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Simplifying Fractions 3
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Properties of Negative Exponents
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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 “one-fourth” 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 “one-fourth’s” of the whole:

Thus, we can think of


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 “one-fourth” parts. They can be grouped into four groups of 3.

Earlier, we labelled such groups of 3 pieces of one-fourth of the whole by the fraction . Thus, of one whole is equivalent to one-fourth 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 twenty-three 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


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.

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