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### Properties of Subtraction

```
Date: 04/08/97 at 17:22:35
From: kim Sebastian
Subject: Properties of Subtraction

1) Is subtraction commutative? How can this be justified?
2) Is subtraction associative? and how can this be justified?

Thanks,
Mrs. Sebastian
```

```
Date: 04/09/97 at 15:01:08
From: Doctor Mike
Subject: Re: Properties of Subtraction

Dear Mrs. Sebastian,

This is an interesting question, and the answer is subtle, but
there IS an answer.  It's sort of a "yes and no" answer.  Here goes.

1. Subtraction is NOT commutative.  If you evaluate 3-2 and 2-3
you get +1 and -1, respectively.  NOT the same thing.

2. Subtraction is NOT associative.  If you evaluate (3-2)-1 and
3-(2-1) you get 0 and 2, respectively.  NOT the same thing.

3. However, **addition** is commutative for all numbers, even
including negative numbers.  An example related to (1.) is:

3 + -2  =  -2 + 3  =  +1

4. Also, **addition** is associative for all numbers, even the
negative numbers.  An example related to (2.) above is:

(3 + -2) + -1  =  3 + (-2 + -1)  =  0

To see this better, evaluate what is inside parentheses, getting

(  1  ) + -1  =  3 + (  -3  )  =  0

I hope this helps.  Feel free to write back if you're not yet
completely comfortable with this.  I'm glad you want to get it.

-Doctor Mike,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Subtraction

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