Question

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

	rational	irrational
$\sqrt{49}$	$\circ$	$\circ$
$2\sqrt{2}$	$\circ$	$\circ$
$-\frac{19}{11}$	$\circ$	$\circ$
$19\pi$	$\circ$	$\circ$

Explanation:

Step1: Recall definitions

A rational number is a number that can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. It includes terminating, repeating decimals, integers, fractions, perfect square roots. An irrational number is non - repeating, non - terminating decimal, and square roots of non - perfect squares, multiples of $\pi$ (except when $\pi$ is canceled out, but $\pi$ itself is irrational).

Step2: Classify $29.\overline{59}$

$29.\overline{59}$ is a repeating decimal. Repeating decimals can be expressed as a fraction. Let $x = 29.\overline{59}=29.595959\cdots$. Then $100x=2959.595959\cdots$. Subtract $x$ from $100x$: $100x - x=2959.5959\cdots - 29.5959\cdots$, $99x = 2930$, $x=\frac{2930}{99}$. So it is rational.

Step3: Classify $\sqrt{49}$

$\sqrt{49}=7$, and $7=\frac{7}{1}$, where $7$ and $1$ are integers and $1
eq0$. So $\sqrt{49}$ is rational.

Step4: Classify $2\sqrt{2}$

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

Step5: Classify $-\frac{19}{11}$

$-\frac{19}{11}$ is in the form $\frac{p}{q}$ where $p=- 19$, $q = 11$ are integers and $q
eq0$. So it is rational.

Step6: Classify $19\pi$

$\pi$ is an irrational number. When we multiply an irrational number by a non - zero rational number (19 is rational), the result is still irrational. So $19\pi$ is irrational.

Answer:

• $29.\overline{59}$: rational (select the circle under "rational" for $29.\overline{59}$)
• $\sqrt{49}$: rational (select the circle under "rational" for $\sqrt{49}$)
• $2\sqrt{2}$: irrational (select the circle under "irrational" for $2\sqrt{2}$)
• $-\frac{19}{11}$: rational (select the circle under "rational" for $-\frac{19}{11}$)
• $19\pi$: irrational (select the circle under "irrational" for $19\pi$)