Adding and Multiplying FractionsDate: 05/31/2001 at 22:18:29 From: Warren Soward Subject: Fractions WHY do I have to find a common denominator to add fractions? WHY don't I have to find a common denominator to multiply fractions? This is not dealt with in lists of algebraic axioms. I think the answer is that the usual rules for adding and multiplying fractions are notational. They don't have to do with fundamental mathematical principles. Thanks for your help. Date: 06/01/2001 at 08:56:36 From: Doctor Peterson Subject: Re: Fractions Hi, Warren. I suppose in a sense you could say this is just "notational," but only because fractions are essentially a notation. If we only used decimals to indicate fractions, we would have different methods. (The Babylonians did essentially this, and the Egyptians had a more restricted notation for fractions that made addition more complex.) But as long as you express a rational number as a fraction (and even if the fraction is hidden in decimal notation), the concept of a common denominator is fundamental in working with fractions. I'm not sure why you would expect to find anything explicitly about common denominators among algebraic axioms; it is a technique for calculation (or algebraic manipulation), not an axiom in itself. But the technique is certainly founded on axioms. I can write out the technique for adding fractions algebraically: a c ad bc ad + bc --- + --- = ---- + ---- = ------- b d bd bd bd The first step depends on the fact that a/b = (ad)/(bd) which can be traced back to axioms; and the second is based on a/c + b/c = (a+b)/c which is essentially the distributive property. The fact that we HAVE to use a common denominator simply reflects the fact that this particular sequence of operations is the one that gives the desired result, a single fraction. I searched our archives to see what we've said about the reason for common denominators (a common question); here are a couple of my explanations that might be of interest to you: When to Add or Multiply Denominators? http://mathforum.org/dr.math/problems/laurie.04.02.01.html Dividing Fractions http://mathforum.org/dr.math/problems/alex03.16.99.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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