- K B I - Know Best Inference

Rational approximations (approssimazioni razionali)

Tavola numerica controllata

Dott. p.i. Kofler Massimo

Socio ordinario S.I.S. – Società Italiana di Statistica

Socio ordinario U.M.I. – Unione Matematica Italiana

Pigreco - (3.1415926535897932384626...)

3/1, 13/4, 16/5, 19/6, 22/7, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 355/113, 52.163/16.604, 52.518/16.717, 52.873/16.830, 53.228/16.943, 53.583/17.056, 53.938/17.169, 54.293/17.282, 54.648/17.395, 55.003/17.508, 55.358/17.621, 55.713/17.734, 56.068/17.847, 56.423/17.960, 56.778/18.073, 57.133/18.186, 57.488/18.299, 57.843/18.412,

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e - (2.718281828459045235360287...)

3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1.071/394, 1.264/465, 1.457/536, 1.264/465, 1.457/536, 2.721/1.001, 12.341/4.540, 15.062/5.541, 17.783/6.542, 20.504/7.543, 23.225/8.544, 25.946/9.545,

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SQRT 2 - Radice di 2 - (1.4142135623730950488...)

1/1, 3/2, 4/ 3, 7/5, 17/12, 24/17 , 41/29, 99/70, 140/99 , 239/169, 577 /408, 816/577, 1.393/985, 3.363/2.378, 4.756/ 3.363, 8.119/5.741, 19.601/13.860, 27.720/ 19.601, 47.321/33.461, 114.243/80.782, 161.564/ 114.243, 275.807/195.025, 665.857/470.832, 941.664/ 665.857, 1.607.561/1.136.689, 3.880.899 /2744210, 5.488.420/3.880.899,

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SQRT 3 - Radice di 3 - (1.7320508075688772935...)

2/1, 5/3, 7 /4, 12/7, 19/11, 26 /15, 45/26, 71/41, 97 /56, 168/97, 265/153, 362 /209, 627/362 , 989/571,1.351/780, 2.340/1.351, 3.691/2.131, 5.042 /2.911, 8.733/5.042, 13.775/7.953, 18.817 /10.864,32.592/18.817, 51.409/29.681, 70.226 /40.545, 121.635/70.226 , 191.861/110.771, 262.087/ 151.316, 453.948/262.087,

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SQRT 5 - Radice di 5 - (2.2360679774997896964...)

2/1, 5/2, 7/3, 9/4, 20/ 9, 29/13, 38/17, 123/55, 161/72, 360/ 161, 521/233, 682/305, 2.207/987, 2.889/1.292, 6.460/2.889, 9.349/4.181, 12.238/5.473, 39.603/17.711, 51.841/23.184, 115.920/51.841 , 167.761/75.025, 219.602/98.209, 710.647/317.811, 930.249/416.020, 2.080.100/ 930.249, 3.010.349/1.346.269,

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SQRT 6 - Radice di 6 - (2.44948974278318...)

2/1, 5/2, 12/ 5, 17/7, 22/9, 49 /20, 120/49, 169/69, 218/89, 485/198, 1.188/485, 1.673/683, 2.158/881, 4.801/1.960, 11.760/ 4.801, 16.561/6.761, 21.362/8.721, 47.525 /19.402, 116.412/47.525, 163.937/66.927, 211.462/86.329, 470.449 /192.060, 1.152.360/470.449 , 1.622.809/662.509, 2.093.258/854.569,

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SQRT 7 - Radice di 7 - (2.6457513110645905905)

3/1, 5/2, 8/3, 21/ 8, 29/11, 37/14, 45/17, 82/31, 127/48, 336/ 127, 463/175, 590/223, 717/271, 1.307/494, 2.024 /765, 5.355/2.024, 7.379/2.789, 9.403/3.554, 11.427/4.319, 20.830/7.873, 32.257/12.192, 85.344/ 32.257, 117.601/44.449, 149.858/56.641, 182.115/68.833, 331.973/125.474, 514.088 /194.307, 1.360.149/514.088,

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SQRT 10 - Radice di 10 - (3.16227766016838)

