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/
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