Mathematical Meaning of Undefined

Date: 12/19/2002 at 13:33:31
From: Melissa
Subject: Mathematical meaning of the word "undefined"

My math teacher wants us to find out the meaning of the word 
"undefined" in mathematical terms.

Date: 12/19/2002 at 14:18:47
From: Doctor Ian
Subject: Re: Mathematical meaning of the word "undefined"

Hi Melissa,

Something is 'undefined' when there is no good way to define it. The
classic example is division by zero. There is no good definition for 

  n / 0 = ?

because any definition you try to put in there creates problems with
other things in math. In this case, the problem is just that 

  n / 0 = ?         

and

  n = 0 * ?

are just two ways of saying the same thing. So if you decide that you
can multiply something by zero and get something _other_ than zero,
then you have to change the definition of zero, which would be a huge
problem. So we say that division by zero is undefined. 

Another example would be the logarithm of a negative number. 
Logarithms are defined this way:

                                          a
  log x = a        means that        x = b
     b

For example, 

                                          4
  log 16 = 4       means that       16 = 2
     2

Now, suppose x is negative instead of positive:

                                          ?
  log (-16) = ?    means that      -16 = 2
     2

There is just no way to choose a value for '?' that makes sense: 2^x 
is positive when x is positive; and it's positive when x is negative; 
and it's positive when x is zero. So we've run out of possibilities.

The thing you have to keep in mind, though, is that something that is
undefined today may end up with a definition tomorrow. The classic
example of that is the square root of a negative number:

   Complex Numbers
   http://mathforum.org/library/drmath/view/53877.html 

The square root of -1 used to be undefined. Now it's defined as the
'imaginary' number 'i', which lets us deal with the square root of any
negative number:

  sqrt(-4) = sqrt(4 * -1)

           = sqrt(4) * sqrt(-1)

           = 2 * i

Imaginary numbers also let us deal with logarithms of negative 
numbers: 

   The Log of a Negative Number
   http://mathforum.org/library/drmath/view/55564.html 

However, they haven't helped us find a way to define division by zero.
But that's not to say that someday, someone won't come up with one!  

Does this help? 

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