How Can the Product of Two Radicals Be Finite?

Date: 06/27/2006 at 15:54:47
From: Keleigh
Subject: (no subject)

How is it possible that sqrt(5) * sqrt(20) equals exactly 10?

  sqrt(5) equals an infinite number (~2.236067977...)

  sqrt(20) equals an infinite number (~4.472135955...)

  sqrt(5) * sqrt(20) equals 10

I can't comprehend this.  Is it possible that two infinite numbers 
eventually have the correct combination to make 10?  Or is this just 
a man-made decision that the answer is approximately 10?  Please help!

Date: 06/28/2006 at 09:29:36
From: Doctor Ian
Subject: Re: (no subject)

Hi Keleigh,
 
The first thing to do is stop using the word "infinite" to describe
what's going on here.  That by itself might go a long way towards
clearing up your confusion.

A number itself, e.g., 
    ___
  \/ 2

isn't infinite at all--it's somewhere between 1.4 and 1.5, because

  1.4^2     = 1.96
     ___
  (\/ 2 )^2 = 2

  1.5^2     = 2.25

And it's somewhere between 1.41 and 1.42, because

  1.41^2    = 1.9881
     ___
  (\/ 2 )^2 = 2

  1.42^2    = 2.0164

And we can keep narrowing this down as much as we want, limited only
by our patience.  But clearly, a number between 1.41 and 1.42 can't be
said to be "infinite" in any meaningful sense of that word, because we
can hardly expect to find an infinite number bounded by two finite
numbers.  

What we _can_ say is this:  The number 
    ___
  \/ 2

can't be expressed exactly as a fraction whose denominator is a power
of 10.  That is, there are no integers k and n such that

    ___     k
  \/ 2  = ----
          10^n
 
As I said, we can generate a series of approximations to this, 

  14  141  1414  14142  141421  1414213
  --, ---, ----, -----, ------, -------, ...
  10  100  1000  10000  100000  1000000

but none of them will give us the exact value.  Does this mean the
number doesn't exist?  Let's construct an isosceles right triangle,
where the legs have length 1:


     A
     . .
     .   .
     .     .   ?
   1 .       .
     .         .
     .           .
     .             .
     C. .. . . . . . B
            1

What is the length of the hypotenuse?  We can agree that there is some
number that describes the length, right?  The Pythagorean Theorem
tells us that if we call that number x, it will be true that

  x^2 = 1^2 + 1^2

  x^2 = 2

So there is _some_ number that, when multiplied by itself, gives a
product of 2.  The question is, what do we call it?  

Its exact name is just 
        ___
  x = \/ 2
 
When we try to translate that into a different representation, which
we find convenient for dealing with numbers in lots of situations, we
find that it doesn't work very well.  But this is a mismatch between
the number and the representation.  It says nothing about the number
itself.  

In the cases where they "combine perfectly", note that it always
reduces to a root being multiplied by itself, e.g., 
      ___     ____
    \/ 5  * \/ 20
      ___     _______
  = \/ 5  * \/ 5 * 4
      ___     ___     ___
  = \/ 5  * \/ 5  * \/ 4
      ___     ___     
  = \/ 5  * \/ 5  * 2
   
Now, at this point, if we realize that 
    ___
  \/ 5

is just a name we give to 

  the number--whatever it is!--that, when multiplied by 
  itself, gives a product of 5

then there should be no mystery that we end up with an integer for a
result:

       ___     ___     
  = (\/ 5  * \/ 5 ) * 2
  
  = 5 * 2

  = 10

So there's really no "combination" going on.  It's more like
"rejoining".  

Does this help? 

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

Date: 06/28/2006 at 21:13:53
From: Keleigh
Subject: Thank you ((no subject))

Thank you for taking the time to answer my question so thoroughly.  I
think I need to eliminate "infinite" from my vocabulary so I don't get
confused again!  You provided several explanations and scenarios and
you MADE it make sense.  I really appreciate it and it definitely 
helped!  You're great :)