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Which of the following describe 6.111…6.111\ldots? Select all that apply.\newlineMulti-select Choices:\newline(A) natural number\newline(B) rational number\newline(C) irrational number\newline(D) real number

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Q. Which of the following describe 6.111…6.111\ldots? Select all that apply.\newlineMulti-select Choices:\newline(A) natural number\newline(B) rational number\newline(C) irrational number\newline(D) real number
  1. Natural Number Check: Determine if 6.111…6.111\ldots is a natural number.\newlineNatural numbers are the set of positive integers starting from 11, 22, 33, and so on. They do not include fractions or decimals.\newline6.111…6.111\ldots is a decimal number with a repeating pattern, so it is not a natural number.
  2. Rational Number Check: Determine if 6.111…6.111\ldots is a rational number.\newlineA rational number is any number that can be expressed as the quotient or fraction pq\frac{p}{q} of two integers, with the denominator qq not equal to zero. Since 6.111…6.111\ldots is a repeating decimal, it can be expressed as a fraction.\newlineTo express 6.111…6.111\ldots as a fraction, let x=6.111…x = 6.111\ldots Then, 10x=61.111…10x = 61.111\ldots Subtracting the original equation from this, we get 9x=559x = 55, which means x=559x = \frac{55}{9}. Therefore, 6.111…6.111\ldots is a rational number.
  3. Irrational Number Check: Determine if 6.111…6.111\ldots is an irrational number. An irrational number is a number that cannot be expressed as a simple fraction. It's a non-terminating, non-repeating decimal. Since we have already determined that 6.111…6.111\ldots can be expressed as a fraction (559)(\frac{55}{9}), it is not an irrational number.
  4. Real Number Check: Determine if 6.111…6.111\ldots is a real number.\newlineReal numbers include all the numbers on the number line, including rational and irrational numbers, positive and negative numbers, and zero. Since 6.111…6.111\ldots is a rational number, it is also a real number.

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