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### 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/
```
Associated Topics:
Elementary Division
Middle School Division

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