2a/5 or 2a Divided by 5?

Date: 12/13/2001 at 14:18:49
From: Jessie6
Subject: Math question or algebra expressions 3-2

Hi, 

In my 8th grade class we are studying algebra. Why, when writing a 
division problem, do you write 2a over 5 instead of 2a divided by 5?

Date: 12/13/2001 at 23:00:03
From: Doctor Peterson
Subject: Re: Math question or algebra expressions 3-2

Hi, Jessie.

This is a good question! I just answered someone else about why you 
divide the numerator by the denominator to convert a fraction to a 
decimal, and referred to this answer in our archives:

   Dividing Fractions to find Decimals
   http://mathforum.org/dr.math/problems/erin.03.12.01.html   

The answer to your question is really the same: a fraction is the same 
thing as a division, so we use the fraction bar, in either the slanted 
form "/" or the horizontal form "___", as a division sign. (In 
particular, we write "6 divided by 3" as "6/3" in our e-mail, and 
computer programming languages always use it that way. Yet I don't 
know that I've ever heard or seen this explained.

Let's look at my example. If I have 6 apples and share them among 3 
people, how many will each get? The answer, 2, is the result of 
dividing 6 by 3. But if I look at the question in terms of fractions, 
I can divide each apple in 3 equal parts, and give each person 6 of 
them, one from each apple. Then each person gets 6/3 of an apple; he 
can then put them back together to form 2 whole apples, if he feels 
like it. So "6/3" means the same thing as dividing 6 by 3.

You can also notice that dividing by 3 is the same as multiplying by 
1/3; so I can write 6 divided by 3 as 6 * 1/3, which written as a 
fraction is 6/3 (before simplifying). Maybe that says it more clearly.

In abstract situations, like 2a/5, it's less obvious what a fraction 
means; but then, everything is less obvious when you get into algebra. 
What we are doing is taking what you are used to doing with actual 
numbers, and just indicating the operations we are doing, without 
being able to actually do them until we evaluate the expression for 
particular numbers. This is really just what we do when we write a 
fraction; one dictionary definition I have seen, which I really like, 
says that a fraction is "an indicated division." So, though you never 
knew it before, when you work with fractions you are really doing a 
sort of algebra, manipulating an expression that represents a division 
you haven't actually done. When you simplify a fraction by dividing 
the numerator and denominator by the same number, you're really using 
the fact that ax divided by ay gives the same result as x divided by y 
to make an equivalent division.

So in algebra, we think of all divisions as fractions, since there's 
no real difference.

Now, I have explained why you CAN write a division as a fraction; but 
why do we PREFER to do so, and particularly to use the horizontal 
fraction bar?

The fraction form, in my mind, is much clearer and easier to read, 
especially in complicated expressions. If I write

    ((x - (2x - 3)/(3x - 2))/(3x + 4)

and its fractional equivalent

        2x - 3
    x - ------
        3x - 2
    ----------
      3x + 4

it is easier to see the relation between the various parts, and how it 
might be simplified, when you see it in the latter form. Notice that I 
didn't have to use parentheses at all; that's because the bar serves 
the double purpose of indicating division and of separating the 
divisor from the dividend. In fact, this usage derives in part from an 
old use of the bar (called a vinculum) to do what we usually do with 
parentheses today, by putting a bar above any group of symbols that 
are to be kept together:

    ___________________
        ______   ______   ______
    x - 2x - 3 / 3x - 2 / 3x + 4

In particular, this lets us avoid the question (never completely 
settled, despite what textbooks tell you) whether

    ab/cd

means

    ab
    --
    cd

or

         b
    a * --- * d
         c

When you write with horizontal fraction bars, you never have to decide 
whether parentheses are necessary in order to say what you mean. And 
anything that lets us say just what we mean without ambiguity is a 
good thing. Much of mathematics is a matter of clear communication.

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