Fractional Numbers: Basic operations with fractional numbers

 

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Objectives

Upon completion of this unit you shall be able to:

• Represent  and establish relationships between fractional numbers
• Effectuate operations with rational numbers           
• Evaluate percents and ratios
• Apply fractional numbers in real life domains

 

Introduction and representation

Let us recall that, so far, we have defined the set of integers as

For these numbers we have defined operations such as addition, subtraction, multiplication and subtraction. Out of all these four operations, division is the only operation that cannot be effectuated between any integer numbers.
When dividing integers we can get to a point where the number divided is not exactly dividable by the second, and in the division algorithm we get a remainder.
For instance, if we try to divide 16 by 5 we get 3 and a remainder of 1. We have thus the representation

or, in general

Let us now imagine that we have a segment of length 15, and we divide it in three. Since 15 can be divided exactly by 3, we get three segments of length 5. But what happens if we divide a 16 cm segment in 3 equally long pieces? Clearly it can be done physically, but what about the length of that segment? We define that length by the notation:

In general, for any two integers, we define

where  cannot be 0. These numbers, which are not integers, are fractions of integers, or fractional numbers, and they are positioned on the line between the integer numbers, as in the figure below:

      -3       -2      -1       0       1        2        3

Let us notice that  is between

 and .
We also note that the integer number 2 can be represented as a fractional number in more than one way:  and the same is true for any fractional number.
In general

for any  integers.

For any fractional number  ,  is called the numerator and  is called the denominator. Any two fractions can be written as to have the same denominator, by multiplying each fraction’s numerator and denominator with the denominator of the other fraction:

 can be written as .

This way we can define the larger number as the fraction with the largest numerator when we write the fractions as to have the same denominator. For instance let us compare the numbers  and  :

Since 15 is smaller than 28, we get that  is greater than .

Basic operations

The set of all fractional numbers is denoted by  and contains all integers, and all fractions obtained by dividing any two integers. We define addition on this set in the following manner:

For instance

Subtraction is defined in an analogue manner

Multiplications follow the pattern below

as in the next example

Division takes a slightly different form,  as it is replaced with the multiplication of the first fraction with the reversed second fraction:

The example below illustrates that:

Decimal representation

We noticed earlier that any rational number may be represented in more than one form. However, there is one form that is unique for each rational number. We introduced the rational numbers as the result of divisions that cannot be completed, because of the remainder. In the following we shall see what happens if we continue the division after we reach the remainder:
    123        15
    120        8,2         
     = 30
        30
=
After we obtain 3, which is smaller than 15, we add a zero to the left, and continue the division, the following digits in the result being preceded by a coma.
We get that  , which is a fractional number between  8 and 9. This representation is the decimal representation of a rational number. The part situated on the right of the dot represents how much of the divider is the remainder of the integer division. We can picture this decimal part as a fraction of a whole. If we divide 1 by 2, we get 0,5 which represents one half of 1, corresponding to the  division of a segment of length one in two equal parts. The same can be said about the result of dividing 1 by 4, which is 0,25, that is a quarter of a segment of length one.

0,25                 0,5                   0,75                    1
 

We can perform additions with rational numbers in the decimal representation, just as if we were adding integers, except that we must align the corresponding commas, and we must complete the shorter number with trailing zeros, as in the following example, where we add 12,35 and 1,6: