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Undefined Fractions

Date: 02/19/2002 at 01:47:35
From: Sandy Morris
Subject: Undefined numbers

My students always ask, Why is a fraction with a denominator of 
zero called "undefined"?

None of my explanations seem to satisfy the kids.  Any ideas?

Date: 02/19/2002 at 02:39:12
From: Doctor Jeremiah
Subject: Re: Undefined numbers

Hi Sandy,

A fraction bar is a division symbol, so you are really dividing the 
top number by the bottom number. But you can't divide by zero, so zero 
cannot be the bottom number.

The reason you can't divide by zero is that no one has bothered to 
define what it means (thats why it undefined). There is no definition, 
so dividing by zero has no answer.

We could define it to mean something, but we would have to be very 
careful not to contradict any other math laws along the way.

If you define division by zero to mean the same thing as division by 
any other number, then you can do things like making the number 2 
equal to the number 1:

   False Proofs, Classic Fallacies   

So we need to come up with a definition of what it means to divide by 
zero that is not the same as what division means for every other 
number. And it would be confusing if division meant two different 
things (especially in algebra where you have a variable that you don't 
know the value of!).

So to avoid the confusion of having division mean two things, we just 
don't have a definition for division by zero at all.

See the Dr. Math FAQ:

   Dividing by 0   

Does that help?

- Doctor Jeremiah, The Math Forum   
Associated Topics:
High School Number Theory
Middle School Fractions

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