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Old 2021-11-14, 05:58   #1
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

2·503 Posts
Default √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
File Type: pdf aproximation of square root of two.pdf (139.8 KB, 38 views)
File Type: pdf Scan_0044.pdf (1.30 MB, 47 views)
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Old 2021-11-14, 09:28   #2
Batalov
 
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Mar 2008
Phi(4,2^7658614+1)/2

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@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)
...
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Old 2021-11-14, 12:59   #3
xilman
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Quote:
Originally Posted by MattcAnderson View Post
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.
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Old 2021-11-14, 18:33   #4
Dr Sardonicus
 
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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,...
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Old 2021-11-15, 07:23   #5
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

17568 Posts
Thumbs up

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
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Old 2021-11-15, 07:30   #6
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

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Talking

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 + ...)).
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Old 2021-11-15, 09:00   #7
xilman
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Quote:
Originally Posted by MattcAnderson View Post
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.
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Old 2021-11-15, 09:03   #8
xilman
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Quote:
Originally Posted by Dr Sardonicus View Post
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.
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Old 2021-12-05, 03:51   #9
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

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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
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Old 2021-12-12, 20:44   #10
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

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Thumbs up some more data

look.

Cheers

Matt
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File Type: pdf 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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