Question

determine whether each number is rational or irrational.

	rational	irrational
\\(\sqrt{11}\\)	\\(\circ\\)	\\(\circ\\)
\\(\sqrt{100}\\)	\\(\circ\\)	\\(\circ\\)
\\(0.\overline{1}\\)	\\(\circ\\)	\\(\circ\\)

Explanation:

Step1: Analyze $\frac{1}{11}$

A rational number is a number that can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$. $\frac{1}{11}$ is in the form of $\frac{p}{q}$ with $p = 1$, $q=11$ (both integers, $q
eq0$), so it is rational.

Step2: Analyze $\sqrt{11}$

The square root of a non - perfect square positive integer is irrational. Since 11 is not a perfect square (there is no integer $n$ such that $n^{2}=11$), $\sqrt{11}$ is irrational.

Step3: Analyze $\sqrt{100}$

We know that $\sqrt{100}=10$, and 10 can be written as $\frac{10}{1}$ (where $p = 10$, $q = 1$ are integers and $q
eq0$), so $\sqrt{100}$ is rational.

Step4: Analyze $0.\overline{1}$

A repeating decimal is a rational number because it can be expressed as a fraction. Let $x=0.\overline{1}=0.111\cdots$. Then $10x = 1.111\cdots$. Subtracting $x$ from $10x$: $10x - x=(1.111\cdots)-(0.111\cdots)\Rightarrow9x = 1\Rightarrow x=\frac{1}{9}$, which is in the form of $\frac{p}{q}$ with $p = 1$, $q = 9$ (integers, $q
eq0$), so it is rational.

Answer:

• $\frac{1}{11}$: Rational (select the circle under Rational)
• $\sqrt{11}$: Irrational (select the circle under Irrational)
• $\sqrt{100}$: Rational (select the circle under Rational)
• $0.\overline{1}$: Rational (select the circle under Rational)