Why Does Division Convert a Fraction to a Decimal?

Date: 01/04/2009 at 20:18:24
From: Claudia
Subject: Fractions to decimals

Why do you divide the denominator into the numerator to get a decimal?
 What is the math behind this?

It just seems odd that there is no explanation as to why this
operation transfers a fraction to a base 10 decimal.  I want to
understand WHY this happens and the reasoning behind it.

I am thinking that you are trying to get the denominator as close to
100 as possible so that you can make the fractions equivalent to
decimals.  I am just not sure why division gives the appropriate
decimal to us.

Date: 01/04/2009 at 22:58:53
From: Doctor Peterson
Subject: Re: Fractions to decimals

Hi, Claudia.

A fraction is essentially a division; in algebra we use the fraction
bar as the standard division sign, and for good reason.  Let's take a
look.

First, consider a fraction like 1/4.  In pictures, you can think of
that as a pie divided into 4 equal parts, and we just take one of
them.  Can you see how that is 1 divided by 4?  Just as dividing 12 by
4 would mean dividing 12 objects into 4 equal parts (3 each) and
taking one of them, dividing 1 by 4 means dividing 1 object (our pie)
into 4 equal parts (now pieces, not wholes) and taking one of them.

Now, how about when the numerator is not 1?  Consider 3/4.  We can
think of this as starting with ONE pie, dividing it into 4 pieces, and
taking 3 of them; but we can also think of starting with THREE pies,
dividing EACH of them into 4 pieces (making 12 in all), and taking one
from EACH pie.  The 3 pieces we took are 1/4 of the 12 pieces; we've
divided the 3 pies by 4 (and could, if we wished, give each of 4
people 3 quarters, so we've divided the 3 pies among 4 people).

A more sophisticated way of looking at fractions is not to think in
terms of pies, but multiplication and division of numbers.  When we
write "12 divided by 4 is 3", we mean that 3 is the number by which
you have to multiply 4 to get 12.  Similarly, what does it mean to
talk about 3 divided by 4?  It's the number by which you can multiply
4 to get 3.  And 3/4 times 4 is 3, so that's the answer: 3 divided by
4 is 3/4.

Now, when we convert a fraction to a decimal, all we are doing is
carrying out, in decimal form, the division that the fraction
represents.  That is, in my example, we've said that 3/4 means 3
divided by 4, so we go ahead and divide 3.00 by 4 and get 0.75.  Since
both 3/4 and 0.75 are obtained by dividing 3 by 4, they are two ways
to express the same number!

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

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

Date: 01/04/2009 at 23:22:50
From: Claudia
Subject: Thank you (Fractions to decimals)

Thank you so much for your explanation.  I have a better understanding
now!  I really appreciate it.  Claudia