3/1, 13/4, 16/5, 19/6, 60/ 19, 79/25, 98/31, 117/37, 487/154, 604/191, 721/228, 2.280/721, 3.001/949, 3.722/1.177, 4.443/1.405, 18.493/5.848, 22.936/7.253, 27.379/8.658, 86.580/27.379,

Nota: la frazione 86.580/27.379 non migliora ma uguaglia la precedente precisione ottenuta dalla 27.379/8.658

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SQRT 11 - Radice di 11 - (3.3166247903554)

3/1, 7/2, 10/3, 33/ 10, 43/13, 53/16, 63/19, 136/41, 199/60, 660/199 , 859/259, 1.058/319, 1.257/379, 2.713/818, 3.970/1.197, 13.167/ 3.970, 17.137/5.167, 21.107/6.364, 25.077/7.561, 79.201/23.880,

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SQRT 13 - Radice di 13 - (3.605551275463989)

4/1, 7/2, 11/3, 18/5, 83/23, 101/28, 119/33, 137/38, 256/71, 393/109, 649/180, 2.340/649 , 2.989/829, 3.638/1.009, 4.287/1.189, 4.936/1.369, 9.223/2.558, 14.159/3.927, 23.382/6.485, 84.305/23.382,

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SQRT 17 - Radice di 17 - (4.12310562561766)

4/1, 17/4, 20/5, 21/5, 25/6, 29/7, 33/8, 136/33, 169/41, 202/49, 235/57, 268/65, 1.373/333, 1.641/398, 1.919/463, 2.177 /528, 8.976/2.177 , 11.153/2.705, 13.330/3.233, 15.507/3.761, 17.684/4.289,

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SQRT 19 - Radice di 19 - (4.35889894354067)

4/1, 17/4, 22/5, 26/6, 35/8, 48/11, 61/14, 109/25, 170/39, 741/170 , 911/209, 1.081/248, 1.251/287, 1.421/326, 4.433/1.017, 3.012/691, 11.878/2.725, 16.311/3.742, 20.744/4.759, 37.055/8.501, 57.799/13.620, 251.940/57.799,

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SQRT 23 - Radice di 23 - (4.795831523312719...)

5/1, 19/4, 24/5, 115/24, 139/29, 163/34, 187/39, 211/44, 235/49, 681/142, 916/191, 1.151/240, 5.520/1.151, 6.671/1.391, 7.822/1.631, 8.973/1.871, 10.124/2.111, 11.275/2.351, 32.674/6.813, 43.949/9.164,

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SQRT 29 - Radice di 29 - (5.385164807134504...)

5/1, 11/2, 16/3, 27/5, 43/8, 70/13, 447/83, 517/96, 587/109, 657/122, 727/135, 1.524/283, 2.251/418, 3.775/701, 6.026/1.119, 9.801/1.820, 52.780/9.801,

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SQRT 31 - Radice di 31 - (5.567764362830022...)

6/1, 11/2, 28/5, 39/7, 128/23, 167/30, 206/37, 451/81, 657/118, 663/155, 1.520/273, 8.463/1.520 , 9.983/1.793, 11.503/2.066, 13.023/2.339, 14.543/2.612, 16.063/2.885, 17.583/3.158,

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SQRT 37 - Radice di 37 - (6.082762530298219...)

6/1, 43/7, 39/7, 49/8, 55/9, 61/10, 67/11, 73/12, 444/73, 517/85, 590/97, 663/109, 736/121, 809/133, 882/145, 6.247/1.027, 7.129/1.172, 8.011/1.317, 8.893/1.462, 9.775/1.607, 10.657/1.752, 64.824/10.657,

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SQRT 41 - Radice di 41 - (6.403124237432848...)

6/1, 13/2, 19/3, 32/5, 237/37, 269/42, 301/47, 333/52, 365/57, 397/62, 826/129, 1.223/191, 2.049/320, 13.120/ 2.049, 15.169/2.369,

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SQRT 43 - Radice di 43 - (6.557438524302...)

7/1, 13/2, 33/5, 46/7, 59/9, 223/34, 282/43, 341/52, 400/61, 1.141/174, 1.541/235, 1.941/296, 3.482/531, 22.833/ 3.482, 26.315/4.013, 29.797/4.544, 33.279/5.075, 36.761/5.606, 40.423/6.137, 43.725/6.668,

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SQRT 47 - Radice di 47 - (6.85565460040104...)

