Question

which number is irrational?
○ $\frac{1}{6}$
○ $\sqrt{17}$
○ $\sqrt{49}$
○ $0.\overline{06}$

Explanation:

Step1: Recall the definition of irrational numbers

An irrational number is a number that cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$, and it has a non - repeating, non - terminating decimal expansion. Rational numbers include fractions, terminating decimals, repeating decimals, and perfect square roots.

Step2: Analyze $\frac{1}{6}$

$\frac{1}{6}$ is a fraction of two integers, so it is a rational number. Its decimal expansion is $0.1\overline{6}$, which is a repeating decimal.

Step3: Analyze $\sqrt{17}$

To determine if $\sqrt{17}$ is rational or irrational, we check if 17 is a perfect square. The perfect squares around 17 are $4^2 = 16$ and $5^2=25$. Since 17 is not a perfect square, $\sqrt{17}$ has a non - repeating, non - terminating decimal expansion. So $\sqrt{17}$ cannot be written as a fraction of two integers, and it is an irrational number.

Step4: Analyze $\sqrt{49}$

We know that $7\times7 = 49$, so $\sqrt{49}=7$. 7 can be written as $\frac{7}{1}$, where 7 and 1 are integers and $1
eq0$. So $\sqrt{49}$ is a rational number.

Step5: Analyze $0.\overline{06}$

$0.\overline{06}$ is a repeating decimal. A repeating decimal can be expressed as a fraction. Let $x = 0.060606\cdots$. Then $100x=6.060606\cdots$. Subtract $x$ from $100x$: $100x - x=6.0606\cdots - 0.0606\cdots$, which gives $99x = 6$, so $x=\frac{6}{99}=\frac{2}{33}$. Since it can be written as a fraction, it is a rational number.

Answer:

$\sqrt{17}$ (the option corresponding to $\sqrt{17}$)