Collection of approximations for p

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1  Approximation formulae

The history of p is full of more or less good approximations.

1.1  Rational approximations

The first estimations of the ratio of the circumference to its diameter are found in the ancient times. The symbol p was not yet used to design this ratio.
p
3+  1

8
=3.1(25)       (Babylonians)
p

 4

3

4

 
=3.1(604...)       (Egyptians)
p
 22

7
=3.14(285...)       (Archimedes 287-212 B.C.)
p
3+  8

60
+  30

602
=3.141(666...)       (C.Ptolemy 100-170)
p
 62832

20000
=3.141(6)       (India around 500)
p
 333

106
=3.1415(094...)
p
 355

113
=3.141592(920...)       (Zu Chongzhi,Adriaen Métius, ...)

Ptolemy gave his approximate value of p in his Almagest and he used sexagesimal fractions [1].

The famous and remarquable value 355/113 was published in 1625 by Adriaen Metius but it was already used in China around 480 and also known to the Japanese.

1.1.1  Continued fractions

The theory of continued fractions provides the sequence of the best rational approximations for the number p, the first terms are given recursively by (starting with x=p and n=0)
in
=
[x]
x
=
 1

x-in
n
=
n+1.

It follows that i0=3,i1=7,i2=15,... and it's usual to write it as :
p = [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,2,1,1,15,3,13,...]
which becomes in term of fractions

3,  22

7
,  333

106
,  355

113
,  103993

33102
,  104348

33215
,  208341

66317
,  312689

99532
,  833719

265381
,  1146408

364913
,  4272943

1360120
,...
and for example the fraction
 428224593349304

136308121570117
=3.14159265358979323846264338327(569...)
has 29 correct digits and it was found by the the Japenese mathematician Arima in 1766. We know from an important theorem of continued fraction (see [2] for a proof) that the error of the approximation pn/qn satisfies

 pn

qn
-p
 1

qnqn+1
<  1

qn2
.

1.2  With square roots


p


 

10
 
=3.1(622...)       (India around 600)
p
 3

4
( 3+6) = 3.1(361...)      (Nicolaus de Cusa-1464)
p
2+3=3.14(626...)
p
 88




785
=3.14(085...)       (TychoBrahe-1580)
p
102-11=3.14(213...)       (Grosvenor-1868)
p


 

51
 
-4=3.141(428...)
p
  


 40

3
-23
 
=3.1415(333...)       (Kochansky-1685)
p
 13

50


 

146
 
=3.14159(195...)       (Specht-1836)
p
 141

1232

5+6

 

14
 

=3.14159265358(015...)

Kochansky's value (Poland) was published in 1685 and is the result of a geometrical construction for p. Another construction was given by Specht (Germany) for it's value.

1.3  Ramanujan's approximations

In his work on modular equations Ramanujan found impressive approximations, this is just a small selection [4].


p

92+  192

22

1/4

 
=3.14159265(258...)
p
 9

5
+   


 9

5
 
=3.141(640...)
p
 9801

4412
2=3.141592(730...)
p
 63

25

 17+155

7+155

=3.141592653(805...)
p
 12




130
log

(2+5)(3+

 

13
 
)

2


=3.14159265358979(265...)
p
 12




190
log
(22+

 

10
 
)(3+

 

10
 
)
=3.141592653589793238(190...)

The last approximations are of the form
p  2

n
log( 8gn12)
where usually n is an integer and gn a Ramanujan's invariant.

To conclude this section here is a remarkable approximation of p published in 1984 by Morris Newman and Daniel Shanks [3]. Set
a
=
 1071

2
+92

 

34
 
+  3

2
  


255349+43792


34
 
,
b
=
 1553

2
+133

 

34
 
+  1

2
  


4817509+826196


34
 
,
c
=
429+3042+2

 

92218+652082
 
,
d
=
 627

2
+2212+  1

2


 

783853+5542682,
 
then


p-  6




3502
log(2abcd)

< 7.4.10-82.

1.4  Other approximations


p

 2143

22

1/4

 
=3.14159265(258...)      (Plouffe)
p

 77729

254

1/5

 
=3.14159265(411...)       (Castellanos)
p
 6

5
log( 7+35) =  6

5
log(2)+  24

5
log(j)=3.14159(337...)
p
 e2

3
+  19

28
=3.14159(012...)
p
 4




j
=3.14(460...)       (Ghyka)
p
 6

5
(1+j)=3.141(640...)
where
j =  1+5

2
is the Golden ratio.

References

[1]
F. Cajori, A History of Mathematical notations, Dover, (republication 1993, original 1928-1929)

[2]
G.H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, (1979)

[3]
M. Newman, D. Shanks, On a Sequence Arising in Series for p, Math. of Comp., (1984), vol. 42, p. 199-217

[4]
S. Ramanujan, Modular equations and approximations to p, Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372




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