Square Roots

Date: 07/17/98 at 12:54:28
From: Ted
Subject: Problem

Dear Dr.Math,

My dad taught me how to do decimals, but now I am confused - what are 
square roots and what do they look like? Can you tell me?

Date: 07/17/98 at 16:46:49
From: Doctor Pat
Subject: Re: Problem

Ted, 

A square root is a simple idea, but some of them are hard to express 
in decimal form. First, to answer the question about what they are.

If you multiply a number by itself, the answer is called the square of 
the number. For example, since 5x5 = 25 we call 25 the SQUARE of 5.  
In the same way 36 is the square of 6, etc. This probably came about 
because the area of a square with sides of 6 units is 36 square units.
  
Now to work the other way, to find out what number times itself 
= 49 for instance, is called finding the square root. SOOOOOOOO the 
SQUARE ROOT of 25 is 5 because 5 SQUARED is 25; and the square root 
of 49 is 7, etc.   

So far so good, but sometimes it doesn't come out so nicely. For 
instance, what is the square root of 5? Well, it must be MORE than 2 
because 2 is the square root of 4, and it must be LESS than 3 because 
3 is the square root of 9, and 5 is between 4 and 9, so its square 
root must be between 2 and 3.  

We can guess 2.5 and see if that works. It's easy to test since we 
know we are right if 2.5 x 2.5 is equal to 5. Yikes, I got 6.25 which 
is too much, so we'd better guess a little lower.

I could continue guessing and getting closer and closer to the answer, 
but I could NEVER get it exactly because it does not end and it does 
not repeat. Numbers like this are called irrational (which means NOT a 
fraction) because there is NO fraction of whole numbers that is 
exactly the same). A CLOSE answer for the square root of 5 is 
2.236067977.... , my calculator ran out of digits, but that is still 
not EXACTLY right.
  
I think there is a lot of new language here and you may need some time 
to think about all this. After you have spent some time on the square 
roots that have whole number answers, you can search the Dr. Math 
archives for other information on irrational numbers; there are a lot 
of them (actually an infinite number).  

Good luck,

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