Example 1: Recognizing Rational Numbers
6.6 Rational Numbers and Decimals
Goal 1: Identifying Rational Numbers
A number is rational if it can be written as the quotient of two
integers. Numbers that cannot be rewritten as the quotient of two
integers are called irrational.
Rational Numbers: 1/2, -3/5, 9/4, 5/1
Show that the following numbers are rational.
a. 4
4 is rational because it can be written as 4 = 4/1.
b. 0.5
0.5 is rational because it can be written as 0.5 = 1/2.
Irrational Numbers: √2, √3, √5
c. -3
-3 is rational because it can be written as -3 = -3/1.
*Natural numbers (1, 2, 3, ...), whole numbers (0, 1, 2, ...), and
integers (..., -2, -1, 0, 1, 2, ...) are examples of rational numbers.
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Example 2: Showing that a Mixed Number is Rational
Goal 2: Writing Decimals as Fractions
Show that the mixed number 1 1/4 is rational.
Decimals can be terminating, repeating, or nonrepeating.
Solution:
1 1/4 =
*In decimal form, every rational number is either terminating or
repeating, and every irrational number is nonrepeating.
1 + 1/4 =
Number
3/8
Decimal
0.375
Comment
Rational, terminating
16/11
1.454545... = 1.45
Rational, repeating
√2
1.414213562...
Irrational, nonrepeating
4/4 + 1/4 =
5/4 (written as a quotient of two integers)
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(Example 3: Writing Decimals as Fractions
On Your Own:
Write the decimal as a fraction.
Rational or Irrational?
*If rational, write as the quotient of two integers (fraction).
a. 0.45
1. -3
0.45 = 45/100
(move decimal two places, put number over 100)
5 * 9/5 *20
(factor)
9/20
(simplify)
2. √6
3. 0.4
Rational or Irrational? Terminating, Repeating, or nonrepeating?
1. 3/5
b. 0.8
2. √8
80/100
2 * 2 * 2 * 2 * 5 / 2* 2 * 5 * 5
3. 13/12
Write the decimal as a fraction.
2*2/5
1. 0.35
4/5
2. 0.64
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