Why Were Fractions Invented?

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

Why were fractions invented?

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/