Are These Properties of Real Numbers?

Date: 04/22/2002 at 21:49:38
From: Chris Olsowsky
Subject: Are these properties of real numbers?

We (at college) have been discussing properties of real numbers and 
have come up with a list that we feel are properties, but Internet 
searches and searches through math books at the library and math 
journals have turned up nothing.  Are we just not looking in the 
right place?  The "properties" in question are:

 (a*b)/a = b
 a - (-b)= a + b
 -a/-b = a/b
 a + (-b) = a - b
 a^b * a^c = a^(b+c)

They all seem so intuitive we feel they must be properties.

Date: 04/22/2002 at 22:59:44
From: Doctor Peterson
Subject: Re: Are these properties of real numbers?

Hi, Chris.

These are all true, but the first three are not basic enough to be 
considered fundamental properties and given names, since they can be 
easily demonstrated using more basic properties:

    (a*b)/a = (b*a)/a = b*(a/a) = b*1 = b

    a - (-b) = a + -(-b) = a + b

    (-a)/(-b) = (-1*a)/(-1*b) = (-1)/(-1) * a/b = a/b

    a + (-b) = a - b  can be considered a definition of subtraction

    a^b * a^c = a^(b+c) is called the addition property of exponents
                   (I think! Maybe it's the multiplication property)

I suspect the reason you don't find the last in common lists of 
properties is only that it is not used in more general concepts (like 
groups, rings, and fields) in abstract algebra. I notice, for 
example, that Dr. Ian mentions it as a property here

   Properties of Exponents
   http://mathforum.org/library/drmath/view/57293.html 

but doesn't give it a name. Why? Because mathematicians have named the 
properties that they use in varied contexts and need to refer to often 
- such as the commutative property, which holds for addition and 
multiplication of numbers but not for multiplication of matrices. None 
of the familiar mathematical objects have three levels of operations 
(addition, multiplication, and exponentiation) so as to need this 
_kind_ of property; it is unique to exponents, so we just use it 
rather than name it.

So perhaps that's the real issue here; it's not so much that only the 
most basic properties are called properties, as I said at first, but 
that only the most generally useful properties are named. Textbook 
authors often do have to name all the little properties, but after 
high school those names get forgotten!

All of your "properties" are certainly very basic and should seem 
intuitive to anyone who works with algebra, so at least practically 
speaking, they can be thought of as properties.

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