Zero in the Denominator

Date: 8/14/96 at 10:31:35
From: Anonymous
Subject: A Problem Regarding Zero in the Denominator

What is the value of 4/0 * 0/4?

Thanks,
Sg

Date: 8/15/96 at 0:55:25
From: Doctor Mike
Subject: A Problem Regarding Zero in the 
Denominator

Hello Sg,
  
The correct answer to this question is that this expression does not 
have a value. The reason for this is the 4/0 part.  Why? 

Think of a typical fraction with a numerator bigger than its 
denominator, like 20/5. This is equivalent to 4, because 5 goes into 
20 a total of 4 times. 22/5 is a little larger than that, namely 4.4 , 
since 5 goes into 20 a total of 4 times, with 2 left over. 

I imagine that this kind of thing is very familiar to you. For 5 to 
"go into" a number a certain number of times just means that if you 
take 5 away from that number again and again you will eventually 
arrive at zero or get a negative result.
  
Now think about what happens if the numerator stays the same but the 
denominator is zero, 20/0 . How many times does zero go into 20? 
Equivalently, how many times do you have to take 0 away from 20 so 
that the result will get down to zero or negative? 

It should be clear that you would never get down to zero, since no
matter how many times you subtract 0 from 20, you still get 20.  
Usually, a mathematician will say that an expression like 20/0 or 4/0 
does not have a value, or has an infinite value. Certainly it is not 
equal to any ordinary number.
   
Now for the other 0/4 part. That is just an ordinary fraction that is 
equal to zero. 

Think of a pie cut into 4 equal parts. The fraction 1/4 represents one 
piece, 2/4 represents 2 pieces or half of the pie, 4/4 represents all 
of the pie, and 0/4 represents none of the pie.  In an expression like 
"X * 0/4" where X is any number, the value would be X times zero, or 
zero.  The reason that this is not true for what you asked about is 
that X has to be a number, and 4/0 is just not a number.
  
I hope this helps.    

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/