Adding and Multiplying Fractions

Date: 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/