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Square roots

Date: 27 Nov 94 12:11:58 EST 
From: Kathy Charner
Subject: For Sydney

Dear Dr.Math,

What are square roots and how do they work?

P.S. Thanks!

Date: Sun, 27 Nov 1994 21:59:20 -0500 (EST)
From: Dr. Sydney
Subject: Re: For Sydney

Dear Sam,

Hey there!  How are you doing?  I'm so glad you wrote to Dr. Math --
you asked a GREAT question.  I'm very impressed that you are already
thinking about square roots--you are indeed a budding mathematician. 

Before I answer your question about square roots, I think it would be best
to first talk about square numbers.  Do you know what a square number is?

A square number is what you get when you multiply a whole number by 
itself. For instance, since 3 x 3 = 9, 9 is a square number.  Likewise, 
16 is a square number since 4 x 4 = 16.  Can you think of some other 
square numbers?  

Taking the square root of a number is the opposite operation (in math lingo
it is called the "inverse") of squaring a number.  What that means is that to
find the square root of a number, you ask yourself what number times itself
will yield that number.  So, say you wanted to know what the square root of
9 was.  You would ask yourself what number multiplied times itself yields
9?  Then you would say, aha!... 3 x 3 = 9, so 3 is the square root of 9.
And, you would be right.  

But there is also another square root... -3.  -3 x -3 = 9 so -3 is also a 
square root of nine.  

I don't know how much you know about negative numbers, but briefly, you 
can think of a negative number as the opposite of a positive number.  If I give 
you 3 apples, I have given you +3 apples.  You have given me -3 apples.  
Another way to think of negative numbers is that if you add a negative number 
and its "opposite" (the corresponding positive number), you get 0.  
So, 3 + (-3) = 0.  

One of the most fundamental properties of real numbers is that a negative 
number times a negative number yields a positive number.  That is why
 -3 x -3 = 9.

Anyway, back to what we were talking about with square roots... every
positive number has two square roots -- one is the negative of the other.
We usually only worry about the positive square root, though.  So, if someone
were to ask you what the square root of 9 is, you could say just 3 (and not
worry about the -3).  

What are the square roots of 16?  

If you move a step beyond this, though, and think about the square roots of
numbers that are not square, then things get a little messy, but not to
fear, the same ideas will still work.  When we try and find the square root
of a number that is not square, we can either approximate it using a
calculator (or doing calculations by hand using trial and error) or use the
square root sign and leave the square root sign in the reduced form of the
number.  Have you seen the square root sign before?  If you haven't, maybe
one of your parents or your teacher could show it to you.  Unfortunately, I
can't write it on the computer--there is no key for square root!  Say for
instance you were wondering how you would write the square root of 3.  
The symbol for this is a 3 under the square root sign. That indicates that
when that number is squared it yields 3.  

Square roots are lots of fun, and you can do a lot of neat stuff with them.
Here's something to think about:  what if you want to take the square root
of a negative number?  This is a very difficult concept, so if it doesn't
make sense for a while, don't sweat it... it took mathematicians a very long
time (in fact it was not until the 1500's and 1600's that mathematicians
even began to get a grasp on this concept) to figure it out.

I hope some of this helps.  Please do write back if you have any questions
about what I said or if you have any other math questions.  These are very
difficult concepts, so don't worry if it seems overwhelming.  Math is full
of neat concepts like this that are at first overwhelming but once you learn
more about them they are pretty neat.  Write back soon!

--Sydney "the doctor is IN!!" Foster
Associated Topics:
Elementary Square Roots

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