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Select all the numbers that are irrational.\newlineMulti-select Choices:\newline(A) −67-\frac{6}{7}\newline(B) 454\sqrt{5}\newline(C) 0.3330.333\newline(D) 0.7‾0.\overline{7}\newline(E) −6.11-6.11

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Checkpoint: Rational and irrational numbersright chevron icon

Full solution

Q. Select all the numbers that are irrational.\newlineMulti-select Choices:\newline(A) −67-\frac{6}{7}\newline(B) 454\sqrt{5}\newline(C) 0.3330.333\newline(D) 0.7‾0.\overline{7}\newline(E) −6.11-6.11
  1. Analyze −67-\frac{6}{7}: Step 11: Analyze number (A) −67-\frac{6}{7}.−67-\frac{6}{7} is a fraction that can be expressed as a ratio of two integers, −6-6 and 77. Since it can be expressed as a ratio, it is a rational number.
  2. Analyze 454\sqrt{5}: Step 22: Analyze number (B) 454\sqrt{5}. 454\sqrt{5} involves the square root of 55, which cannot be expressed as a ratio of two integers. Since 5\sqrt{5} is an irrational number, multiplying it by 44 does not change its irrational nature.
  3. Analyze 0.3330.333: Step 33: Analyze number (C) 0.3330.333.\newline0.3330.333 can be expressed as 13\frac{1}{3}, which is a repeating decimal.\newlineRepeating decimals are considered rational numbers.
  4. Analyze 0.7‾0.\overline{7}: Step 44: Analyze number (D) 0.7‾0.\overline{7}. 0.7‾0.\overline{7} represents the repeating decimal 0.7777…0.7777\ldots, which can be expressed as 79\frac{7}{9}. Repeating decimals are rational numbers.
  5. Analyze −6.11-6.11: Step 55: Analyze number (E) −6.11-6.11.\newline−6.11-6.11 is a terminating decimal, which can be expressed as −611100-\frac{611}{100}.\newlineTerminating decimals are rational numbers.

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