Select all the numbers that are rational. Multi-select Choices: (A)-1/3 (B)sqrt 3 (C)0.{6} (D)2 π (E)1.8989

Analyze option (A): Step 1: Analyze option (A) -1/3. -1/3 is a fraction, which can be expressed as a ratio of two integers (-1 and 3). This is a characteristic of rational numbers.Analyze option (B): Step 2: Analyze option (B) sqrt(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.Analyze option (C): Step 3: Analyze option (C) 0.{6}. 0.{6} represents 0.6666..., which is a repeating decimal. Repeating decimals are rational because they can be expressed as a fraction.Analyze option (D): Step 4: Analyze option (D) 2 π. 2 π (2 times pi) is irrational because π is an irrational number, and multiplying an irrational number by a rational number (2) results in an irrational number.Analyze option (E): Step 5: Analyze option (E) 1.8989. 1.8989 is a terminating decimal, which means it can be expressed as a fraction. Terminating decimals are rational.

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Select all the numbers that are rational. $\newline$ Multi-select Choices: $\newline$ (A) $-\frac{1}{3}$ $\newline$ (B) $\sqrt{3}$ $\newline$ (C) $0.\overline{6}$ $\newline$ (D) $2 \pi$ $\newline$ (E) $1.8989$

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Math Problems

Grade 8

Checkpoint: Rational and irrational numbers

Full solution

Q. Select all the numbers that are rational.

\newline

Multi-select Choices:

\newline

(A)

-\frac{1}{3}

\newline

(B)

\sqrt{3}

\newline

(C)

0.\overline{6}

\newline

(D)

2 \pi

\newline

(E)

1.8989

Analyze option (A): Step $1$ : Analyze option (A) $-\frac{1}{3}$ . $\newline$ $-\frac{1}{3}$ is a fraction, which can be expressed as a ratio of two integers ( $-1$ and $3$ ). $\newline$ This is a characteristic of rational numbers.
Analyze option (B): Step $2$ : Analyze option (B) $\sqrt{3}$ . $\sqrt{3}$ is an irrational number because it cannot be expressed as a fraction of two integers; its decimal expansion is non-terminating and non-repeating.
Analyze option (C): Step $3$ : Analyze option (C) $0.\overline{6}$ . $0.\overline{6}$ represents $0.6666\ldots$ , which is a repeating decimal. Repeating decimals are rational because they can be expressed as a fraction.
Analyze option (D): Step $4$ : Analyze option (D) $2 \pi$ . $2 \pi$ ( $2$ times $\pi$ ) is irrational because $\pi$ is an irrational number, and multiplying an irrational number by a rational number ( $2$ ) results in an irrational number.
Analyze option (E): Step $5$ : Analyze option (E) $1.8989$ . $1.8989$ is a terminating decimal, which means it can be expressed as a fraction. Terminating decimals are rational.