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Instruction 2-4 Integers and the Number Line | Adding and Subtracting Integers | Inequalities and the Number Line | Comparing and Ordering Rational Numbers | Dividing Rational Numbers | Summary |
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| Comparing and Ordering Rational
Numbers http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=300 |
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| CCSTD HS Grades Algebra 13.0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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What are
Rational Numbers?
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
Look at some rational numbers below formed using integers –7, –4, 0, 5, and
9.  ,
 ,
 ,
 ,
 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. Solution.  ,
 ,
 ,
 ,
 ,
 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  and – . 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 
As you see, –  is to the right
of . So, – > . Practice 15. Compare each pair of
numbers. (a)
 and
- (b) –  and – (c) –  and - (d)
 and – Using Common Denominator http://www.aaaknow.com/fra66jx2.htm 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
 > – .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:
Example 1. Using the fractions
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.
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.
Another way to find the lowest
common denominator is to factor both denominators. Using the
fractions,
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,
You see that both numbers now have
the same denominator. The numerator of
Example 3. Compare
–  and – .Solution. The denominators are 15 and 20. Factor these
numbers. 15 = 3 × 5 20 = 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 = 4 60 ÷ 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. –
 = – –  = – . Then, –
 = – –  = – .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 (a)
 and – (b)
 and
 (c) –  and –  ,
 , – , –
 ,
 Practical Exercise 3. Compare and
Order  , – ,
 , –
Answer
Finding numbers that are between two other numbers. Example: Find a number between
Solution: The lowest common denominator is 12.
Choose a number between 4 and 9. Your resulting choices of fractions
between Links for Students, Parents and Teachers Now let's do Practice Exercise 2-4 (top).
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and
the
denominators are 6 and 9. These denominators can be put in the table
below.
and 
. Since -16
is to the left of -15 on the number line,
and
.
and
.
is larger than
the numerator of
. So,
is greater
than 
. That is,
.
and
.
are:
which reduces to
which reduces to
.