Math Problem Statement
Select all the rational numbers. A.5/6
B.11−−√
C.6.565656…
D.0.23
E.0.32416…
F.−538
Solution
Let's evaluate each option to determine whether it is a rational number.
A.
- This is a fraction with both the numerator and denominator as integers. Since it can be written as a ratio of two integers, it is a rational number.
B.
- The square root of 11 is an irrational number because it cannot be expressed as a fraction of integers. The decimal expansion of is non-terminating and non-repeating.
C.
- This is a repeating decimal (the "56" repeats indefinitely). Repeating decimals can be expressed as fractions, so this is a rational number.
D.
- This is a terminating decimal, which can be written as . Therefore, it is a rational number.
E.
- The decimal expansion goes on indefinitely without repeating, suggesting that this is an irrational number. The decimal does not terminate or repeat, so it cannot be expressed as a fraction of integers.
F.
- This is an integer. All integers are rational numbers because they can be written as fractions (e.g., ).
Conclusion:
The rational numbers from the list are:
- A.
- C.
- D.
- F.
Would you like further explanations or clarifications on any of these? Here are some related questions to explore:
- How can we convert a repeating decimal into a fraction?
- What are some methods for identifying irrational numbers?
- Can an integer be both rational and irrational?
- How do we determine whether a decimal will terminate or repeat?
- What is the significance of rational and irrational numbers in real-life applications?
Tip: A number is rational if it can be written as a fraction , where and are integers, and .
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Math Problem Analysis
Mathematical Concepts
Rational Numbers
Irrational Numbers
Decimals
Fractions
Formulas
A rational number can be expressed as a fraction of the form p/q where p and q are integers, and q ≠ 0
Theorems
Rational numbers can be represented as fractions
Repeating decimals are rational numbers
Non-terminating, non-repeating decimals are irrational numbers
Suitable Grade Level
Grades 6-8
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