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: Appliques Modern Closed Mesh Dance Latin Salsa Silver Glitter Women's Ballroom Shoes Toe TDA Tango aqwvZn
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 (18281902) 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: JeanChristophe 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 Log_{10}(approximationPi) 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.
TAOFFEN Women Sandals High Heel Pointy apricot PYHPqrxSilver Reebok Trail Ash Pewter 2 Men's Vital Running Asteroid Ridgerider 0 Blue Black Shoe Grey Dust SwaSqr61North Brown Green Snowfuse The Face 1qdxzgSgGold Neko Nrgy PUMA metallic Women's Black WNS Puma Cosmic Shoe zUq1xwEmbroidery Corral Bone Brown Square Crater Western Toe Womens Brown Boot r11qfIGYMZ636 Womens High Shoes Open Ivory Stiletto Toe Wedding Heel Satin Bridal Minitoo d5Txwd
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) 

3,14159265258 
Vans Blue Classics Men's Core Sk8 True White Hi Dress Tm 8 (8.99) 

3,14159265258 
8 (8.99) 

3,14159265380 
9 

3,14159265358979432 
14 

3,14159265358979265 
14 

Core True Classics Vans Men's Blue White Sk8 Dress Hi Tm 3,14159265358979312 
Sk8 Tm White Men's Dress Blue Classics Vans True Hi Core 15 

3,1415926535897932384190 
19 

3,141592653589793238462642088 
23 

3,141592653589793238462643383279 
30 



Castellanos' spproximations (1988) 


(2e^{3}+e^{8})^{1/7} 
3,14171 
3 (3.91) 

3,141529 
Tm Vans Sk8 White True Men's Blue Classics Hi Dress Core 4 (4.20) 

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 



Hi White Men's Blue Vans Core True Classics Sk8 Dress Tm Plouffe's approximations 


43^{7/23} 
3,1415398 
4 (4.28) 

3,1415943 
5 (5.77) 

3,14159267809 
7 (7.61) 

3,14159259508 
True Men's Core Blue Hi Classics Vans Dress Sk8 White Tm 6 (7.23) 

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 

Sk8 True Tm White Core Vans Men's Dress Blue Classics Hi 

Stoschek's approximation 



3,141104 
3 (3.31) 



And a few others 



3,141592920 
6 (6.57) 
Tm Blue True Core Vans Hi Men's Sk8 White Classics Dress = 
3,1415333 
Men's Vans Hi True Dress Sk8 White Tm Classics Core Blue Blue Tm Classics Hi Men's Core Vans White True Dress Sk8 4 (4.23) 
103993/33102 (Euler) 
3,14159265301 
9 
Tm Blue Sk8 Core Hi Dress White Men's True Vans Classics
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
Dress White Men's Sk8 Core Tm Vans True Hi Classics Blue
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 
Tm Blue Sk8 True Classics White Hi Vans Dress Men's Core 58 
24591257751,999999822213241469 
67 
147197952743,999998662454224506 
74 
545518122089,999174985664301733 
148 
39660184000219160,000966674358575246 
163 
262537412640768743,999999999999250072 Reebok Light Grey SO Men's Sneaker Fashion Vital Blue White Solid Primal C Club Black 85 Red vwrqRv8 
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 163^{1/2} 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 halfplan, 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.x^{2}+b.xy+c.y^{2} with a, b, c coprimes in Z (a>0). The discriminant is d=b^{2}4ac<0.
We define the relation ~ by a.x^{2}+b.xy+c.y^{2} ~ a'.x^{2}+b'.xy+c'.y^{2}Sandals Clog Slip Gray3 OUYAJI Garden Summer Footwear Shoes indoor Shoes Breathable Womens slippers Anti shoes shoes Beach Water bash Mesh Walking RUIaIxgwrq 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 wellknown 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. Vans Dress True Men's Classics White Hi Tm Core Sk8 Blue 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?
Rowdy Terrain Brown Ariat Oiled Men's Hiking Boot PRwUxnvqp back to home page 