Question

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

	rational	irrational
-√26	○	○
86.\overline{87}	○	○
69.21	○	○
-11π	○	○
-√81	○	○

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 $q
eq0$. This includes integers, fractions, terminating decimals, and repeating decimals. An irrational number is a non - repeating, non - terminating decimal, and cannot be expressed as a fraction of two integers.

Step2: Classify $-\sqrt{26}$

We know that $\sqrt{26}\approx5.099$, and 26 is not a perfect square. So $-\sqrt{26}$ is a non - repeating, non - terminating decimal. So $-\sqrt{26}$ is irrational.

Step3: Classify $86.\overline{87}$

The bar over 87 indicates that 87 repeats indefinitely. A repeating decimal can be expressed as a fraction. Let $x = 86.\overline{87}=86.878787\cdots$. Then $100x=8687.878787\cdots$. Subtract $x$ from $100x$: $100x - x=8687.8787\cdots - 86.8787\cdots$, $99x = 8601$, $x=\frac{8601}{99}$. So $86.\overline{87}$ is rational.

Step4: Classify $69.21$

69.21 is a terminating decimal. A terminating decimal can be written as a fraction (e.g., $69.21=\frac{6921}{100}$). So 69.21 is rational.

Step5: Classify $- 11\pi$

$\pi\approx3.1415926\cdots$ is an irrational number. When we multiply an irrational number by a non - zero integer ($- 11$ in this case), the result is still irrational. So $-11\pi$ is irrational.

Step6: Classify $-\sqrt{81}$

We know that $\sqrt{81} = 9$, so $-\sqrt{81}=-9$. - 9 is an integer, and integers are rational (since $-9=\frac{-9}{1}$). So $-\sqrt{81}$ is rational.

Answer:

• For $-\sqrt{26}$: irrational
• For $86.\overline{87}$: rational
• For $69.21$: rational
• For $-11\pi$: irrational
• For $-\sqrt{81}$: rational

So the classification is:

Number	Rational	Irrational
$86.\overline{87}$	$\bullet$ (selected)	$\circ$ (unselected)
$69.21$	$\bullet$ (selected)	$\circ$ (unselected)
$-11\pi$	$\circ$ (unselected)	$\bullet$ (selected)
$-\sqrt{81}$	$\bullet$ (selected)	$\circ$ (unselected)