Undefined Fractions

Date: 06/10/2005 at 15:13:03
From: Jon
Subject: Difference Between Undefined and No Solution

Hi Dr. Math,

I am having trouble determining the difference between an undefined 
fraction and a fraction with no solution.  Maybe you can help me 
determine the difference.

The thing I find most confusing is that I just can't seem to find in
my textbook or remember whether the statements above mean the fraction
has a denominator of zero or equals zero.

So far I have been thinking about whether the denominator equals zero
in an undefined fraction or possibly in the fraction with no solution,
or if the undefined fraction means that it equals zero or possibly the
no solution one equals zero. 

Hopefully you can help me out on this topic. Thanks, Jon.

Date: 06/10/2005 at 22:55:28
From: Doctor Peterson
Subject: Re: Difference Between Undefined and No Solution

Hi, Jon.

If something is undefined, that means it is not defined.

If something has no solution, that means there is no number that is a
solution.  (That is, we can't solve it.)

So the question is, what do we mean by "defined" and by "solution".

A fraction represents a division, and we can divide by any number
except zero.  Division by zero is not defined.  That is, we can define
the fraction

  6 / 2 = 3

because

  6 = 2 * 3

Division is the inverse of multiplication; it tells you what number
can be multiplied to get the given number.  But if you divided by 0:

  6 / 0 = ?

that would mean

  6 = 0 * ?

There is no number that you could put there; so this division is
undefined.

Now, a fraction doesn't have a solution; only an equation does!  A
solution is a value that makes an equation true.  So if we have an
EQUATION containing a fraction, and you find that the only possible
solution makes the fraction undefined, then it can't really be a
solution, right?  You can't evaluate the two sides of the equation, so
you can't say it's true.  That's how undefined fractions relate to
equations with no solution.

So, the fraction 0/1 is defined (as zero), while the fraction 1/0 is
undefined (has no value).

The equation

   0
  --- = 1
   x

has no solution, because 0 divided by anything is 0, and can't be
equal to 1.  If you multiply both sides by x, you get

  0 = x

but x = 0 can't really be a solution because 0/x on the left is
undefined, so the equation is not true when x is 0.

Does that help?  If not, maybe you can send me some examples of the
kind of problem you are trying to solve.


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