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Machin's Formula
The hunt for PI began in
Egypt and in Babylon about two thousand years before Christ. The
Egyptians obtained the value (4/3)4=3.160493827 and the
Babylonians the value 3 1/8=3.125 for PI.About
the same time, the Indians used the square root of 10 for PI.
In
the 3rd century B.C. Archimedes considered inscribed and
circumscribed polygons of 96 sides and deduced that 3 + 10/71 < PI < 3 + 1/7
.
Various people calculated PI, including:
Ptolemy (c.150 AD) 3.1416
Tsu Ch`ung Chi (430-501 AD) 355/113
Al`Khwarizmi (c.800) 14 places
Vičte (1540-1603) 9 places
Roomen (1561-1615) 17 places
Van Ceulen (c.1600) 35 places There was no theoretical
progress involved in these improvements, only greater stamina in
calculation.
The European Renaissance brought about in due course
a whole new mathematical world. Among the first emergence of
mathematical formulae for PI.One of the best-known was that of James
Gregory (1638-1675):
Gregoru also showed the more general
result:
from which the first series if we put x=1.
Archimedes`constant PI has many infinite series, infinite
product, definite integral and continued fraction representations.
It was proved to be irrational by Lambert and transcendenal by
Lindemann. The first truly attractive formula for coputing decimal
digit of PI was found by John Machin ( 1680-1752), in 1706.
Gauss, Stirling and others used the following
formulae for PI/4:
The adventage of Machin`s formula is that the
second term converes very rapidly and the first is nice for decimal
arithmetic.
Here are the computations: All computations to 15 decimal places:
arctan(1/5) arctan(1/239):
1/5 = 0·200000000000000 1/239 = 0·004184100418410
1/375 =-0·002666666666666 1/40955757 =-0·000000024416591
1/15625 = 0·000064000000000 1/3899056325995= 0·000000000000256
1/546875 =-0·000001828571428
1/17578125 = 0·000000056888889
1/537109375 =-0·000000001861818
1/15869140625 = 0·000000000063015
1/457763671875 =-0·000000000002184
1/12969970703125 = 0·000000000000077
1/362396240234375=-0·000000000000002
SUMMING:
arctan(1/5) = 0·1973955598498807 and arctan(1/239) = 0·004184076002074
Putting these in the Machin's formula gives:
Pi/4= 4xarctan(1/5 ) - arctan( 1/239 )
or Pi = 16xarctan(1/5 ) - 4xarctan( 1/239 )
= 16x0·1973955598498807 - 4x0·004184076002074
= 3·1415926535897922
Using this, Machin became the first individual to
correctly compute 100 digits of PI.
John Machin was born 1689
in England. We know that he acted as a private tutor to Brook Taylor
teaching him mathematics in 1701. He continued to correspond with
Taylor for many years and this is a useful source for understanding
his mathematical thinking. The two met in coffeehouses, a standard
place where mathematical discussions were held during this period.
We also know that Machin was friendly with Keill, who taught at
Oxford, and with de Moivre who like Machin was a private tutor of
mathematics at this time. In 1706 William Jones reports that Mr John
Machin`s formula allows PI be calculated:- ... to above 100
places.
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No indication is
given in Jones's work, however, as to how Machin discovered
his series expansion for p so when de Moivre wrote to Johann
Bernoulli on 8 July 1706 telling him about Machin's series for
PI he suggested that Johann Bernoulli might tell Jakob Hermann
about Machin's unproved result. He did so and Hermann quickly
discovered a proof that Machin's series converges to PI. He
produced techniques that show other similar series also
converge rapidly to PI and he wrote on 21 August 1706 to
Leibniz giving details. Two years later, on 6 July 1708, de
Moivre wrote again to Johann Bernoulli about Machin's series,
on this occasion giving two different proofs that it converged
to PI.
On 16 May 1713 Machin was appointed as Professor of
Astronomy at Gresham College, London. He succeed Dr Torriano
and went on to hold the chair until his death 38 years later.
For nearly 30 of these years he acted as Secretary to the
Royal Society, being appointed in 1718 and holding the post
until 1747. John Machin died 9 June 1751 in London.
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An Englishman named Shanks
used Machin`s formula to calculate PI to 707 places, publishing the
results of many years of labour in 1873.
Here is a summary of
how the calculating of PI went:
1699: Sharp used Gregory`s result to get 71 correct digits.
1701: Machin used an improvement to get 100 digits and the
following used his methods:
1791: De Lagny found 112 correct digits.
1789: Vega got 126 places and in 1794 got 136.
1841: Rutherfor calculated 152 digits and in 1853 got 440.
1873: Shanks calculated 707 places of which 527 were
correct.
1949: A computer was used to calculate PI to 2000 places.
1973: Guilloud and Bouyer used a version of it to compute one
million digits of Pi on a CDC 7600.
1997: Y.Kanada calculated PI to over 51 billion
digits.
Here, we shall give a proof for the John
Machin`s formula:
Well-known trigonometrical identity that can be used to prove Machin`s formula is:
We shall put α = β
For α=2α
In general case
Here are valid identities for polynomials
Pn(x) and Qn(x) :
Using former reccurency relations we get for
n=4 the next polynomials:
Finally we have:
For tg( 4α - β ) we get
or:
If we introduce supstitations:
where
We have:
Now we get for t the next
relation:
In case of x=1 and n=4 we get
t=4/956=1/239 and arctg(1/1)=PI/4:
So, we have
proved Machin`s formula:
Let us take n=1,2,3,4,5,... - we get
series of Machin`s formulae:
We have just seen that there are infinitely
many formulae for PI/4 using arcustanges function.
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