Continued Fraction Approximations for Sqrt(2)

Last Up date: 2017 May 5 on nuMacMini /work/gd/lafarr/math

    Latest updates are are at the top.
  1. 2017 May 4 Better headings.
  2. 2017 April 12 Used \ for too large fractions. They alternate from too samll to too large.
  3. 2003 November 23 Started

First 100 Rational Approximations for Square Root of 2
Computed Sunday 2003 November 11 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.

  1. I think most people are more interested in Pi; that interest may motivate them to try to do Pi.
    If I had done Pi they would probably just look and nod, but to learn one has to do more than just observe.
  2. It is quite remarkable: If you square any of the fractions above the square of the numerator is always just one off from being exactly twice the square of the denominator. Showing how close the approximation is to the square root of 2.
    • It is quite easy to recognize when one number is nearly twice another which makes me prefer sqrt(2) to other Square Roots.
    • If you are not too lazy:Try squaring some of the smaller ones like: 17/12 or 41/29 if you are really ambitious try some bigger ones! Like 1,393/985 most pocket calculators can handle it.
  3. Square Root of 2 has a very simple C.F. which repeats for ever. A repeating sequence occurs for the Square Root of any integer. The repeating sequence often is more than a single digit.
    Examples: I put the repeating part in ( )'s
    Sqrt(2)=1,(2)
    Sqrt(3)=1,(1,2)
    Sqrt(5)=2,(4)
    Sqrt(6)=2,(2,4)
    Sqrt(7)=2,(1,1,1,4)
    Sqrt(10)=3,(6)
    Sqrt(17)=4,(8)
    Sqrt(26)=5,(10)
    • I included the last three above to illustrate the C.F. sequence for sqrt(n**2+1) which is: n,(2n)

    My previous page explain how to find "Best" rational approximations

    E-mail me The phone block links to my BIO page.

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