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0 As the Denominator

Date: 01/19/2002 at 17:20:43
From: Kyle Stout
Subject: 0 as the denominator

Can zero over zero equal anything?  My brother, who is a freshman in 
college, says it can, but my math teacher says it can't. At first I 
agreed with my teacher, but my brother showed me this whacked-out 
equation that made it look as if zero over zero can equal anything.  
Even so, I told him that anything divided by zero is undefined and 
can't be done, but he stayed with his idea. My math teacher 
recommended I write you this question.

Thank you.

Date: 01/19/2002 at 17:58:24
From: Doctor Ian
Subject: Re: 0 as the denominator

Hi Kyle,

A lot depends on what you mean by 'zero over zero'.  

If you're trying to divide something by the constant value zero, 
you're right - it can't be done. The operation is undefined. 

The 'whacked-out equation' that your brother was showing you probably 
had to do with limits, that is, with predictions about what the ratio 
of two functions would be if you looked at their values at infinity -  
which, of course, you can't.  

Let's say you have a function like f(x) = 1/x.  As you get farther 
from the origin, the value of the function gets closer to zero:

  f(1)             = 1
  f(2)             = 0.5
  f(10)            = 0.1
  f(100)           = 0.01
  f(1,000,000,000) = 0.000000001

In fact, although the function never actually reaches zero, we can 
make it get as close as we want by going far enough out. So we say 
that 'the function goes to zero as x goes to infinity'. But that's a 
lot to write, so we abbreviate it

  limit  f(x) = 0         "The limit of f(x) as x approaches 
  x->inf                   infinity is zero."

Now, suppose we have a second function, g(x) = 2/x.  This also goes to 
zero as x goes to infinity.  So we have

  limit  g(x) = 0

Here's the fun part. What if we divide one limit by the other? On the 
one hand, this looks like

  limit  f(x) 
  x->inf         0
  ------------ = -          Undefined?
  limit  g(x)    0

But that's somewhat misleading, because of course, neither the 
numerator nor the denominator ever really _gets_ to zero.  

In fact, for any value of x that we choose, it's pretty clear that 
g(x) will be twice f(x), so the ratio is really 1/2.

If you think about this for a while, it should become clear that you 
can, indeed, come up with two functions, the ratios of whose limits
would be any number you want.  (Would you like the ratio to be 12.3?  
Okay, let's let f(x) = 12.3/x, and g(x) = 1/x.) 

So I would say this: division by _zero_ is undefined, but division by 
a _limit_ that goes to zero is indeterminate (that is, the value you 
get depends on the limits you start with). As long as you're careful 
to make the distinction, there shouldn't be any confusion.

Does this help? 

- Doctor Ian, The Math Forum   
Associated Topics:
High School Number Theory

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