Select the rational numbers. Select all that apply.
-1.42372
0.5 \overline{8}
\sqrt{64}
0
2.23607 \ldots
0.25
0.58558555855558 \ldots
0.12345 \ldots
Answer (2)
Identify rational numbers as those expressible as a fraction q p , including terminating and repeating decimals.
Analyze each number: − 1.42372 , 0.5 8 , 64 , 0 , and 0.25 are rational, while 2.23607 … and 0.58558555855558 … and 0.12345 … are irrational.
Confirm that − 1.42372 = − 100000 142372 , 0.5 8 = 90 53 , 64 = 8 , 0 = 1 0 , and 0.25 = 4 1 .
Select the rational numbers: − 1.42372 , 0.5 8 , 64 , 0 , 0.25 .
Explanation
Understanding Rational Numbers We are asked to identify the rational numbers from the given list.
Rational numbers are numbers that can be expressed in the form q p , where p and q are integers and q = 0 . This includes terminating decimals, repeating decimals, integers, and fractions. Irrational numbers, on the other hand, cannot be expressed in this form; they are non-terminating and non-repeating decimals.
Analyzing Each Number -1.42372 is a terminating decimal, so it can be written as a fraction. Specifically, − 1.42372 = − 100000 142372 . Therefore, it is a rational number.
0.5 8 is a repeating decimal, which can be expressed as a fraction. To convert it, let x = 0.5 8 . Then 10 x = 5. 8 and 100 x = 58. 8 . Subtracting 10 x from 100 x , we get 90 x = 58 − 5 = 53 , so x = 90 53 . Therefore, it is a rational number.
64 is the square root of 64, which is 8. Since 8 = 1 8 , it is a rational number.
0 can be expressed as 1 0 , so it is a rational number.
2.23607 … is a non-terminating, non-repeating decimal, which is approximately equal to 5 . Therefore, it is an irrational number.
0.25 is a terminating decimal, which can be expressed as 4 1 . Therefore, it is a rational number.
0.58558555855558 … is a non-terminating, non-repeating decimal. Therefore, it is an irrational number.
0.12345 … is a non-terminating, non-repeating decimal. Therefore, it is an irrational number.
Identifying Rational Numbers Based on the analysis, the rational numbers are: -1.42372, 0.5 8 , 64 , 0, and 0.25.
Final Answer The rational numbers from the list are − 1.42372 , 0.5 8 , 64 , 0 , and 0.25 .
Examples
Rational numbers are used in everyday life, such as when calculating proportions, percentages, or dividing quantities. For example, if you want to split a bill of $25.50 equally among 3 people, you would use rational numbers to determine each person's share. The share would be $\frac{25.50}{3} = $8.50, which is a rational number. Understanding rational numbers helps in managing finances, cooking, and many other practical situations.
The rational numbers identified from the list are -1.42372, 0.5̅8, √64, 0, and 0.25. These numbers can be expressed in the form of fractions, which confirms their rational nature.
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