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Select all the numbers that are irrational. $\newline$ Multi-select Choices: $\newline$ (A) $\sqrt{2}$ $\newline$ (B) $\frac{2}{3}$ $\newline$ (C) $0.\overline{8}$ $\newline$ (D) $\frac{\pi}{2}$ $\newline$ (E) $-4.99$

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Simplify rational expressions

Full solution

Q. Select all the numbers that are irrational.

\newline

Multi-select Choices:

\newline

(A)

\sqrt{2}

\newline

(B)

\frac{2}{3}

\newline

(C)

0.\overline{8}

\newline

(D)

\frac{\pi}{2}

\newline

(E)

-4.99

Analyze $\sqrt{2}$ : Step $1$ : Analyze the first choice, $\sqrt{2}$ . $\sqrt{2}$ is known to be an irrational number because it cannot be expressed as a fraction of two integers.
Consider $\frac{2}{3}$ : Step $2$ : Look at choice (B), which is $\frac{2}{3}$ . This is a rational number because it is expressed as a fraction where both numerator and denominator are integers.
Examine $0.\overline{8}$ : Step $3$ : Consider choice (C), which is $0.\overline{8}$ . This represents the repeating decimal $0.888\ldots$ , which can be expressed as a fraction $(\frac{8}{9})$ , making it a rational number.
Evaluate $\pi/2$ : Step $4$ : Evaluate choice (D), $\pi/2$ . Pi ( $\pi$ ) is an irrational number, and dividing it by $2$ does not change its nature; thus, $\pi/2$ is also irrational.
Examine $-4.99$ : Step $5$ : Examine choice (E), $-4.99$ . This is a decimal number that can be exactly expressed as $-\frac{499}{100}$ , which is a rational number.

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