Full solution

Q. Which of the following describe

6.111\ldots

? Select all that apply.

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Multi-select Choices:

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(A) natural number

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(B) rational number

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(D) real number

Natural Number Check: Determine if $6.111\ldots$ is a natural number. $\newline$ Natural numbers are the set of positive integers starting from $1$ , $2$ , $3$ , and so on. They do not include fractions or decimals. $\newline$ $6.111\ldots$ is a decimal number with a repeating pattern, so it is not a natural number.
Rational Number Check: Determine if $6.111\ldots$ is a rational number. $\newline$ A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, with the denominator $q$ not equal to zero. Since $6.111\ldots$ is a repeating decimal, it can be expressed as a fraction. $\newline$ To express $6.111\ldots$ as a fraction, let $x = 6.111\ldots$ Then, $10x = 61.111\ldots$ Subtracting the original equation from this, we get $9x = 55$ , which means $x = \frac{55}{9}$ . Therefore, $6.111\ldots$ is a rational number.
Irrational Number Check: Determine if $6.111\ldots$ 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 $6.111\ldots$ can be expressed as a fraction $(\frac{55}{9})$ , it is not an irrational number.
Real Number Check: Determine if $6.111\ldots$ is a real number. $\newline$ Real numbers include all the numbers on the number line, including rational and irrational numbers, positive and negative numbers, and zero. Since $6.111\ldots$ is a rational number, it is also a real number.