mersenneforum.org √2 as a fraction
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2021-11-14, 05:58   #1
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

3F016 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, 40 views) Scan_0044.pdf (1.30 MB, 48 views)

 2021-11-14, 09:28 #2 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 969510 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

3·5·743 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 41·131 Posts If n is a positive integer and d is a divisor of n, the simple continued fraction for $\sqrt{n^{2}+d}$ is n, 2n/d, 2n, 2n/d, 2n, 2n/d,...
 2021-11-15, 07:23 #5 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 24×32×7 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 24×32×7 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

3·5·743 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

3×5×743 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 $\sqrt{n^{2}+d}$ 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 24·32·7 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

17608 Posts
some more data

look.

Cheers

Matt
Attached Files
 Matt types stood on the knowledge.pdf (60.9 KB, 17 views)

Last fiddled with by MattcAnderson on 2021-12-12 at 20:45

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