Question

use the drop-down menus to classify each number as rational or irrational.
click the arrows to choose an answer from each menu.
$4.overline{79}$ is choose...
$sqrt{24}$ is choose...
$\frac{pi}{2}$ is choose...

Explanation:

Step1: Classify \( 4.\overline{79} \)

A repeating decimal can be expressed as a fraction of two integers. Let \( x = 4.\overline{79} = 4.797979\ldots \). Then \( 100x = 479.7979\ldots \). Subtract \( x \) from \( 100x \): \( 100x - x = 479.7979\ldots - 4.7979\ldots \), so \( 99x = 475 \), and \( x=\frac{475}{99} \). Since it can be written as a fraction, \( 4.\overline{79} \) is rational.

Step2: Classify \( \sqrt{24} \)

Simplify \( \sqrt{24}=\sqrt{4\times6} = 2\sqrt{6} \). \( \sqrt{6} \) is irrational (it cannot be expressed as a fraction of two integers), and multiplying an irrational number by a non - zero rational number (2) gives an irrational number. So \( \sqrt{24} \) is irrational.

Step3: Classify \( \frac{\pi}{2} \)

\( \pi \) is an irrational number (it has a non - repeating, non - terminating decimal expansion). Dividing an irrational number by a non - zero rational number (2) results in an irrational number. So \( \frac{\pi}{2} \) is irrational.

Answer:

\( 4.\overline{79} \) is rational; \( \sqrt{24} \) is irrational; \( \frac{\pi}{2} \) is irrational