Is Division Sharing?

Date: 09/08/2001 at 18:58:00
From: Emily
Subject: Fraction

Hi, Dr Math,

Why when I do a fraction division (e.g. 1/2 divided by 1/2 = 1) is the 
answer bigger than the first fraction in question (i.e. 1/2)? 
Shouldn't division mean sharing, so logically the answer is smaller 
than the first fraction?

Thanks, Emily

Date: 09/09/2001 at 00:15:48
From: Doctor Ian
Subject: Re: Fraction

Hi Emily,

It's not strictly true that 'division means sharing'. In fact, 
division is just another way of looking at multiplication. For 
example, all of the following are just different ways of saying the 
same thing:

  3 times 4 equals 12. 
  4 times 3 equals 12.
  12 divided by 3 equals 4.
  12 divided by 4 equals 3. 

It's sort of like looking at this picture,

     /\
    /  \
   /    \
  +------+
  |      |
  |      |
  +------+

and knowing that both of the following descriptions

  The triangle is above the square. 

  The square is below the triangle. 

are equally true, since they are really just different ways of saying 
the same thing. 

In other words, whenever it is true that

  a * b = c

it must also be true that

  c / a = b

and

  c / b = a

(unless either a or b is zero, in which case all bets are off).  
That's what we _mean_ by division. 

Now, let's think about what this means for division by 1/2.  If

  c / (1/2) = a

then it must be true that

          c = a * (1/2)

which means that a must be _larger_ than c.  

Thinking about division as 'sharing' is one way that teachers try to 
make the concept simpler for students to understand. But it's 
important to realize that these simplifications usually lead you in 
the wrong direction once you start looking at things more carefully.  

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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