Question

which number is irrational?

• $\frac{1}{6}$
• $sqrt{17}$
• $sqrt{49}$
• $0.overline{06}$

Explanation:

Step1: Recall irrational number definition

An irrational number is a number that cannot be expressed as a fraction of two integers and has a non - repeating, non - terminating decimal expansion. Rational numbers include fractions, terminating decimals, repeating decimals, and perfect square roots.

Step2: Analyze \(\frac{1}{6}\)

\(\frac{1}{6}=0.1666\cdots\), which is a repeating decimal. Repeating decimals can be expressed as a fraction of two integers, so \(\frac{1}{6}\) is rational.

Step3: Analyze \(\sqrt{17}\)

To determine if \(\sqrt{n}\) is rational, we check if \(n\) is a perfect square. The perfect squares around 17 are \(4 = 2^2\), \(9=3^2\), \(16 = 4^2\), \(25=5^2\). Since 17 is not a perfect square, \(\sqrt{17}\) has a non - repeating, non - terminating decimal expansion. So \(\sqrt{17}\) is irrational.

Step4: Analyze \(\sqrt{49}\)

We know that \(7\times7 = 49\), so \(\sqrt{49}=7\), which is an integer. Integers are rational numbers (can be written as \(\frac{7}{1}\)).

Step5: Analyze \(0.\overline{06}\)

A repeating decimal \(0.\overline{06}=0.060606\cdots\) can be expressed as a fraction. Let \(x = 0.060606\cdots\), then \(100x=6.060606\cdots\). Subtract \(x\) from \(100x\): \(100x - x=6.0606\cdots - 0.0606\cdots\), \(99x = 6\), \(x=\frac{6}{99}=\frac{2}{33}\). So it is rational.

Answer:

\(\boldsymbol{\sqrt{17}}\) (the option with \(\sqrt{17}\))