Square Roots Taken Repeatedly

Date: 11/21/2002 at 22:43:29
From: John
Subject: Square Roots

Why, when you repeatedly take the square root of a whole number, do 
you end up with 1?

For example, if you take the square root of 10, and repeat taking the 
square root, you will end up with 1.

Thanks.

Date: 11/21/2002 at 23:00:00
From: Doctor Ian
Subject: Re: Square Roots

Hi John,

Actually, you won't get all the way to one.  The only way you can get
_exactly_ one by taking a square root is if you start with 1. That is, 

  sqrt(1) = 1

but 

  sqrt(1 + something) = (1 + something smaller) >  1

However, if you keep taking square roots, a calculator will eventually 
be unable to represent 'something smaller', so it will _look_ as if 
it's telling you that the value is 1. But it's not. It's just running 
out of space to store all the zeros. 

That is, suppose you're taking square roots, and you get to the point
where the next square root is 

  1.0000000000000000000000000000000000000000000000000000000000000001

Most calculators would just throw away the digits at the right and 
give you something like 

  1.000000000000000000000000

And when you take the square root of _this_, you'll just get 1 again,
because the calculator thinks you're taking the square root of 1.  

But suppose a calculator was smart enough to represent the number as

              -64
  1 + 1.0 x 10

or even as

                 6
              -(2 )
  1 + 1.0 x 10

Then you could keep taking square roots for the rest of your life, and
you'd never get all the way to 1.  You'd just keep getting 

  1 + (something r-e-a-l-l-y small)

Does that make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/