Decimals Versus Fractions: Pluses and Minuses

Date: 08/19/2011 at 12:22:34
From: Terri
Subject: Why do we sometimes use decimals instead of fractions?

I have just started teaching middle school math. My students' questions
routinely take the form of, "Why do they write it that way"? My answer is
usually to equate "math shorthand" with the texting lingo of kids -- just
a quicker way to write something.

When asked what they thought the hardest part of math is, many answered
fractions and decimals. So when we talked about how a decimal represents
the same thing as a fraction, they asked, "Why use decimals?" I really
don't know that answer, unless it too is simply another shorter, quicker
way to express quantities.

My thought is that .023 is easier, quicker, and neater than writing out 23
over 1,000. However, I don't want to pass that along if there is more to
it than that. Is that why, or is there another reason?

I guess I'm searching for a deeper meaning of decimals and their use.

Date: 08/19/2011 at 14:47:52
From: Doctor Ian
Subject: Re: Why do we sometimes use decimals instead of fractions?


Hi Terri,

> I have just started teaching middle school math.

Congratulations!

> My students' questions routinely take the form of, "Why do they write
> it that way"? My answer is usually to equate "math shorthand" with the
> texting lingo of kids -- just a quicker way to write something.

That's a nice way to tie it to something they know.

But there's also more to it than just writing things more quickly. And
sometimes the fraction is actually quicker to write.

For example, I'd rather write this ...

  1/7 

... than this ...
    ______
  0.142857

... wouldn't you? 

> When asked what they thought the hardest part of math is, many answered
> fractions and decimals. So when we talked about how a decimal
> represents the same thing as a fraction, they asked, "Why use
> decimals?" I really don't know that answer, unless it too is simply
> another shorter, quicker way to express quantities.

One way to get them to answer this for themselves is to ask them to do
some basic operations using the two different notations. What they will
find, I think, is that for the most part, decimals make it easier to add
things:

   1/4  +  1/5 = 5/20 + 4/20
               = 9/20

   0.25 + 0.20 = 0.45 

But in some sense, that's just because decimals already have common
denominators (or are nearly there if you just tack some zeros on the end):

     0.12   + 0.3456 
   = 0.1200 + 0.3456 
   = 0.4656

So if you have fractions with the same denominator, there's really no
advantage. In fact, sometimes the fractions are easier, e.g.,

        1/7 + 2/7      = 3/7
  
   0.142857 + 0.285714 = ... I don't even want to add those!

Using decimals can also make comparisons easier:

   3/5 < 4/7 ?         Uh, maybe.

   0.6 < 0.57 ?        Clearly not!

On the other hand, with fractions, multiplication is easy, while with
decimals it's more work:

    3/4 * 2/3   = 6/12
                = 1/2

   0.75 * 0.667 = 0.50025

And fractions have the nice property that you can often cancel numerators
and denominators, so you don't even have to multiply:

       /   /   /
   2   3   4   5   2   1
   - * - * - * - = - = -      
   3   4   5   6   6   3
   /   /   /

Compare that to the decimal alternative, 

  0.667 * 0.75 * 0.8 * 0.833 = 0.3333666

Note, too, that in all these cases, the result from the fraction is exact,
while the result with the decimal has to be rounded off somewhere... which
may actually have adverse consequences in a later calculation.

The invention of calculators skewed things in favor of decimals, since
they're usually easier to enter and to show on a simple display. But the
tide is turning back as systems like Mathematica, which can do exact
calculations, become more common.

So in some sense, the reason we have both fractions and decimals is the
same reason we have different kinds of hammers, and different kinds of
saws, and different kinds of screwdrivers.... Any notation is a kind of
tool, and not every tool is the best for every job.

And that's a good lesson for your students to learn, because it's really
quite rare in mathematics for there to be one best way to solve a problem.
Part of the art of mathematics is in deciding which approach is likely to
be best, before jumping in with the first one that occurs to you, or the
one you habitually use.

Does this help? 

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

Date: 08/19/2011 at 14:59:06
From: Terri
Subject: Thank you (Why do we sometimes use decimals instead of fractions?)

Yes, that helped tremendously! (That's why I love this site!)

I really like comparing the various representations to saws, hammers, and
drills.... different tools for different jobs. That will help them so
much.

My goal is to help these kiddos truly understand numbers, not just
memorize steps and procedures. I think your analogy and examples will help
with that a lot.