Fractions or Decimals?

Date: 05/18/99 at 21:04:30
From: Alysia LaGambina
Subject: fractions and decimals

Hi,

My name is Alysia. My question is, why do we need both decimals and 
fractions to represent amounts less than one?

Date: 05/19/99 at 11:46:32
From: Doctor Rick
Subject: Re: fractions and decimals

Hi, Alysia. This is an interesting question.

Both decimals and fractions go back a long way. The Babylonians had 
something like our decimals, except their numbers were based on 60 
instead of 10. The Egyptians used fractions, but their fractions all 
had 1 in the numerator (that is, they would if they had written them 
the way we do). Fractions as we know them were used by the Greeks. But 
not always - sometimes the Greeks preferred Babylonian-style 
"hexagesimals," and sometimes they preferred Egyptian-style "unit 
fractions."

If I had to pick either decimals or fractions and never use the other 
again, which would I choose? That would be a hard choice.

Decimals are better than fractions when I need to do a lot of 
calculations. You add and multiply decimals just the same way you do 
whole numbers, except you have to keep track of where the decimal 
point goes.

We have chosen decimals over fractions in designing computers and 
computer languages, and calculators, too. Computers and calculators 
don't understand fractions.

But decimals pose a problem. When you write 1/8 as a decimal, it's 
0.125 - it has more digits than 1/8. Multiplying 24 times 0.125 by 
hand takes more work than multiplying 24 by 1/8. This will happen 
with a lot of fractions.

Now that we have calculators and computers, the number of digits 
doesn't matter so much. We can live with 0.125 instead of 1/8. But it 
gets a lot worse. When you try to write 1/3 as a decimal, you get 
0.33333333333333333333333333333333333333333333333333333333333333333333
3333333333333333333333333333333333333333333333333333333333... I'll 
stop there, but you get the idea. You can't write the decimal 
EXACTLY, because it would go on forever.

This problem shows up occasionally on calculators. You do some 
calculation and you know the answer should be 1, but it comes out as 
0.999999999. That's very close to 1; in fact, if the 9's went on 
forever, it would be exactly the same as 1 - but it doesn't look right 
at all.

For a long time people preferred to work with fractions rather than 
decimals. That's part of the reason for all the strange unit 
conversions we have: 1 quart is 1/4 gallon, 1 inch is 1/12 foot. 
Decimals must have started gaining the upper hand by the time the 
metric system was developed, in the 1790's. With the advent of 
computers, decimals really took over. Fractions are not nearly as 
important now as they once were.

But there is still a place for fractions. In math, we often want to 
keep exact results. Since 1/3 is an exact number but 0.3333 is only 
approximately the same number, we have to write the fraction 1/3 in 
order to keep it exact. I hope your teacher insists on this - it's 
hard to tell whether you really understand what you are doing if you 
just punch numbers into a calculator and write down the decimal 
result. If you're wrong, it's hard to tell what mistake you made. 
Exactness is good.

If I had to choose, I'd have to go along with decimals as long as I 
need to use a calculator. But I would really miss fractions when I 
need to do math by hand (I don't always have a calculator) and when 
I want to do math exactly. I'm glad I don't have to make that choice. 
You'll be glad, too, if you learn to work with both decimals and 
fractions well.

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