π


...


and the area of a square of side 8 is 64, we easily see that the effective value
for is
13
≈ 3 1605 or 3
81
Most modern histories give this value to be
π

π


...


From the Bible:
“Also he made a molten sea of ten cubits from brim to brim, round in
compass, and five cubits the height thereof; and a line of thirty cubits did
compass it round about”
...
Located in the
Mesopotamian River Valley, they created large monuments to their gods and
kings while engaging in commerce throughout the Mediterranian
...
Excellent
records of their whole society, including their mathematics has been preserved
in the tens of thousands of clay tablets
...

For example, they seemed to be well aware of the Pythagorean theorem
a thousand years before Pythagoras
...
Generally, they used the
approximation
π = 3
...
These tablet were on whole concerned with the
ratios of areas and perimeters of regular polygons to their respective side
lengths
...
9600
60 60

The correct value of this ratio is
6
2π
and this gives an effective value of π to be
3

1
8


...


Three problems vexed (and blessed) their mathematical development:
• How to square the circle
...

• How to trisect an angle
...

What they didn’t know was that these problems would require more than
two millenia to solve
...

(It’s proof is based on the Eudoxus Method of Exhaustion, the Greek version
of taking limits, without formally defining them
...

From this we know that the Greek mathematicians were well aware that
there must be a mathematical model equivalent to the one we know as

A = π r2

Archimedes
The greatest mathematician of antiquity, Archimedes
(287-212 B
...
), proved two fundamental relations about the perimeters and
areas of inscribed and circumscribed regular polygons
...
bn denote the regular inscribed 6 · 2
...
Bn for the circumscribed polygons
...

2pn Pn
pn + Pn
p
= an An

Pn+1 =

pn+1 =

an+1

An+1 =

p
pn Pn+1

2an+1 An
an+1 + An

Using n-gons up to 96 sides he derives

3

10
1
<π<3
71
7

Following Archemedes, Apollonius (250-175 B
...
), in his book, now lost,
on rapid calculation techniques, Quick Delivery, approximates π as
π ≈ 3
...
Apollonius is considered one
of the last great mathematicians of antiquity
...

The great astronomer and the father of trigonometry Claudius Ptolemy
(100-178 A
...
) gives
377
π≈
= 3
...

120
although his method of derivation is unknown
...
1500)
Rome
The Romans were warriors, engineers, authors, and rulers
...
Even with all the Greek mathematics available,
they computed circular areas using proportion and the statement
A wheel of diameter 4 feet has a perimeter of 12 1 feet
...
C
...
They
√
also used the alternative approximation, π ≈ 10 ≈ 3
...
By the 5th century
more accurate approximations were known
...
14159292035
accurate →
113
He also shows
3
...
1415927

Note that to eleven decimal places π = 3
...
Results of this accuracy were not achieved in the West until the 15th century
...
D
...

The result is approximately the circumference of
a circle of which the diameter is 20,000
...
1416
20, 000
1250
Yet, the Hindu approximation to π was usually

√
10
...
D
...
1416
20, 000

In about 1436 Al-Kashi gives
2π ≈ 6
...


The Modern History of Pi
France The French mathematician Francois Vieta (1540-1603), the first
mathematician to us symbolic notation for knowns (consonants) and unknowns (vowels) discovered the first direct formula for the calculation of
π
...
Using
the notation an for the area of the inscribed n-gon, apply the formula
π
a2n = an sec
n
recursively, beginning with n = 4 (the square)
...

Germany
Ludolph van Ceulen (1540-1610) computed π accurate to
20 places
...
So proud was he of this
accomplishment that he had the digits engraved on his tombstone
...
(It is not yet called π
...


England
...
The first approximation to π came from the clergyman, cryptographer and mathematician John Wallis (1616-1703)
...
, he produced
2
1 · 3 · 3 · 5 · 5 · 7···
=
π
2 · 2 · 4 · 4 · 6 · 6···

From this Lord William Brouncker (1620?-1684), the first president of
the Royal Society, discovered the formula
4
=1+
π

1
9

2+
2+

25
49
2+
2 + ···

James Gregory (1637-1675) derived the well known (to us) formula
Z

0

x

x5
x7
dt
x3
+
−
+ ···,
= arctanx = x −
1 + t2
3
5
7

now called Gregory’s formula
...

(Note the convergence is very slow
...
”
In 1874, Machin’s formula was used by William Shanks (1812-1882) to
achieve 707 places of accuracy for π
...
Only about 510 terms
are required to achieve this accuracy
...

Perhaps the greatest mathematician that ever lived, Leonhard Euler (1707-1783), was born in Basel and studied under James Bernoulli
...
Petersburg, originally to serve on the
faculty of medicine and physiology
...
His contributions to mathematics have been well documented and are prodigious
...

