Question

classify each number below as a rational number or an irrational num

	rational	irrational
$-\frac{6}{7}$	$\circ$	$\circ$
$-12\pi$	$\circ$	$\circ$
$-\sqrt{4}$	$\circ$	$\circ$
$2\sqrt{6}$	$\circ$	$\circ$
$30.\overline{7}$	$\circ$	$\circ$

Explanation:

Step1: Recall definitions

A rational number is a number that can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. An irrational number is a non - repeating, non - terminating decimal, or a number that cannot be expressed as a fraction of two integers.

Step2: Classify $-\frac{6}{7}$

$-\frac{6}{7}$ is in the form of $\frac{p}{q}$ where $p = - 6$ and $q=7$ (both integers, $q
eq0$). So $-\frac{6}{7}$ is rational.

Step3: Classify $- 12\pi$

$\pi$ is an irrational number (non - repeating, non - terminating decimal). When we multiply an irrational number by a non - zero integer ($- 12$ here), the result is still irrational. So $-12\pi$ is irrational.

Step4: Classify $-\sqrt{4}$

First, calculate $\sqrt{4}=2$, so $-\sqrt{4}=- 2$. $-2$ can be written as $\frac{-2}{1}$, where $p=-2$ and $q = 1$ are integers, $q
eq0$. So $-\sqrt{4}$ is rational.

Step5: Classify $2\sqrt{6}$

$\sqrt{6}$ is an irrational number (since 6 is not a perfect square). When we multiply an irrational number by a non - zero integer (2 here), the result is still irrational. So $2\sqrt{6}$ is irrational.

Step6: Classify $30.\overline{7}$

A repeating decimal can be expressed as a fraction. Let $x = 30.\overline{7}$, then $10x=307.\overline{7}$. Subtract $x$ from $10x$: $10x - x=307.\overline{7}-30.\overline{7}$, $9x = 277$, $x=\frac{277}{9}$. Since it can be expressed as a fraction of two integers, $30.\overline{7}$ is rational.

Answer:

• For $-\frac{6}{7}$: rational (select the circle under "rational")
• For $-12\pi$: irrational (select the circle under "irrational")
• For $-\sqrt{4}$: rational (select the circle under "rational")
• For $2\sqrt{6}$: irrational (select the circle under "irrational")
• For $30.\overline{7}$: rational (select the circle under "rational")