Rational Numbers (Grades 9-12)
|Comparing and Ordering Rational Numbers|
|CCSTD HS Grades Algebra 13.0|
dividing each pair of integers, we can define a new set of numbers. This is
called the set of rational numbers. If a & b are two integers, then a number is called a rational number if b does not equal 0.
Look at some rational numbers below formed using integers 7, 4, 0, 5, and
Integers are also included in the set of rational numbers. That is, any integer is a rational number as well. For example, rational numbers, , and are the same as integers 17, 12, and 36.
In a rational number, the integer above the line is called the numerator. The integer below the line is called the denominator.
Practice 14. Form six rational numbers using 3, 4, and 7.
A good way to compare two rational numbers is graphing them on a number line. On a number line, the rational number to the right of another rational number is greater.
As you see, is to the right of. So, >.
Practice 15. Compare each pair of numbers.
(a) and -
(c) and -
Besides using number line, and decimals, you can use the common denominator method to compare and order rational numbers. In this method, we group the problems into two cases.
Case 1. Two rational numbers have unlike signs. The positive number is always greater than the negative number. For example if is compared with , then > .
Example 1. Using the fractions and the denominators are 6 and 9. These denominators can be put in the table below.
Solution. Notice that the
“X1, X2, etc” shows you what numbers you are going to multiply the
denominators by. As soon as you have a number in the top row that
matches a number in the bottom row, you have the lowest common
denominator. Here the lowest common denominator is 18.
To find out which number is the largest, you will have and .
Solution. Here the lowest common denominator is 72. To convert each fraction to fractions with the same denominator, multiply the numerator and denominator of by 8 and the numerator and denominator of by 3.
and . Since -16 is to the left of -15 on the number line, and .
Another way to find the lowest common denominator is to factor both denominators. Using the fractions, and .
Notice that both denominators have 6 as a factor. By multiplying the common factor by the other factors, we will find that the least common multiple of 12 and 18 is 2 × 3 × 6 = 36. So, the common denominator is 36. Divide the common denominator by each original denominator.
We must change the denominators to 36. In order to do this, multiply the numerator and denominator of the first number,, by 2. Likewise, multiply the numerator and denominator of the second number, by 3.
You see that both numbers now have the same denominator. The numerator of is larger than the numerator of . So, is greater than . That is, .
Example 3. Compare
15 = 3 × 520 = 4 × 5
The least common multiple of 15 and 20 is 3 × 4 × 5 = 60. That is, the common denominator is 60. Divide 60 by each denominator.
60 ÷ 15 = 460 ÷ 20 = 3
Numerator and denominator of the first number must be multiplied by 4. The numerator and the denominator of the second number must be multiplied by 3. These operations change the rational numbers to equivalent ones which they have the same denominator.
Since both and have the same denominator. The number with smaller numerator is to the right of the number with the greater numerator on the number line.. That is, is greater than . So, > .
Practice 16. Compare
Practice 17. Compare and order.
, , , ,
Practical Exercise 3. Compare and Order , ,,
Finding Numbers that are Between Two Other Numbers
You know that if you have the numbers 5 and 10 and want to find a whole number between them, you have a choice of 6, 7, 8, or 9. Similarly, if you have two fractions and want to find a number between them you can begin by finding the lowest common denominator, changing the fractions so they are fractions with the same denominator, then choosing a fraction between the two original fractions.
Example: Find a number between and .
The lowest common denominator is 12.
Choose a number between 4 and 9. Your resulting choices of fractions between are: , which reduces to , which reduces to .