Question

identify whether each number is rational or irrational.
drag each number into the correct box of the table to show whether it is rational or irrational.
rational numbers
irrational numbers
$-\frac{11}{5}$
$sqrt{\frac{1}{4}}$
$4.2$
$sqrt{8}$
$pi$
$0.overline{36}$
$sqrt{\frac{5}{4}}$

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 can be a fraction, terminating decimal, repeating decimal, or a square root of a perfect square. An irrational number is a non - repeating, non - terminating decimal, or a square root of a non - perfect square (except perfect squares), or special numbers like $\pi$.

Step2: Analyze $-\frac{11}{5}$

$-\frac{11}{5}$ is a fraction with integer numerator and non - zero integer denominator. So, it is rational.

Step3: Analyze $\sqrt{\frac{1}{4}}$

$\sqrt{\frac{1}{4}}=\frac{1}{2}$, which is a fraction of two integers. So, it is rational.

Step4: Analyze $4.2$

$4.2=\frac{42}{10}=\frac{21}{5}$, a fraction of two integers. So, it is rational.

Step5: Analyze $\sqrt{8}$

$\sqrt{8} = 2\sqrt{2}$, and $\sqrt{2}$ is irrational. So, $\sqrt{8}$ is irrational.

Step6: Analyze $\pi$

$\pi$ is a non - repeating, non - terminating decimal (approximately $3.14159\cdots$), so it is irrational.

Step7: Analyze $0.\overline{36}$

$0.\overline{36}$ is a repeating decimal. Let $x = 0.3636\cdots$, then $100x=36.3636\cdots$, $100x - x=36$, $99x = 36$, $x=\frac{36}{99}=\frac{4}{11}$, a fraction of two integers. So, it is rational.

Step8: Analyze $\sqrt{\frac{5}{4}}$

$\sqrt{\frac{5}{4}}=\frac{\sqrt{5}}{2}$, and $\sqrt{5}$ is irrational. So, $\sqrt{\frac{5}{4}}$ is irrational.

Answer:

Rational Numbers

$-\frac{11}{5}$, $\sqrt{\frac{1}{4}}$, $4.2$, $0.\overline{36}$

Irrational Numbers

$\sqrt{8}$, $\pi$, $\sqrt{\frac{5}{4}}$