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## Main Question or Discussion Point

Hi guys,

I am a 1st year Engineering/Maths student. I have recently started reading "journey through genius"; which so far is great! After reading chapter 3 on Archimedes i was very curious about how he approximated a lower and upper bound for sqrt(3). I looked at the translated proof in my "great western books - vol. 11", however, there is no mention of how he arrived at such figures 265/153 < sqrt(3) < 1351/780.

After searching the net, i realise there is no way to know exactly how he came to these numbers. However, i did come across an interesting article at http://www.mathpages.com/home/kmath038.htm [Broken]

This article suggests starting with (5/3) as the estimate. I found by using 5/3 and the bisecting method?

a_{n + 1} = (1/2)(a_n + (N/a_n)), with (a_0)^2 = (5/3)^2 < 3 = N

the first 3 numbers are

5/3, 26/15, 1351/780, ...

However this method seems to skip the lower bound of 265/153.

In the above mentioned article it states "if we imagine that their first estimate for the square root of 3 was 5/3, perhaps based on the fact that 5^2 = 25 is close to 3(3^2) = 27. From here it isn't hard to see that if x is a bound on the square root of 3, then

(5x+9)/(3x+5) is a closer bound on the opposite side."

The last sentence is not "easy for me to see"; I am wandering if anybody can help me understand how (5x+9)/(3x+5) was derived? It seems related to a_{n + 1} = (1/2)(5/3) + (9/5)?

If you use the article's formula and x_1 = 5/3, you get 26/15; allow x_2 = 26/15, you get 265/153; and finally, allow x_3 = 265/153, you get 1351/780. Interesting results!

Thanks for your help,

Brendan

I am a 1st year Engineering/Maths student. I have recently started reading "journey through genius"; which so far is great! After reading chapter 3 on Archimedes i was very curious about how he approximated a lower and upper bound for sqrt(3). I looked at the translated proof in my "great western books - vol. 11", however, there is no mention of how he arrived at such figures 265/153 < sqrt(3) < 1351/780.

After searching the net, i realise there is no way to know exactly how he came to these numbers. However, i did come across an interesting article at http://www.mathpages.com/home/kmath038.htm [Broken]

This article suggests starting with (5/3) as the estimate. I found by using 5/3 and the bisecting method?

a_{n + 1} = (1/2)(a_n + (N/a_n)), with (a_0)^2 = (5/3)^2 < 3 = N

the first 3 numbers are

5/3, 26/15, 1351/780, ...

However this method seems to skip the lower bound of 265/153.

In the above mentioned article it states "if we imagine that their first estimate for the square root of 3 was 5/3, perhaps based on the fact that 5^2 = 25 is close to 3(3^2) = 27. From here it isn't hard to see that if x is a bound on the square root of 3, then

(5x+9)/(3x+5) is a closer bound on the opposite side."

The last sentence is not "easy for me to see"; I am wandering if anybody can help me understand how (5x+9)/(3x+5) was derived? It seems related to a_{n + 1} = (1/2)(5/3) + (9/5)?

If you use the article's formula and x_1 = 5/3, you get 26/15; allow x_2 = 26/15, you get 265/153; and finally, allow x_3 = 265/153, you get 1351/780. Interesting results!

Thanks for your help,

Brendan

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