Question

classify each number below as a rational number or an irrational number.
-(sqrt{29}) (sqrt{1}) (\frac{9}{10}) (11pi) (43.overline{79})

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 non - repeating, non - terminating decimal and cannot be expressed as a fraction of two integers.

Step2: Classify $-\sqrt{29}$

29 is not a perfect square. So $\sqrt{29}$ is a non - repeating, non - terminating decimal. Multiplying it by - 1 does not change its nature. So $-\sqrt{29}$ is irrational.

Step3: Classify $\sqrt{1}$

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

Step4: Classify $\frac{9}{10}$

$\frac{9}{10}$ is in the form $\frac{p}{q}$ where $p = 9$ and $q = 10$ are integers and $q
eq0$. So $\frac{9}{10}$ is rational.

Step5: Classify $11\pi$

$\pi$ is an irrational number (non - repeating, non - terminating decimal). 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 $43.\overline{79}$

A repeating decimal can be expressed as a fraction. Let $x = 43.\overline{79}=43.797979\cdots$. Then $100x=4379.7979\cdots$. Subtract $x$ from $100x$: $100x - x=4379.7979\cdots - 43.7979\cdots$, $99x = 4336$, $x=\frac{4336}{99}$, which is a fraction of two integers. So $43.\overline{79}$ is rational.

Answer:

Number	Rational	Irrational
$\sqrt{1}$	$\circ$
$\frac{9}{10}$	$\circ$
$11\pi$		$\circ$
$43.\overline{79}$	$\circ$