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Square Roots of Numbers between 0 and 1


Date: 07/08/99 at 11:00:32
From: eric s adair
Subject: Square Roots of Numbers between 0 and 1

Why do square roots always come closer to the number one? If I take 
the square root of 5 and keep taking the square root of the square 
root of the square root etc., it eventually equals one.

Take the square root of .5 and keep taking the square root of the 
square root etc., it also eventually equals one.

One number gets smaller until it equals one and one number gets bigger 
until it equals one.

Ouch.


Date: 07/08/99 at 13:02:47
From: Doctor Rick
Subject: Re: Square Roots of Numbers between 0 and 1

Hi, Eric.

I don't personally find this painful, but it is interesting. I will 
show you a graphical way to think about what you are doing.

This graph shows the functions y = x (the straight line) and
y = sqrt(x) (the curve).

  y                                           /
  |                                         /
  |                                       /
  |                                     /
  |                                   /
  |                                 /
  |                               /
  |                             / ...............         * 
  |                           /..:         *    :
  |                         /.:  *              :
  |                       *                     :
  |                 *.../ :                     :
  |              * :  /   :                     :
  |           *...../     :                     :
  |        *  :   /       :                     :
  |      *    : /         :                     :
  |     *     /           :                     :
  |   *     / :           :                     :
  |  *    /   :           :                     :
  | *   /     :           :                     :
  |*  /       :           :                     :
  | /         :           :                     :
  +-----------------------+---------------------------------- x
             0.5          1                     2

Your operation of taking the square root of a number repeatedly is 
depicted graphically by the dotted lines. Start at 2 and take its 
square root: the height of the dotted line at x = 2 is y = sqrt(2). 
Set x to this value, that is, x = y. This corresponds to moving 
horizontally until you hit the line y = x. Then take the square root 
of x again: move vertically until you hit the curve, y = sqrt(x).

Repeat this operation and you can see that the horizontal and vertical 
lines bounce around between the straight line and the curve, getting 
closer and closer to 1. If you start at x = 0.5, the same thing 
happens, but this time you move right towards 1.

You can see that the reason for this convergence is that

1. y = sqrt(x) is above y = x for x between 0 and 1, and it is below
   y = x for x greater than 1.

2. The slope of y = sqrt(x) is positive, that is, sqrt(x) increases as 
   x increases.

You always converge on 1 because that is where the lines y = x and
y = sqrt(x) cross.

Are you surprised that the square root of a number less than 1 is 
greater than the number? The graph shows that it is. You can also see 
it this way:

If you multiply a number by a number less than 1, the result is 
smaller than the original number. For instance, 8 * 0.5 = 4, which is 
less than 8.

If you square a number less than 1, you are multiplying it by a number 
less than 1 (namely, itself), so the result is less than the original 
number. For instance, 0.5 * 0.5 = 0.25, which is less than 0.5.

The square root of a number is the number whose square is the first 
number. If the number is less than 1, then the original number (which 
is the square root squared) must be less than the square root. Thus, 
the square root is greater than the original number. For instance, the 
square root of 0.25 (which is 0.5) is greater than 0.25.

Does either of these approaches to the question shed more light on it 
for you? Maybe they raise more questions - if so, feel free to ask 
them!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Middle School Square Roots

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