Commutative Property and Division

Date: 02/21/2002 at 11:12:51
From: Danielle
Subject: Commutative Property *(Division)*

I am trying to prove that the commutative property following certain 
rules will work in all circumstances for division. The rules are:

  - no using Zero
  - the first number has to stay the same
  - you have to use more than two numbers (ex. 26/4/2 = 26/2/4) 

Can you find a way to disprove this? Has anyone ever discovered this? 
Why does it work? 

Thanks- 
Danielle

Date: 02/21/2002 at 11:33:25
From: Doctor Peterson
Subject: Re: Commutative Property *(Division)*

Hi, Danielle.

Well, it's not a genuine commutative property, but I can tell you why 
it works. It's actually sort of interesting.

What's happening is that repeated division is the same as a 
multiplication and a division:

    (26/2)/4 = (26/2) * (1/4) = 26/(2*4)

Since the 2 and 4 (the two divisors) are really being multiplied 
together, you can commute them:

    (26/4)/2 = (26/4) * (1/2) = 26/(4*2)

You can do the same thing to prove this in general:

    (a/b)/c = (a/b)(1/c) = a/(bc) = a/(cb) = (a/c)(1/b) = (a/c)/b

A true commutative property of division would have to say that

    4/2 = 2/4

which of course is not true. And, when you see your equation expressed 
with the parentheses in place, you can see that you are never really 
operating on the 2 and the 4; no property of an operation can be 
applied across a parenthesis.

What you've done is sort of like a magic trick, hiding what's really 
going on. I would not use this as a "property," but would do exactly 
what I did above when I need to work with such a repeated division. I 
also prefer to use parentheses to make it clear what I mean, since 
repeated division is easy to misinterpret.

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

Date: 02/21/2002 at 12:11:35
From: Danielle
Subject: Commutative Property *(Division)*

Has anyone discovered that before? So it can't it or can it be proved 
wrong with those rules? 

Thanks so much for the help -
Danielle

Date: 02/21/2002 at 12:26:48
From: Doctor Peterson
Subject: Re: Commutative Property *(Division)*

Hi, Danielle.

It's been discovered many times before, I'm sure; it just isn't a 
"property" that really deserves a name, so no one pays much attention 
to it. But I showed you how to prove that it is always true for any a, 
b, and c, as long as b and c are non-zero. 

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