Square Root Theory

Date: 11/16/2001 at 00:11:54
From: Timothy
Subject: Square root theory

Hello,

When I enter any positive number in the calculator, then take the 
square root of that number, then take the square root of that number, 
and keep pressing the square root button over and over, I eventually 
get to number 1. 

When I start with a fraction like 0.1, I also end up with a number 1.

Can you explain to me why this happens?

Sincerely,
Tim

Date: 11/16/2001 at 11:50:11
From: Doctor Rob
Subject: Re: Square root theory

Thanks for writing to Ask Dr. Math, Tim.

You are generating a sequence of numbers x[0], x[1], x[2], ...,
using the equation

   x[n] = sqrt(x[n-1]),  n = 1, 2, ...

starting with a value x[0] which is nonzero.  In your example with
x[0] = 0.1, you get

   x[0] = 0.1,
   x[1] = sqrt(x[0]) = 0.3162277660,
   x[2] = sqrt(x[1]) = 0.5623413252,
   x[3] = sqrt(x[2]) = 0.7498942093,
   ...
   x[32] = sqrt(x[31]) = 0.99999999

If these values of
x[n] converge to a limit L, that limit must satisfy the equation

   L = sqrt(L).

The only solutions to this equation are L = 0 and L = 1. If you 
started with x[0] = 0, you would converge to the limit L = 0, but
since you started with a nonzero x[0], you must converge to the
limit L = 1. You can see that by showing that if 0 < x[n-1] < 1, then

   0 < x[n-1] < sqrt(x[n-1]) = x[n] < 1,

and if 1 < x[n-1], then

   1 < x[n] = sqrt(x[n-1]) < x[n-1],

so that each x[n] is closer to 1 than its predecessor.

Now the actual values of x[n] never reach 1, but the values a 
calculator shows you are just approximations to x[n], rounded off to a 
certain accuracy. Those approximations do eventually reach 1.

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