Ankle Platform Heels Latasa High Pumps strap White Women's apvHxO Approximations and strange things about Pi
This page is devoted to the rational and irrational approximations which are nearest to Pi. And there are some great ones ! Not to mention some approximate curiosities concerning Pi.
Indeed, since people realised that there was a constant that defined the ratio of the perimeter of a circle over its diameter, a discovery dating from the Egyptians and the Babylonians, mathematicians have tried to give the exact value of Pi or, after the XVIIIth century, tried to give simple approximations of Pi.
During ancient times
Of course during antiquity mathematicians didn't know Pi was irrational, not even transcendant. So they believed some approximations of Pi were exact values ! This progression during antiquity is related on the geometric era page , I will just remind you of the values: 91 Remonte Wedge Multi Red Sandal D4569 F6wqR6zx
The Babylonians had estimated:
which was quite remarquable for that time. The Egyptian scribe Ahmès had obtained :
=(16/9)2=3,16...
The Bible has remained famous for its hazardous approximation of : 3 !! Let's not forget that Archimedes had set the limits of Pi, which showed that he was conscious of not being able to get the exact value.
In China, Tsu Chung Chih went ahead of the West, and suggested :
355/113 = 3,14159292...
But let's get back to the Western world. Still trying to get an exact value for Pi, Nicolas of Cues suggested :
The best of all was certainly Edward Johnston Goodwin (1828-1902) who in 1897 in Indiana (US), put forward a bill where several formulae, once applied, would have assigned to Pi the values 4 then 3.1604, then 3.2, and finally 3.232 ! Of course, this ingenious author had resolved the squaring of the circle, the trisection of an angle, and the duplication of the cube, which had been proved impossible a few years before, and he had allowed his discoveries to be used free of charge! Even the famous review American Mathematical Monthly, new at the time, had accepted two of his articles ! Fortunately, this project which was nearly accepted was finally refused because it was considered at the time that the law should not decide scientific truth... and a good job too!...
After the "renaissance"
The demonstration of the irrationality in 1761 then of the trascendence in 1882, destroyed howeverthe amateurs' and mathematicians' hope of getting the exact value of Pi, apart from the scatty man from Indiana. At that time, the fun was about to start, because people started to search for simple rational or irrational approximations of Pi . And the imagination and patience of mathematicians was fertile ! Check it out : A few approximations :
Note from myself: Jean-Christophe Benoist noticed with pertinence that books about Pi (Delahaye, Warufsel...) used to sort the approximations of Pi as a function of the number of correct digits they provide. From this tenet, 3.14151 has a better rank than 3.1416 whereas the second approximation is the best. Moreover, two approximations having the same number of correct digits can't be discriminated. He then proposed me to consider the score -Log10(|approximation-Pi|) as a sorting criterion, which has the advantage of taking into account the real distance with Pi ! Thus, I included some values of this score for the smallest approximations into parenthesis.
Formulae |
Approximated value |
Correct digits of Pi |
|
|
|
|
3,1409 |
2 (3.21) |
|
|
|
Kochansky's Approximation |
|
|
|
3,141533 |
4 (4.23) |
|
|
|
Ramanujan's approximations |
|
|
= |
3,14164 |
3 (4.32) |
|
3,14182 |
3 (3.63) |
|
3,141623 |
3 (4.51) |
|
3,14164 |
3 (4.32) |
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3,14159265258 |
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|
3,14159265258 | Kuruka Estate Running Blue Shoe Fluo Green Diadora Men's xgSEwqAI
8 (8.99) |
|
3,14159265380 |
9 |
|
3,14159265358979432 |
14 |
|
3,14159265358979265 |
14 |
|
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|
3,1415926535897932384190 |
19 |
|
3,141592653589793238462642088 |
23 |
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|
3,141592653589793238462643383279 |
30 |
|
|
|
Castellanos' spproximations (1988) |
|
|
(2e3+e8)1/7 |
3,14171 |
3 (3.91) |
|
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|
3,141575 |
4 (4.76) |
|
3,14159704 |
5 (5.36) |
|
3,141592452 |
6 (6.70) |
|
3,14159259259 |
6 (7.21) |
|
3,141592649 |
7 (8.41) |
|
3,141592654111 |
8 (9.28) |
|
3,14159265363 |
9 |
|
3,14159265358817 |
11 |
|
3,14159265359037 |
10 |
|
3,141592653589780419 |
13 |
|
|
|
Logan Women Platform High Black Thick Winter Shoes Heels Black Motorcycle On Slip Jerald Snow Boots Plouffe's approximations |
|
|
437/23 |
3,1415398 |
4 (4.28) |
|
3,1415943 |
5 (5.77) |
|
3,14159267809 |
7 (7.61) |
|
3,14159259508 |
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|
3,14159265258 |
8 (8.99) |
|
3,14159265297 |
8 |
|
3,141592653492 |
9 |
|
3,1415926535949 |
10 |
|
3,141592653586778 |
11 |
|
3,1415926535912 |
10 |
|
3,141592653589793238462649201 |
23 |
|
3,1415926535897932384626433832797 |
30 |
|
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|
Stoschek's approximation |
|
|
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|
3,141104 |
3 (3.31) |
|
|
|
And a few others |
|
|
|
3,141592920 |
6 (6.57) |
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3,1415333 |
Jerald Heels Shoes Logan Boots Black Black Motorcycle Platform Slip Women Winter Snow Thick High On Slip Logan Black Shoes Jerald Motorcycle On Women Platform Thick Snow Heels Winter Boots Black High 4 (4.23) |
103993/33102 (Euler) |
3,14159265301 |
9 |
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It is worth noting that Navy Blanc Blanc De Skool VN000D3HNVY Black Old Gum Vans qvpTn (1913) and Olds (1963) gave a geometrical justification of the approximation 355/113. Gardner (1966) looked after . As for Dixon (1991) he was interested in and=.
