Dividing by Zero and Dividing Zero by Something

Date: 11/14/2006 at 19:43:11
From: Sabrina
Subject: Zero principle in division

I am trying to understand the difference between dividing by zero and 
dividing zero by something.  One of them is zero and the other is 
"can't be done".  Which is which and why???  Please help!

Date: 11/15/2006 at 10:45:16
From: Doctor Ian
Subject: Re: Zero principle in division

Hi Sabrina,

When we 'divide A by B', we're doing this:

     __
  B ) A

So if we're 'dividing zero by something', it looks like 

     __
  ? ) 0

And if we're 'dividing by zero', it looks like

     __
  0 ) ?

So far, so good?  Now, note that
      
       4           3
     ___         ___
  3 ) 12      4 ) 12      3 x 4 = 12

are three ways of saying exactly the same thing.  In other words, any
division problem can be rewritten as a multiplication problem.  So if
we have something like 

      ?
     __
  5 ) 0

it means the same thing as 

  5 * ? = 0

And the only value that makes that true is 0:

                          0
                         __
  5 * 0 = 0    --->   5 ) 0

So if we divide 0 by anything (except 0, as we'll see in a moment), we
get 0... BECAUSE if we multiply anything by 0, we get 0.  That's just
the same fact, stated two different ways. 

Does that make sense?  (Write back if it doesn't.)

So, what about dividing BY zero?  Well, if we have something like

      ?
     __
  0 ) 5

it has to mean the same thing as

  0 * ? = 5

right?  But there ISN'T ANY POSSIBLE VALUE that can make that true
since anything times 0 will still be 0.  So we say that division BY
zero is 'undefined'.  It doesn't make sense, and we just can't do it. 

Does this help?

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

Date: 11/28/2006 at 19:15:31
From: Sabrina
Subject: Thank you (Zero principle in division)

Thanks so much for clearing that up.  It really helped.  It is still a
bit confusing but your answer explains it all.  I just hope I can
remember that on my next test!!!