# Collection of approximations for p

## 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
 »
 æè 4 3 öø 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
 éë 3, 22 7 , 333 106 , 355 113 , 103993 33102 , 104348 33215 , 208341 66317 , 312689 99532 , 833719 265381 , 1146408 364913 , 4272943 1360120 ,... ùû
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
 êê pn qn -p êê £ 1 qnqn+1 < 1 qn2 .

### 1.2  With square roots

 p
 »
 Ö 10 =3.1(622...)       (India around 600)
 p
 »
 3 4 ( Ö3+Ö6) = 3.1(361...)      (Nicolaus de Cusa-1464)
 p
 »
 Ö2+Ö3=3.14(626...)
 p
 »
88

 Ö 785
=3.14(085...)       (TychoBrahe-1580)
 p
 »
 10Ö2-11=3.14(213...)       (Grosvenor-1868)
 p
 »
 Ö 51 -4=3.141(428...)
 p
 »
æ
Ö

 40 3 -2Ö3

=3.1415(333...)       (Kochansky-1685)
 p
 »
 13 50 Ö 146 =3.14159(195...)       (Specht-1836)
 p
 »
 141 1232 æè 5+6 Ö 14 öø =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
 »
 æè 92+ 192 22 öø 1/4 =3.14159265(258...)
 p
 »
9

5
+   æ
Ö

 9 5

=3.141(640...)
 p
 »
 9801 4412 Ö2=3.141592(730...)
 p
 »
 63 25 æè 17+15Ö5 7+15Ö5 öø =3.141592653(805...)
 p
 »
12

 Ö 130
log æ
ç
è
 (2+Ö5)(3+ Ö 13 )

Ö2
ö
÷
ø
=3.14159265358979(265...)
 p
 »
12

 Ö 190
log æ
è
(2Ö2+
Ö

10

)(3+
Ö

10

) ö
ø
=3.141592653589793238(190...)

The last approximations are of the form
 p » 2 Ön 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
Ö

34

+  3

2
æ
Ö

 255349+43792 Ö 34

,
 b
 =
1553

2
+133
Ö

34

+  1

2
æ
Ö

 4817509+826196 Ö 34

,
 c
 =
 429+304Ö2+2 Ö 92218+65208Ö2 ,
 d
 =
 627 2 +221Ö2+ 1 2 Ö 783853+554268Ö2,
then
ê
ê
ê
p-  6

 Ö 3502
log(2abcd) ê
ê
ê
< 7.4.10-82.

### 1.4  Other approximations

 p
 »
 æè 2143 22 öø 1/4 =3.14159265(258...)      (Plouffe)
 p
 »
 æè 77729 254 öø 1/5 =3.14159265(411...)       (Castellanos)
 p
 »
 6 5 log( 7+3Ö5) = 6 5 log(2)+ 24 5 log(j)=3.14159(337...)
 p
 »
 e2 3 + 19 28 =3.14159(012...)
 p
 »
4

 Ö j
=3.14(460...)       (Ghyka)
 p
 »
 6 5 (1+j)=3.141(640...)
where
 j = 1+Ö5 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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On 6 May 2002, 15:56.