Strange facts
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All the above formulae are themselves curiosities because by inverting a formula giving Pi, you nearly get an integer or a rational or irrational fraction. The best example is that of Roy Williams' expression with n a natural integer : For certain values of n (particularly among Heegner's numbers 1,2,3,7,11,19,43,67 and 163), we almost find an integer :
n |
Value of |
|
|
25 |
6635623,999341134233266067 |
37 |
199148647,999978046551856766 |
43 |
884736743,999777466034906661 |
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24591257751,999999822213241469 |
67 |
147197952743,999998662454224506 |
74 |
545518122089,999174985664301733 |
148 |
39660184000219160,000966674358575246 |
163 |
262537412640768743,999999999999250072 Brown Solid WeiPoot Closed Women's Low Pumps Shoes Round Buckle Toe Heels T7nvtBr7 |
232 |
604729957825300084759,999992171526856431 |
268 |
21667237292024856735768,000292038842412959 |
522 |
14871070263238043663567627879007,999848772648279480 |
652 |
68925893036109279891085639286943768,000000000163738644 |
719 |
3842614373539548891490294377805829192,999987249566012187 |
The "Fascinant nombre Pi" (see Slides Red studs Gold Studded Dress Pan Sandals Rivets Flats Mules Slippers Strappy Womens Rockstud Caitlin Backless XO4Uqw) tells us that Martin Gardner had used the case n=163 as an April fool's joke in 1975 saying that was an integer, which at the time was no easy to contradict ! In fact, this extraordinary particular case apparently comes from the fact that the algebric numbers field given by 1631/2 has a unique factorisation proprety, as Heegner's numbers have particular arithmetic qualities.
Some precisions about this fact: Navy Blanc Blanc De Skool VN000D3HNVY Black Old Gum Vans qvpTn, (obviously !) had noticed that is close to an integer from less than 10-12 which is really unexpected. His argument from the theory of modular functions was only understood dozens of years later.
This argument deals with the same function j as in the Borwein's page: we first consider (t complex number). This is a modular function, which means that it is defined on the superior complex half-plan, it has a limit in , it is meromorph and above all invariant under the transformation (a, b, c, d entiers).
Now, let's define an equivalence relation on the quadratic forms a.x2+b.xy+c.y2 with a, b, c coprimes in Z (a>0). The discriminant is d=b2-4ac<0.
We define the relation ~ by a.x2+b.xy+c.y2 ~ a'.x2+b'.xy+c'.y2Sunset Stacy Dress Adams Black Men's HrwHE1q if one form is the result of the other (and reciprocally) under the transformation :
This equivalence relation keeps intact the discriminant d. The number of equivalence classes is noted h(d) and a well-known theorem from Heegner (52), Baker et Stark (66) says that :
These integers are called Heegner's numbers.
Moreover, Webershowed that:
We then know that .
On the other side, being a meromorph function, j has a Laurent's development in q which is:
From the previous value , we get :
n being integer, which completes the proof.
The algebrical approach shows that Heegner's numbers are very important. The generated imaginary quadratic fields like have some fascinating factorization properties. For instance, the number of class of is 1 which is equivalent to the fact that the decomposition in prime factors in this field is unique. Black Black Logan Boots Slip Thick Heels Platform Shoes Women High Snow Motorcycle On Jerald Winter Another curiosity is the number : Less than 1 thousandth of an error with 20 for such a simple number, that's surprising isn't it?
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