The Meaning of Fractions
The following illustrates important properties of fractions,
we will give a brief review of the concept here.
Fractionsare written in the form of two whole numbers, aligned
vertically, and separated by a horizontal line:
The upper number is called the numerator, and the lower number
is called the denominator. The origin of these words will become
clear as we go along. Fractions in which the numerator is smaller
than the denominator are called proper fractions. A simple, but
not very intuitive, interpretation is that a fraction indicates
division. For example
A more intuitive understanding comes from viewing fractions as
dealing with parts of a whole. For example, the fraction
has a denominator of 4, indicating it refers to what you get
when you take an item and divide it into four equal parts. If we
let a circle represent “the whole thing,” then the four
equal parts that we can picture making up the circle are easy to
visualize:
We refer to each of these pieces as being
“onefourth” of the circle (and in this case, sometimes
reflecting the Latin influences on the English language, people
also say “one quarter” to mean the same thing), and we
write in symbols the phrase “one fourth” as the
fraction
So, the denominator of this (and any other) fraction indicates
into how many equal pieces the entire object is being separated.
Then, when we write
we are referring to three of these
“onefourth’s” of the whole:
Thus, we can think of
as
You can see from this that the denominator tells us how many
smaller pieces the whole is being divided into. The numerator
tells us how many (the “number”) of those pieces
represented by the fraction. Note that if we have three
“wholes” and divide each of them up into four equal
parts,
we get 12 (= 3 x 4) of these “onefourth” parts.
They can be grouped into four groups of 3.
Earlier, we labelled such groups of 3 pieces of onefourth of
the whole by the fraction . Thus, of one
whole is equivalent to onefourth of 3 wholes:
So, we have two alternative visualizations of the fraction :
 Take a whole and divide it into four equal parts. Then
represents how much of the original whole is included in
three of those parts.
 Take three wholes and divide them together into four
equal parts. Then represents how much of a whole
you get when you divide those three wholes into four
equal parts.
Because fractions and division are so closely related,
there’s one more interpretation of fractions that is
sometimes helpful. Consider the improper fraction (numerator
bigger than denominator)
which we know equates to the decimal value
You can think of the statement
as meaning that there are 5 fours in 20. That is, if you start
removing groups of four things from a pool of twenty things, you
will find that there are enough items in the pool to give five
groups of four things. This understanding of a fraction is used
in the next note in this series to understand what fractions with
zeros in the numerator or denominator might represent.
When an improper fraction does not reduce to a whole number,
as, for example,
it just means that after you remove five complete groups of
four things from the original pool of twentythree things, you
are left with just three items in the pool. This is not enough to
be able to remove another complete group of four. In fact,
what’s left is of a complete group of four. Hence
in conventional notation. Numbers written as
meaning
are often called mixed numbers because they represent the
combination of a whole number part and a fractional part. When
you do arithmetic with mixed numbers, you often need to convert
them to pure fractions first:
Before leaving this attempt to give intuitive meaning to
fractions, we note just one more (rather obvious, but important)
fact. Even whole numbers can be written as fractions, using a
denominator value of 1. Thus technically,
and for any number b,
This is just another way of saying that there are 5 ones in 5,
or b ones in b. Sometimes it is helpful to use this identity
between numbers and fractions to make whole numbers in arithmetic
statements look like fractions.