In 1779 he determined the formula
1
3
π = 20 arctan − 8 arctan
7
79
from which the computation of π can be made using Gregory’s formula
...
He
also derived the formulas
∞
X 1
π2
=
6
n2
n=1
∞
X 1
π4
=
90 n=1 n4

∞
X 1
π6
=
945 n=1 n6

∞
X 1
π8
=
9, 450 n=1 n8

∞
X 1
π10
=
93, 955 n=1 n10


...


...


...


as well as formulas for the sum of the remaining reciprocal (even) integers
up to to order 26
...
In 1801 he also used the modern symbols for e and i = −1
...
Managing to teach himself elementary mathematics and pursue
his mathematical studies while a tutor for a Swiss family, he eventually was
able to enjoy the company of the day’s greatest mathematicians including
Lagrange and Euler
...

He proves this using a continued fraction expansion of e given by Euler in
1737
...

At this time there were so many circle squarers that in 1775 the
Paris Academy passed a resolution that no more solutions on the quadrature
of the circle should be examined by its officials
...
The conviction was
growing that the solution of squaring the circle was impossible
...
For example,
r
q
q
√
√
√
2,
3 + 4,
5 + 6 + 7, etc
...
The set of all solutions, real and complex, of polynomial
equations with rational coefficients are called the algebraic numbers
...

Transcendental numbers were discovered by Joseph Liouville (1809-1882)
in 1844 with the number
1
1
1
+ 2! + 3! + · · ·
10 10
10
In 1873, Charles Hermite (1822-1901) proved that e was transcendental
by showing that it cannot be the solution of a polynomial equation with
rational coefficients
...
L
...
Lindemann (1852-1939) proved that π is transcendental by showing that if a, b, c,
...
are algebraic numbers
the the equation
aem + ben + cer + · · · = 0

has no rational solutions
...
Using Euler’s celebrated formula
eiπ + 1 = 0
it follows that π is not algebraic
...
246, 1897
...
”
“In further proof of the value of the author’s (E
...
Goodman, M
...
) proposed
contribution to education, and offered as a gift to the State of Indiana, is
the fact of his solutions of the trisection of the angle, duplication of the cube
and quadrature of the circle having been already accepted as contributions
to science by the American Mathematical Monthly, the leading exponent of
mathematical thought in this country
...

(via Committee on Swamp Lands)
Passes first reading, Indiana Senate, 1897
...
If we pass this bill which establishes a new
and correct value of π, the author offer our state without cost the use of this
discovery and its free publication in our school textbooks, while everyone
else must pay him a royalty
...
A
...

What we do know!
?
?
?
?

π is irrational
...


Lambert (1761)
Lindemann (1882)
Baker (1966)

Sophisticated Computations
?
?
?
?
?
?

(or what can be done with really fast computers)
Besides the computations already mentioned Machin-like formulae have produced
2037 Digits – Eniac (1949) – (von Neumann, et
...
)
1,000,000 Digits – CDC 7600 (1973) – (Guilloud and Bouyer)
17,000,000 Digits – Symbolics 3670 (1985), (Gosper)
29,000,000 Digits – Cray-2 (1986), (Bailey)
1-4 Billion Digits – (Chudnovskys)
51 Billion Digits – Hitachi’s (1997) – (Kanada)
India
In 1914 the Indian mathematician Srinivasa Ramanujan proved
the following representation of π
...

1/π =
9801 n=0
(n!)4 3964n

Canada
Even better formulas have been more recently developed
...
Compute to 99 terms and you will obtain π to 2,500 digits!!!
Iterative Formulas are available also
...


With n at 15 you are guaranteed over 2 billion places of accuracy for π
...
20th September 1999
Japan
The current record is held by Dr
...
g
...
Then we are declaring 206,158,430,000 decimal digits
as the new world record
...
758 GB * 128)
Algorithm : Gauss-Legendre algorithm
Optimized Verification program run:
Job start : 26th June 1999 01:22:50 (JST)
Job end : 27th June 1999 23:30:40 (JST)
Elapsed time : 46:07:10
Main memory : 817 GB (= 6
...
0

Programs are consisted of two sets of routines, e
...
calculation routines and
message passing routines
...
Daisuke
TAKAHASHI, a Research Associate at our Centre and rather speed sensitive
message passing routines were written by myself
...
5 billion record establishment
...
g
...
CPU used was HITACHI SR8000 at the Information Technology
Center, Computer Centre Division (old Computer Centre,) University of
Tokyo
...
g
...
One
trillion floating point operations per second for all PE’s), were definitely used
as single job and parallel processing for both of programs run
...
lupi
...
html
Yasumasa KANADA
Computer Centre, University of Tokyo
Bunkyo-ku Yayoi 2-11-16
Tokyo 113 Japan
Fax : +81-3-3814-7231 (office)
E-mail: kanada@pi
...
u-tokyo
...
jp

Is that it?

I believe so!



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