First 100 Best Rational Approximations for Square Root of 2
Computed Sun 11-23-2003 17:15:50 LaFarr Stuart
1 / 1
3 / 2
7 / 5
17 / 12
41 / 29
99 / 70
239 / 169
577 / 408
1,393 / 985
3,363 / 2,378
8,119 / 5,741
19,601 / 13,860
47,321 / 33,461
114,243 / 80,782
275,807 / 195,025
665,857 / 470,832
1,607,521 / 1,136,689
3,880,899 / 2,744,210
9,369,319 / 6,625,109
22,619,537 / 15,994,428
54,608,393 / 38,613,965
131,836,323 / 93,222,358
318,281,039 / 225,058,681
768,398,401 / 543,339,720
1,855,077,841 / 1,311,738,121
4,478,554,083 / 3,166,815,962
10,812,186,007 / 7,645,370,045
26,102,926,097 / 18,457,556,052
63,018,038,201 / 44,560,482,149
152,139,002,499 / 107,578,520,350
367,296,043,199 / 259,717,522,849
886,731,088,897 / 627,013,566,048
2,140,758,220,993 / 1,513,744,654,945
5,168,247,530,883 / 3,654,502,875,938
12,477,253,282,759 / 8,822,750,406,821
30,122,754,096,401 / 21,300,003,689,580
72,722,761,475,561 / 51,422,757,785,981
175,568,277,047,523 / 124,145,519,261,542
423,859,315,570,607 / 299,713,796,309,065
1,023,286,908,188,737 / 723,573,111,879,672
2,470,433,131,948,081 / 1,746,860,020,068,409
5,964,153,172,084,899 / 4,217,293,152,016,490
14,398,739,476,117,879 / 10,181,446,324,101,389
34,761,632,124,320,657 / 24,580,185,800,219,268
83,922,003,724,759,193 / 59,341,817,924,539,925
202,605,639,573,839,043 / 143,263,821,649,299,118
489,133,282,872,437,279 / 345,869,461,223,138,161
1,180,872,205,318,713,601 / 835,002,744,095,575,440
2,850,877,693,509,864,481 / 2,015,874,949,414,289,041
6,882,627,592,338,442,563 / 4,866,752,642,924,153,522
16,616,132,878,186,749,607 / 11,749,380,235,262,596,085
40,114,893,348,711,941,777 / 28,365,513,113,449,345,692
96,845,919,575,610,633,161 / 68,480,406,462,161,287,469
233,806,732,499,933,208,099 / 165,326,326,037,771,920,630
564,459,384,575,477,049,359 / 399,133,058,537,705,128,729
1,362,725,501,650,887,306,817 / 963,592,443,113,182,178,088
3,289,910,387,877,251,662,993 / 2,326,317,944,764,069,484,905
7,942,546,277,405,390,632,803 / 5,616,228,332,641,321,147,898
19,175,002,942,688,032,928,599 / 13,558,774,610,046,711,780,701
46,292,552,162,781,456,490,001 / 32,733,777,552,734,744,709,300
111,760,107,268,250,945,908,601 / 79,026,329,715,516,201,199,301
269,812,766,699,283,348,307,203 / 190,786,436,983,767,147,107,902
651,385,640,666,817,642,523,007 / 460,599,203,683,050,495,415,105
1,572,584,048,032,918,633,353,217 / 1,111,984,844,349,868,137,938,112
3,796,553,736,732,654,909,229,441 / 2,684,568,892,382,786,771,291,329
9,165,691,521,498,228,451,812,099 / 6,481,122,629,115,441,680,520,770
22,127,936,779,729,111,812,853,639 / 15,646,814,150,613,670,132,332,869
53,421,565,080,956,452,077,519,377 / 37,774,750,930,342,781,945,186,508
128,971,066,941,642,015,967,892,393 / 91,196,316,011,299,234,022,705,885
311,363,698,964,240,484,013,304,163 / 220,167,382,952,941,249,990,598,278
751,698,464,870,122,983,994,500,719 / 531,531,081,917,181,734,003,902,441
1,814,760,628,704,486,452,002,305,601 / 1,283,229,546,787,304,717,998,403,160
4,381,219,722,279,095,887,999,111,921 / 3,097,990,175,491,791,170,000,708,761
10,577,200,073,262,678,228,000,529,443 / 7,479,209,897,770,887,057,999,820,682
25,535,619,868,804,452,344,000,170,807 / 18,056,409,971,033,565,286,000,350,125
61,648,439,810,871,582,916,000,871,057 / 43,592,029,839,838,017,630,000,520,932
148,832,499,490,547,618,176,001,912,921 / 105,240,469,650,709,600,546,001,391,989
359,313,438,791,966,819,268,004,696,899 / 254,072,969,141,257,218,722,003,304,910
867,459,377,074,481,256,712,011,306,719 / 613,386,407,933,224,037,990,008,001,809
2,094,232,192,940,929,332,692,027,310,337 / 1,480,845,785,007,705,294,702,019,308,528
5,055,923,762,956,339,922,096,065,927,393 / 3,575,077,977,948,634,627,394,046,618,865
12,206,079,718,853,609,176,884,159,165,123 / 8,631,001,740,904,974,549,490,112,546,258
29,468,083,200,663,558,275,864,384,257,639 / 20,837,081,459,758,583,726,374,271,711,381
71,142,246,120,180,725,728,612,927,680,401 / 50,305,164,660,422,142,002,238,655,969,020
171,752,575,441,025,009,733,090,239,618,441 / 121,447,410,780,602,867,730,851,583,649,421
414,647,397,002,230,745,194,793,406,917,283 / 293,199,986,221,627,877,463,941,823,267,862
1,001,047,369,445,486,500,122,677,053,453,007 / 707,847,383,223,858,622,658,735,230,185,145
2,416,742,135,893,203,745,440,147,513,823,297 / 1,708,894,752,669,345,122,781,412,283,638,152
5,834,531,641,231,893,991,002,972,081,099,601 / 4,125,636,888,562,548,868,221,559,797,461,449
14,085,805,418,356,991,727,446,091,676,022,499 / 9,960,168,529,794,442,859,224,531,878,561,050
34,006,142,477,945,877,445,895,155,433,144,599 / 24,045,973,948,151,434,586,670,623,554,583,549
82,098,090,374,248,746,619,236,402,542,311,697 / 58,052,116,426,097,312,032,565,778,987,728,148
198,202,323,226,443,370,684,367,960,517,767,993 / 140,150,206,800,346,058,651,802,181,530,039,845
478,502,736,827,135,487,987,972,323,577,847,683 / 338,352,530,026,789,429,336,170,142,047,807,838
1,155,207,796,880,714,346,660,312,607,673,463,359 / 816,855,266,853,924,917,324,142,465,625,655,521
2,788,918,330,588,564,181,308,597,538,924,774,401 / 1,972,063,063,734,639,263,984,455,073,299,118,880
6,733,044,458,057,842,709,277,507,685,523,012,161 / 4,760,981,394,323,203,445,293,052,612,223,893,281
16,255,007,246,704,249,599,863,612,909,970,798,723 / 11,494,025,852,381,046,154,570,560,297,746,905,442
39,243,058,951,466,341,909,004,733,505,464,609,607 / 27,749,033,099,085,295,754,434,173,207,717,704,165
94,741,125,149,636,933,417,873,079,920,900,017,937 / 66,992,092,050,551,637,663,438,906,713,182,313,772
Yes, the numbers do get big!
Try squaring any of the above as fractions. I think you will
be surprised at how good they are.
I used Square Root of 2, in preference to Pi, for a couple reasons.
