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. 56\frac{5}{6}

  • 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. 11\sqrt{11}

  • The square root of 11 is an irrational number because it cannot be expressed as a fraction of integers. The decimal expansion of 11\sqrt{11} is non-terminating and non-repeating.

C. 6.5656566.565656\ldots

  • This is a repeating decimal (the "56" repeats indefinitely). Repeating decimals can be expressed as fractions, so this is a rational number.

D. 0.230.23

  • This is a terminating decimal, which can be written as 23100\frac{23}{100}. Therefore, it is a rational number.

E. 0.324160.32416\ldots

  • 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. 538-538

  • This is an integer. All integers are rational numbers because they can be written as fractions (e.g., 538=5381-538 = \frac{-538}{1}).

Conclusion:

The rational numbers from the list are:

  • A. 56\frac{5}{6}
  • C. 6.5656566.565656\ldots
  • D. 0.230.23
  • F. 538-538

Would you like further explanations or clarifications on any of these? Here are some related questions to explore:

  1. How can we convert a repeating decimal into a fraction?
  2. What are some methods for identifying irrational numbers?
  3. Can an integer be both rational and irrational?
  4. How do we determine whether a decimal will terminate or repeat?
  5. 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 pq\frac{p}{q}, where pp and qq are integers, and q0q \neq 0.

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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