Heron's Method for Finding Square Roots by HandDate: 12/18/2001 at 03:06:54 From: Alvin Folks Subject: Square Root How do you get the square root of a number manually without a table or calculator? Example: sqrt.root of 62, 588, and 46. Take the sqrt. root of 58; the answer is 7.6157. I got the 7 and the decimal point. I do not know how you get the 6 and 1, etc. Thank you, Rolleralvin Date: 12/18/2001 at 08:46:59 From: Doctor Paul Subject: Re: Square Root I suppose the easiest way to do this is to guess and check. For example, to compute sqrt(62): You know the answer is somewhere between 7 and 8 (closer to 8) so we guess that the answer is 7.8 Now square 7.8 and see if 7.8 is too big or too small. 7.8^2 = 60.84 so 7.8 is too small. Try 7.9 7.9^2 = 62.41 so 7.9 is too big. Try 7.88 7.88^2 = 60.0944 so 7.88 is too big try 7.87 7.87^2 = 61.9369 so 7.87 is too small. Try 7.874. 7.874^2 = 61.999876 which is very close to 62. But of course it's not sqrt(62) so we can repeat the above process basically forever if we so desire. Now, using a calculator, I compute: sqrt(62) = 7.874007874011811019685034448... This verifies that the above method was indeed leading us in the right direction. A much better method was known to Heron (of Alexandria), who is believed to have lived between 150 BC and 250 AD. He noted that if you pick a_1 as a random guess for a possible value of sqrt(n) then a_2 = [a_1 + (n/a_1)]/2 will be a better approximation. Similarly, a_3 = [a_2 + (n/a_2)]/2 will be a better approximation than a_2. These approximations form a sequence of numbers {a_i} that converge very rapidly to sqrt(n). For example: If we desire to compute sqrt(62) and we take a_1 = 7 we compute: a_2 = [7 + (62/7)]/2 = 111/14 = 7.928571428571428571428571428... then a_3 = [111/14 + (62/(111/14))]/2 = 24473/3108 = 7.874195624195624195624195624... continuing, we obtain: a_4 = 1197826897/152124168 = 7.874007876250143238252583244 which is accurate to eight decimal places. As before, we can continue the process indefinitely if we want. I hope this helps. Please write back if you'd like to talk about this some more. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ Date: 12/19/2001 at 23:30:27 From: Alvin Folks Subject: Square Root I would like to thank you Doctor Paul for your help and speedy reply It more than satisfied my understanding of finding the square root of a number by hand. Thank you, Rolleralvin2 |
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