Associated Topics || Dr. Math Home || Search Dr. Math

```
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
are notational. They don't have to do with fundamental mathematical
principles.

```

```
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
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

--- + --- = ---- + ---- = -------
b     d     bd     bd      bd

The first step depends on the fact that

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/
```
Associated Topics:
Middle School Fractions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search