Square Roots Taken RepeatedlyDate: 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/ |
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