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### Why Were Fractions Invented?

```Date: 11/26/2007 at 23:16:10
From: Dominique
Subject: Fractions

Why were fractions invented?

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```
Date: 11/27/2007 at 09:20:28
From: Doctor Peterson
Subject: Re: Fractions

Hi, Dominique.

Fractions were invented to provide a way to work with quantities
smaller than one, such as 2/3 (or quantities between two other whole
numbers, such as 3 2/3).  If we had only whole numbers, then the only
way to talk about such quantities would be to use a smaller unit.  In
fact, that's essentially what the Romans did (who weren't very good
mathematicians in general).  They could talk only about whole numbers
of feet; when they wanted to work with parts of feet, they would
change to inches (of which there were, and are, 12 in each foot).
Then what we might call 1/12 foot would be called 1 inch, using a
whole number; 1/4 foot would be called 3 inches.  (Our word "inch"
comes from their word "uncia", which meant 1/12 of anything--the
smaller unit).

But what if your measurement isn't even a whole number of inches?
Then you'd need a still smaller unit to measure it.  If you want to
be very precise, things get more and more complicated.

Fractions are sort of do-it-yourself units.  You decide what size
unit you need in order to measure something, by choosing how many of
them there will be in each of the existing unit; say you want 8
units per foot.  Then the new unit is called an eighth, and rather
than writing its name we just call it 1/8.  Now we can measure any
number of eighths; 3 of them is written as 3/8.  The numerator tells
how many of the unit you have, and the denominator tells what the
unit is.  (The word "numerator" is Latin for "numberer"--how many; the
word "denominator" is Latin for "namer"--what kind of unit.)

This makes it possible to measure all sorts of things without having
to name lots of new units.  Moreover, by having a nice way to write
them, we can add, subtract, multiply, and divide any fractions easily
(compared to what the Romans would have had to do, if they even
bothered to try).  And that's the benefit of fractions.

This kind of fraction was invented hundreds of years after the end of
the Roman empire, by Hindu mathematicians, and our notation (using the
fraction bar) was established by Arabic mathematicians.

There are other solutions to the same problem.  The decimals we use
(which you may or may not have learned about yet) were invented by the
Arabs, and got their current form around 1500 in Europe.  But the
ancient Babylonians had a very similar idea, based on 60 rather than
10, which was used by ancient astronomers--and is still the basis of
our degree measurement of angles, as well as minutes and seconds of
time.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Elementary Fractions
Elementary Math History/Biography
Middle School Fractions
Middle School History/Biography

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