Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Square Root Theory

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


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?


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
Associated Topics:
High School Sequences, Series

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.