Is Integer Division Associative?

Date: 05/14/2002 at 11:13:43
From: Cody
Subject: Is division of integers associative?

I need to know if the division of integers has the associative 
property. 

Thanks for your help.

Date: 05/14/2002 at 12:39:16
From: Doctor Peterson
Subject: Re: Is division of integers associative?

Hi, Cody.

If you know what an associative property is, you can figure out for 
yourself whether division has the associative property.

Associativity means that it doesn't matter in which order you 
associate (parenthesize) two of the same operation:

    2*(3*4) = 2*12 = 24

    (2*3)*4 = 6*4 = 24

That illustrates the fact that multiplication is associative (though 
of course one example is not enough to _prove_ that it always is). If 
the results had not been equal, you would have a counterexample, an 
example that shows it is _not_ always true.

So try some division problems of this sort and see if you get the 
same answer both ways:

    2/(3/4) = ?

    (2/3)/4 = ?

Of course, since you are asking about division of integers, you have 
to decide what division means when it doesn't come out even -- do you 
say it isn't defined, so that you can only choose an example where 
both divisions give integer results; or do you use the whole-number 
quotient and ignore the remainder (which is what computers mean by 
integer division); or do you just need the numbers you start with to 
be integers, and don't care whether the answer is? In any case, you 
can look for a counterexample, and if you don't find one, think about 
whether you see reason to believe it is always associative.

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