Heron's Method for Finding Square Roots by Hand

Date: 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