The Joy of Fractions

Date: 05/18/2002 at 17:23:36
From: Nicole
Subject: fractions

How come fractions are called fractions?
How come they can be translated into decimals?
Why do they hurt your brain when you first learn them?

Date: 05/18/2002 at 18:45:55
From: Doctor Sarah
Subject: Re: fractions

Hi Nicole - thanks for writing to Dr. Math.

According to Steven Schwartzman's book, _The Words of Mathematics, An 
Etymological Dictionary of Mathematical Terms Used in English_:

  The word 'fraction' comes from Latin 'fractus' which is the past
  participle of 'frangere' which means "to break." 

  A fraction is literally a piece broken off something.  
  In 16th-century English mathematics books, fractions were 
  sometimes referred to as "broken numbers."  

For conversions, see the Dr. Math FAQ:

   Fractions, Decimals, Percentages 
   http://mathforum.org/dr.math/faq/faq.fractions.html 

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

Date: 05/18/2002 at 18:49:29
From: Doctor Ian
Subject: Re: fractions

Hi Nicole,

The word 'fracture' means to break into pieces, and 'fraction' 
comes from the same root.  When you break something into pieces, 
each piece is a 'fraction' of the whole. 

The reason a fraction can be translated into a decimal is that a 
fraction is really just a division that you haven't done yet.  So 
when you see something like 3/4, it's just a way of saying "the 
number that you'd get when you divide 3 by 4".  That is, 3/4 is 
just another name for 0.75, which is just another name for 
75/100, and so on. 

Why would you want to have such a notation?  Well, in many cases 
it turns out that you would divide by something now only to end up 
multiplying by by the same thing later.  By putting off doing the 
multiplications and divisions, you can sometimes avoid doing 
them.  (This is one of the few times in life that procrastination 
can actually pay off.)

For example, you might end up translating a story problem into an 
equation like 

  3   6   12   5   11
  - * - * -- * - * -- = ? 
  5   9   11   2    3

Now, you could go ahead and multiply all the terms in the 
numerator and denominator to get

  3   6   12   5   11   11880
  - * - * -- * - * -- = -----
  5   9   11   2    3    2970

and then you could divide those to get 4!  But that's a lot of 
work, even with a calculator.  It would be much easier to just 
rearrange the terms so that you can see which ones cancel each 
other out:

                            \   \       \
  3   6   12   5   11       3 * 5 * 6 * 11 * 12
  - * - * -- * - * -- = ----------------------
  5   9   11   2    3   2 * 3 * 5 * 9 * 11
                            \   \        \


                        6 * 12
                      = ------
                        2 *  9
                            
And by breaking the remaining terms into factors, we can do the 
same kind of thing again:

                        \   \   \
                        2 * 3 * 3 * 4
                      = -------------
                        2 * 3 * 3
                        \   \   \

                      = 4

So this is one of the things that makes fractions really useful!

As for making your brain hurt, maybe you can learn to think of 
that feeling the way that body builders learn to think of muscle 
soreness: as a sign of growth.  When a body builder lifts a lot 
of weights, his muscles hurt the next day, but that's partly 
because they're growing, so the next time he tries to lift the 
same weights, it will be easier.  Maybe it's the same thing with 
brains? 

I hope this helps.  Write back if you'd like to talk about this 
some more, or if you have any other questions. 

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