20211114, 05:58  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
3×367 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 
20211114, 09:28  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
265D_{16} 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)=x^{2}a=0. You know f'(x). It is 2x. x_{new} = x  f(x)/f'(x) = x  (x^2a)/(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 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 ... x_{new} = x  f(x)/f'(x) = x  (x^{3}a)/(3x^{2}) = (2x^3+a)/(3x^2) ... 
20211114, 12:59  #3 
Bamboozled!
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across
2^{2}·3·941 Posts 

20211114, 18:33  #4 
Feb 2017
Nowhere
3·17·113 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,... 
20211115, 07:23  #5 
"Matthew Anderson"
Dec 2010
Oregon, USA
3×367 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 
20211115, 07:30  #6 
"Matthew Anderson"
Dec 2010
Oregon, USA
3×367 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 + ...)). Last fiddled with by MattcAnderson on 20211115 at 07:31 Reason: fixed continued fraction 
20211115, 09:00  #7 
Bamboozled!
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across
2^{2}·3·941 Posts 

20211115, 09:03  #8 
Bamboozled!
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across
2^{2}·3·941 Posts 

20211205, 03:51  #9 
"Matthew Anderson"
Dec 2010
Oregon, USA
10001001101_{2} 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 20211205 at 03:52 Reason: added the word member 
20211212, 20:44  #10 
"Matthew Anderson"
Dec 2010
Oregon, USA
3×367 Posts 
some more data
look.
Cheers Matt Last fiddled with by MattcAnderson on 20211212 at 20:45 
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