Pi Approximations
Pi Approximations
Pascal Sebah
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 it's diameter are
found in the ancient times. The symbol p was not yet used to design
this ratio.
|
| |
|
|
|
3+ |
1
8
|
= 3.1(25) (Babylonians) |
| |
|
|
( |
4
3
|
)4 = 3.1(604938271...) (Egyptians) |
| |
|
|
|
22
7
|
= 3.14(28571428...) (Archimedes287-212 B.C.) |
| |
|
|
3+ |
8
60
|
+ |
30
602
|
= 3.141(666666...) (C. Ptolemy 100-170) |
| |
|
|
|
62832
20000
|
= 3.141(6) (India around 500) |
| |
|
|
|
333
106
|
= 3.1415(094339...) |
| |
|
|
355
113
|
= 3.141592(9203...) (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)
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
|
|
é
ê
ë
|
3, |
22
7
|
, |
333
106
|
, |
355
113
|
, |
103993
33102
|
, |
104348
33215
|
, |
208341
66317
|
, |
312689
99532
|
, |
833719
265381
|
, |
1146408
364913
|
, |
4272943
1360120
|
,¼ |
ù
ú
û
|
|
|
and for example
|
|
4272943
1360120
|
= 3.141592653589(3891...) |
|
has 12 correct digits. We know from an important theorem of continued
fraction (see [2] for a proof) that the error of the approximation pn/qn satisfies
|
|
ê
ê
ê
|
|
pn
qn
|
-p |
ê
ê
ê
|
£ |
1
qnqn+1
|
< |
1
qn2
|
. |
|
1.2 With square roots
|
| |
|
|
|
|
| __
Ö10
|
= 3.1(622776601...) (India around 600) |
| |
|
|
|
3
4
|
( Ö3+Ö6) = 3.1(3615541276...) (Nicolaus de Cusa-1464) |
| |
|
|
Ö2+Ö3 = 3.14(62643699...) |
| |
|
|
|
88
|
= 3.14(08546850...) (TychoBrahe-1580) |
| |
|
|
10Ö2-11 = 3.14(21356237...) (Grosvenor-1868) |
| |
|
|
|
| __
Ö51
|
-4 = 3.141(4284285...) |
| |
|
|
|
æ
ú
Ö
|
|
= 3.1415(333387...) (Kochansky-1685) |
| |
|
|
|
13
50
|
|
| ___
Ö146
|
= 3.14159(19531...) (Specht-1836) |
| |
|
|
141
1232
|
|
æ
è
|
5+6 |
| __
Ö14
|
ö
ø
|
= 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].
|
| |
|
|
|
|
æ
ç
è
|
92+ |
192
22
|
ö
÷
ø
|
1/4
|
= 3.14159265(2582...) |
| |
|
|
|
9
5
|
+ |
æ
ú
Ö
|
|
= 3.141(6407864...) |
| |
|
|
|
9801
4412
|
Ö2 = 3.141592(7300...) |
| |
|
|
|
63
25
|
|
æ
ç
è
|
|
17+15Ö5
7+15Ö5
|
ö
÷
ø
|
= 3.141592653(805...) |
| |
|
|
|
12
|
log |
æ
ç
ç
ç
è
|
|
Ö2
|
ö
÷
÷
÷
ø
|
= 3.14159265358979(2653...) |
| |
|
|
12
|
log |
æ
è
|
(2Ö2+ |
| __
Ö10
|
)(3+ |
| __
Ö10
|
) |
ö
ø
|
= 3.141592653589793238(1908...) |
|
| |
|
The last approximations are of the form
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
|
| |
|
|
|
|
1071
2
|
+92 |
| __
Ö34
|
+ |
3
2
|
|
æ
Ö
|
|
, |
| |
|
|
|
1553
2
|
+133 |
| __
Ö34
|
+ |
1
2
|
|
æ
Ö
|
|
, |
| |
|
|
429+304Ö2+2 |
Ö
|
92218+65208Ö2
|
, |
| |
|
|
627
2
|
+221Ö2+ |
1
2
|
|
Ö
|
783853+554268Ö2,
|
|
|
| |
|
then
|
|
ê
ê
ê
ê
ê
|
p- |
6
|
log(2abcd) |
ê
ê
ê
ê
ê
|
< 7.4 ×10-82 |
|
1.4 Other approximations
|
| |
|
|
|
|
æ
ç
è
|
|
2143
22
|
ö
÷
ø
|
1/4
|
= 3.14159265(25...) (Plouffe) |
| |
|
|
|
æ
ç
è
|
|
77729
254
|
ö
÷
ø
|
1/5
|
= 3.14159265(41...) (Castellanos) |
| |
|
|
|
6
5
|
log( 7+3Ö5) = 3.14159(33769...) |
| |
|
|
|
e2
3
|
+ |
19
28
|
= 3.14159(01282...) |
| |
|
|
|
6
5
|
log(2)+ |
24
5
|
log(j) = 3.14159(3376...) |
| |
|
|
|
4
|
= 3.14(4605511...) (Ghyka) |
| |
|
|
6
5
|
(1+j) = 3.141(6407864...) |
|
| |
|
where
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
Back to
mathematical
constants and computation
File translated from TEX by TTH, version 2.32.
On 21 May 2000, 23:31.