Multiplying a Number by Itself

Date: 02/27/2001 at 15:57:54
From: Justin Fleming
Subject: Square roots

You have adequately explained how you get a square root, and I 
understand that part of it. My question to you is so what? What does 
it mean to multiply a number by itself?  

I can understand the concept of adding or subtracting; that has 
meaning to me. It represents an order that I can understand in my 
daily life activities. But how did someone just say I am going to 
multiply this number by itself and call it a square root?

Date: 02/27/2001 at 22:52:02
From: Doctor Peterson
Subject: Re: Square roots

Hi, Justin.

At first I thought you were saying the wrong thing in your last 
sentence, but I think you mean "If I take a number and multiply it by 
itself, I can call the original number the square root of the new 
number." That's correct, and I suppose I can see why you might think 
it's a little silly to go through all that. But it's really no 
different from what we do with addition and subtraction. I can add 6 
to 3 and get 9; then I can subtract 6 from 9 and get back to my 
original number, 3. Subtraction is how we "undo" addition; and square 
roots are how we "undo" squaring. We don't subtract just to get back 
to a number we started with; rather, we usually subtract to find out 
what we would have had to start with, but didn't know. It's the 
same with square roots: we use them to find numbers we don't already 
have.

You ask what squaring means. Squares arise in many problems; the 
simplest is the one from which it gets its name. If you make a square 
whose sides are all 5 inches long, then its area is 5x5 = 25 square 
inches. We've multiplied one side by the other to find how many 
one-inch squares it takes to cover it. So if we knew how many square 
inches we had (say, what area a gallon of paint covers), we could find 
the side of the square using a square root.

Another example where the square root is important is the Pythagorean 
theorem. This says, for example, that if I have a 3x4 rectangle, I can 
find the square of the length of its diagonal by squaring the 3, 
squaring the 4, and adding the numbers together: 

   3^2 + 4^2 = 9 + 16 = 25. 

Now what is the length of the diagonal? The square root of this, which 
is 5. Without the square root, we couldn't solve this problem.

I hope this helps you see the meaning and the value of the square root 
a little better.

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