Identifying Algebraic Properties

Date: 12/29/96 at 19:43:19
From: Kenneth L McConnell
Subject: Pre-Algebra Help

Dr. Math,

My daughter is in a pre-algebra class in the eighth grade.  I am 
trying to understand the following problem so I can help her with a 
homework assignment.  Any information you could give me regarding the 
following would be greatly appreciated:

Choose which property is used to get each step of the solution:

	a) Additive Identity of Zero
	b) Associative Property of Addition 
	c) Communitive Property of Addition
	d) Op-Op property
	e) Property of Opposites

	(-3 + 8) + -(-3)

	=(-3 + 8) + 3    Answer__ Why?

	= -3 + (8 + 3)   Answer__ Why?

	= -3 + (3 + 8)   Answer__ Why?

	= (-3 + 3) + 8   Answer__ Why?

	= 0 + 8          Answer__ Why?

	= 8              Answer__ Why?  

My primary request is for a definition of "a" through "e" and how they 
apply to the above.

Thank you,

Ken McConnell

Date: 12/30/96 at 10:23:03
From: Doctor Rob
Subject: Re: Pre-Algebra Help

Glad that you are helping your daughter.  Perhaps you will both learn
something in the process!

Your daughter's textbook should be the source for the definitions of 
the properties you list.  The first three properties are standard, 
while the last two are not, so I can only guess about the meanings of 
the last two.

a) Additive Identity of Zero (STANDARD)

Given any number, if you add zero to it, you get that number back.  In 
equations, for any number a, a + 0 = 0 + a = a.

b) Associative Property of Addition (STANDARD)

Given any three numbers to be added, the result of adding the sum of
the first and second to the third is the same as adding the first to 
the sum of the second and third.  In equations, for any three numbers 
a, b, and c, (a + b) + c = a + (b + c).

c) Commutative Property of Addition (STANDARD)

Given any two numbers, the result of adding the first to the second
is the same as the result of adding the second to the first.  In
equations, for any two numbers a and b, a + b = b + a.

d) Op-Op property  (NON-STANDARD)

Given any number, the opposite of its opposite is the number itself.
In equations, for any number a, -(-a) = a.

e) Property of Opposites (NON-STANDARD)

Given any number, the sum of it and its opposite is zero.  In 
equations, for any number a, a + (-a) = 0.

Now, given this, I think you can figure out which reason goes with
which operation.  Just match the patterns in the properties with the 
patterns in the expressions, and see which ones fit.

By the way, underlying all of the above reasoning is the following
principle:  when two numbers are equal, any expression involving the 
first will remain the same when the second is substituted for it.  For 
example, in the first expression, you find -(-3), and from d) you know 
that -(-3) = 3, so by this substitution principle, you can replace 
-(-3) by 3 without changing the value of the expression.

If we can help you more, please let us know.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/