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/ |
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