7/1, 14/2, 27/4, 34/5, 41/6, 48/7, 329/48, 377/55, 425/62, 473/69, 521/76, 569/83, 617/90, 665/97, 2.612/3.277, 3.942/575, 4.607/672, 31.584/ 4.607, 40.423/6.137, 43.725/6.668,

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SQRT 53 - Radice di 53 - (7.280109889280518...)

7/1, 15/2, 22/3, 29/4, 51/7, 131/18, 182/25, 1.507/207, 1.689/232, 1.871/257, 2.053/282, 2.235/307, 2.417/332, 2.599/357, 5.380/739, 7.979/1.096, 10.578/1.453, 18.557/2.549, 47.692/6.551, 66.249/9.100,

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SQRT 59 - Radice di 59 - (7.681145747868608...)

8/1, 15/2, 23/3, 100/13, 123/16, 146/19, 169/22, 361/47, 530 /69, 4.071/530 , 4.601/599, 5.131/668, 5.661/737, 6.191/806, 6.721/875, 7.251/944, 7.781/1.013, 8.311/1.082, 16.092/2.095, 24.403/3.177,

SQRT 61 - Radice di 61 - (7.810249675906654...)

8/1, 23/3, 31/4, 39/5, 86/11, 125/16, 164/21, 289/37, 453/58, 1.070/137, 1.523/195, 4.116/527, 5.639/722, 18.440/2361, 24.079/3.083, 29.718/3.805,

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SQRT 67 - Radice di 67 - (8.18535277187245...)

8/1, 23/3, 33/4, 41/5, 90/11, 131/16, 221/27, 1.015/124, 1.236/151, 1.457/178, 1.678/205, 1.899/232, 3.577/437, 5.476/669, 9.053/1.106,

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SQRT 71 - Radice di 71 - (8.426149773176358...)

8/1, 17/2, 42/5, 59/7, 278/33, 337/40, 396/47, 455/54, 514/61, 969/115, 1.483/176, 1.997/237, 3.840/413, 29.323/ 3.840, 32.803/3.893, 36.283/4.306, 39.763/4.719, 43.243/5.132,

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SQRT 73 - Radice di 73 - (8.544003745317531...)

9/1, 17/2, 60/7, 77/9, 94/11, 299/35, 393/46, 487/57, 581/68, 1.068/125, 10.193/1.193, 11.261/1.318, 12.329/1.443, 13.397/1.568, 14.465/1.693, 15.533/1.818, 16.601/1.943, 17.669/2.068, 18.737/2.193, 36.406/4.261,

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SQRT 79 - Radice di 79 - (8.888194417315588...)

9/1, 44/5, 53/6, 62/7, 71/8, 80/9, 711/80, 791/89, 871/98, 951/107, 1.031/116, 1.111/125, 1.191/134, 1.271/143, 1.351/152, 1.431/161, 7.075/796, 8.506/957 9.937/1.118, 11.368/1.279, 12.799/1.440,

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SQRT 83 - Radice di 83 - (9.110433579144298...)

9/1, 46/5, 55/6, 64/7, 73/8, 82/9, 747/82, 829/91, 911/100, 993/109, 1.075/118, 1.157/127, 1.239/136, 1.321/145, 1.403/154, 1.485/163, 7.507/824, 8.992/987 10.477/1.150, 11.962/1.313, 13.447/1.476,

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SQRT 89 - Radice di 89 - (9.433981132056603...)

9/1, 19/2, 47/5, 66/7, 151/16, 217/23, 283/30, 500/53, 5.217/553, 5.717/606, 6.217/659, 7.717/712, 7.217/765, 7.717/818, 8.217/871, 8.717/924, 9.217/977, 18.934/2.007, 47.085/4.991, 66.019/6.998,

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SQRT 97 - Radice di 97 - (9.848857801796104...)

10/1, 39/4, 49/5, 59/6, 69/7, 128/13, 197/20, 325/33, 522/53, 847/86, 3.063/311, 3.910/397, 4.757/483, 5.064/569,

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