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Select all the numbers that are rational.\newlineMulti-select Choices:\newline(A) −13-\frac{1}{3}\newline(B) 3\sqrt{3}\newline(C) 0.6‾0.\overline{6}\newline(D) 2π2 \pi\newline(E) 1.89891.8989

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Checkpoint: Rational and irrational numbersright chevron icon

Full solution

Q. Select all the numbers that are rational.\newlineMulti-select Choices:\newline(A) −13-\frac{1}{3}\newline(B) 3\sqrt{3}\newline(C) 0.6‾0.\overline{6}\newline(D) 2π2 \pi\newline(E) 1.89891.8989
  1. Analyze option (A): Step 11: Analyze option (A) −13-\frac{1}{3}.\newline−13-\frac{1}{3} is a fraction, which can be expressed as a ratio of two integers (−1-1 and 33).\newlineThis is a characteristic of rational numbers.
  2. Analyze option (B): Step 22: Analyze option (B) 3\sqrt{3}.3\sqrt{3} is an irrational number because it cannot be expressed as a fraction of two integers; its decimal expansion is non-terminating and non-repeating.
  3. Analyze option (C): Step 33: Analyze option (C) 0.6‾0.\overline{6}. 0.6‾0.\overline{6} represents 0.6666…0.6666\ldots, which is a repeating decimal. Repeating decimals are rational because they can be expressed as a fraction.
  4. Analyze option (D): Step 44: Analyze option (D) 2π2 \pi. 2π2 \pi (22 times π\pi) is irrational because π\pi is an irrational number, and multiplying an irrational number by a rational number (22) results in an irrational number.
  5. Analyze option (E): Step 55: Analyze option (E) 1.89891.8989. 1.89891.8989 is a terminating decimal, which means it can be expressed as a fraction. Terminating decimals are rational.

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