 mersenneforum.org √2 as a fraction
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MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

2×3×191 Posts √2 as a fraction

Hi all,

One of the scanned pages is upside down, but you can print it out if you want.

Regards,
Matt
Attached Files aproximation of square root of two.pdf (139.8 KB, 59 views) Scan_0044.pdf (1.30 MB, 68 views)   2021-11-14, 09:28 #2 Batalov   "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3×19×173 Posts @Matt - Here's an easy construction for square roots approximations of any arbitrary numbers. No need for matrices. Use Newton's method for solving f(x)=x2-a=0. You know f'(x). It is 2x. xnew = x - f(x)/f'(x) = x - (x^2-a)/(2x) = (x^2+a)/2x ...or (x+a/x)/2 as frequently taught in schools For $$\sqrt 2$$: use a=2 and apply this repeatedly: Code: a=2; x=1; x=(x+a/x)/2 3/2 x=(x+a/x)/2 17/12 x=(x+a/x)/2 577/408 x=(x+a/x)/2 665857/470832 x=(x+a/x)/2 886731088897/627013566048 x=(x+a/x)/2 1572584048032918633353217/1111984844349868137938112 For $$\sqrt 10$$: use a=10 and apply this repeatedly: Code: a=10; x=3; x=(x+a/x)/2 19/6 x=(x+a/x)/2 721/228 x=(x+a/x)/2 1039681/328776 x=(x+a/x)/2 2161873163521/683644320912 x=(x+a/x)/2 9347391150304592810234881/2955904621546382351702304 ... Now, try the same to get fast approximation of a cubic root of 2: xnew = x - f(x)/f'(x) = x - (x3-a)/(3x2) = (2x^3+a)/(3x^2) ...   2021-11-14, 12:59   #3
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

11,383 Posts Quote:
 Originally Posted by MattcAnderson Hi all, One of the scanned pages is upside down, but you can print it out if you want. Regards, Matt
Continued fraction for sqrt 2 is 1;2.   2021-11-14, 18:33 #4 Dr Sardonicus   Feb 2017 Nowhere 10110110010102 Posts If n is a positive integer and d is a divisor of n, the simple continued fraction for is n, 2n/d, 2n, 2n/d, 2n, 2n/d,...   2021-11-15, 07:23 #5 MattcAnderson   "Matthew Anderson" Dec 2010 Oregon, USA 2×3×191 Posts Thanks Batalov and others, Some of us are 'into' math and computers. I appreciate the effort. AS a next step. Look at a fraction for square root of 3. I have not memorized that the square root of 3 is shown to be sqrt(3) = 1.732050808. minus some error due to the fact that the square root of 3 is an irrational number. I am not ashamed to share this with you all. Matt   2021-11-15, 07:30 #6 MattcAnderson   "Matthew Anderson" Dec 2010 Oregon, USA 2×3×191 Posts I did a little copying of the definition of continued fraction from Wikipedia. Thank you for showing that to me. Regards, Matt I assume that the infinite continued fraction for the square root of 2 is 1+1/(2 + 1/(2 + ...)). Attached Thumbnails Last fiddled with by MattcAnderson on 2021-11-15 at 07:31 Reason: fixed continued fraction   2021-11-15, 09:00   #7
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

1138310 Posts Quote:
 Originally Posted by MattcAnderson I did a little copying of the definition of continued fraction from Wikipedia. Thank you for showing that to me. Regards, Matt I assume that the infinite continued fraction for the square root of 2 is 1+1/(2 + 1/(2 + ...)).
You assume correctly.   2021-11-15, 09:03   #8
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

11,383 Posts Quote:
 Originally Posted by Dr Sardonicus If n is a positive integer and d is a divisor of n, the simple continued fraction for is n, 2n/d, 2n, 2n/d, 2n, 2n/d,...
That should read n; 2n/d, 2n, ... in conventional notation.

The ; is the continued fraction equivalent to the decimal point.   2021-12-05, 03:51 #9 MattcAnderson   "Matthew Anderson" Dec 2010 Oregon, USA 2×3×191 Posts Thank you for that typing and effort @Batalov I know that requires some effort and learning and typing. As a lifetime member of The Mathematics Association of America, I just thought I would share. Again thanks. For what it's worth, *griz* Last fiddled with by MattcAnderson on 2021-12-05 at 03:52 Reason: added the word member   2021-12-12, 20:44   #10
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

2·3·191 Posts some more data

look.

Cheers

Matt
Attached Files Matt types stood on the knowledge.pdf (60.9 KB, 34 views)

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