Question

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

	rational	irrational
$\sqrt{4}$	$\circ$	$\circ$
$-27.\overline{91}$	$\circ$	$\circ$
$-80.87$	$\circ$	$\circ$
$-\sqrt{14}$	$\circ$	$\circ$

Explanation:

Step1: Recall definitions

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

Step2: Classify $\pi$

The value of $\pi$ is approximately $3.1415926535\cdots$ and its decimal expansion is non - repeating and non - terminating. So $\pi$ is irrational.

Step3: Classify $\sqrt{4}$

We know that $\sqrt{4}=2$, and $2=\frac{2}{1}$, where $2$ and $1$ are integers and $1
eq0$. So $\sqrt{4}$ is rational.

Step4: Classify $- 27.\overline{91}$

A repeating decimal can be expressed as a fraction. Let $x=-27.\overline{91}=-27.919191\cdots$. Then $100x=-2791.919191\cdots$. Subtract $x$ from $100x$: $100x - x=-2791.9191\cdots-(-27.9191\cdots)$, $99x=-2764$, $x =-\frac{2764}{99}$. Since it can be written as a fraction of two integers, $-27.\overline{91}$ is rational.

Step5: Classify $-80.87$

$-80.87=-\frac{8087}{100}$, and $8087$ and $100$ are integers with $100
eq0$. So $-80.87$ is rational.

Step6: Classify $-\sqrt{14}$

The number $14$ is not a perfect square. The decimal expansion of $\sqrt{14}\approx3.7417$ (and the full decimal is non - repeating and non - terminating), so $-\sqrt{14}$ has a non - repeating, non - terminating decimal expansion. Thus, $-\sqrt{14}$ is irrational.

Answer:

• For $\pi$: irrational (select the circle in the 'irrational' column for $\pi$)
• For $\sqrt{4}$: rational (select the circle in the 'rational' column for $\sqrt{4}$)
• For $-27.\overline{91}$: rational (select the circle in the 'rational' column for $-27.\overline{91}$)
• For $-80.87$: rational (select the circle in the 'rational' column for $-80.87$)
• For $-\sqrt{14}$: irrational (select the circle in the 'irrational' column for $-\sqrt{14}$)