Algebra I. Lesson 2
Rational Numbers (Grades 9-12)

Instruction 2-4

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Integers and the Number Line | Adding and Subtracting Integers| Inequalities and the Number Line | Comparing and Ordering Rational Numbers | Dividing Rational Numbers | Summary

Comparing and Ordering Rational Numbers
CCSTD HS Grades Algebra 13.0
What are Rational Numbers? p>By 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 alg1l2_4_files/i0020000.jpg 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 9.

alg1l2_4_files/i0020001.jpg,alg1l2_4_files/i0020002.jpg,alg1l2_4_files/i0020003.jpg, alg1l2_4_files/i0020004.jpg, alg1l2_4_files/i0020005.jpg

Integers are also included in the set of rational numbers. That is, any integer is a rational number as well. For example, rational numbersalg1l2_4_files/i0020006.jpg, alg1l2_4_files/i0020007.jpg, and alg1l2_4_files/i0020008.jpg 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.

Solution.
alg1l2_4_files/i0020009.jpg,alg1l2_4_files/i0020010.jpg,alg1l2_4_files/i0020011.jpg,alg1l2_4_files/i0020012.jpg, alg1l2_4_files/i0020013.jpg, alg1l2_4_files/i0020014.jpg

Comparing and Ordering Rational Numbers

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.


Let compare alg1l2_4_files/i0020015.jpg and –alg1l2_4_files/i0020016.jpg.Graph these points on a number line (Figure 2.15). To find where they go on the number line, it is easier to look at them as mixed numbers rather than improper fractions. Thus, and . To place on the number line, go left from zero 3 units, then continue to halfway between -3 and -4. Similarly, to place on the number line, go left from zero 1 unit, then continue left for another units. Another way to place them accurately on the number line is to make them into decimals.
Using this method, and


Figure 2.15

As you see, –alg1l2_4_files/i0020018.jpg is to the right ofalg1l2_4_files/i0020019.jpg. So, –alg1l2_4_files/i0020020.jpg >alg1l2_4_files/i0020021.jpg.

Practice 15. Compare each pair of numbers.

(a) alg1l2_4_files/i0020022.jpg and -alg1l2_4_files/i0020023.jpg

(b) –alg1l2_4_files/i0020024.jpg and –alg1l2_4_files/i0020025.jpg

(c) –alg1l2_4_files/i0020026.jpg and -alg1l2_4_files/i0020027.jpg

(d) alg1l2_4_files/i0020028.jpg and –alg1l2_4_files/i0020029.jpg

Answers

Using Common Denominator

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 alg1l2_4_files/i0020038.jpg is compared with –alg1l2_4_files/i0020039.jpg, then alg1l2_4_files/i0020040.jpg> –alg1l2_4_files/i0020041.jpg.


Case 2. Two rational numbers have like signs.If these rational numbers are in fraction form, it is easy to tell which is larger and which is smaller by finding the lowest common denominator. To find the common denominator, you can use the following table:

X1 X2 X3 X4 X5
Denominator #1
Denominator #2
 

Example 1. Using the fractions and the denominators are 6 and 9. These denominators can be put in the table below.

X1 X2 X3 X4 X5
6 12 18
9 18

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 .
Notice that you are multiplying the top and the bottom of each fraction by the same number. [This is an allowable operation because any number divided by itself is one, and any number multiplied by one is that number, as stated in the preceding chapter as the identity property of multiplication.] Looking at the numerators of the above two fractions, it is easy to see that since 15 is to the right of 8 on the number line, 15 > 8, therefore, and .


Example 2. Using the fractions and , the denominators are 9 and 24. Put them in the table.
X1 X2 X3 X4 X4 X6 X7 X8
9 18 27 36 45 54 63 72
24 48 72

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 –alg1l2_4_files/i0020061.jpg and –alg1l2_4_files/i0020062.jpg.
Solution. The denominators are 15 and 20. Factor these numbers.

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.

alg1l2_4_files/i0020063.jpg = –alg1l2_4_files/i0020064.jpg


alg1l2_4_files/i0020065.jpg = –alg1l2_4_files/i0020066.jpg.

Then,

alg1l2_4_files/i0020067.jpg = –alg1l2_4_files/i0020068.jpg

alg1l2_4_files/i0020069.jpg = –alg1l2_4_files/i0020070.jpg.

Since both –alg1l2_4_files/i0020071.jpg and –alg1l2_4_files/i0020072.jpg 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, –alg1l2_4_files/i0020073.jpg is greater than –alg1l2_4_files/i0020074.jpg. So, –alg1l2_4_files/i0020075.jpg > –alg1l2_4_files/i0020076.jpg.

Practice 16. Compare

(a) alg1l2_4_files/i0020077.jpg and –alg1l2_4_files/i0020078.jpg

(b) alg1l2_4_files/i0020079.jpg and alg1l2_4_files/i0020080.jpg

(c) –alg1l2_4_files/i0020081.jpg and –alg1l2_4_files/i0020082.jpg

Solution

Practice 17. Compare and order.

alg1l2_4_files/i0020116.jpg, alg1l2_4_files/i0020117.jpg, –alg1l2_4_files/i0020118.jpg, – alg1l2_4_files/i0020119.jpg, alg1l2_4_files/i0020120.jpg

Solution

Practical Exercise 3. Compare and Order alg1l2_4_files/i0020146.jpg , –alg1l2_4_files/i0020147.jpg,alg1l2_4_files/i0020148.jpg,–alg1l2_4_files/i0020149.jpg

Answer

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 .

Solution:

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 .

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