Question

which of the following numbers is irrational?
2×10⁻⁵
0.\overline{23}
\frac{3}{11}
\sqrt{40}

Explanation:

Step1: Recall irrational number definition

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 its decimal expansion is non - repeating and non - terminating. Rational numbers include integers, fractions, terminating decimals, and repeating decimals.

Step2: Analyze $2\times10^{-5}$

We know that $2\times10^{-5}=\frac{2}{10^{5}}=\frac{2}{100000} = 0.00002$, which is a terminating decimal. So, it is a rational number.

Step3: Analyze $0.\overline{23}$

The number $0.\overline{23}$ is a repeating decimal (the bar over 23 indicates that 23 repeats indefinitely). Repeating decimals can be expressed as fractions. Let $x = 0.\overline{23}=0.232323\cdots$. Then $100x=23.232323\cdots$. Subtracting $x$ from $100x$ gives $100x - x=23.2323\cdots - 0.2323\cdots$, so $99x = 23$, and $x=\frac{23}{99}$. So, it is a rational number.

Step4: Analyze $\frac{3}{11}$

$\frac{3}{11}$ is a fraction of two integers with a non - zero denominator, so it is a rational number. Its decimal expansion is $0.272727\cdots$, which is a repeating decimal.

Step5: Analyze $\sqrt{40}$

Simplify $\sqrt{40}=\sqrt{4\times10}=2\sqrt{10}$. Since $\sqrt{10}$ is a non - perfect square root, its decimal expansion is non - repeating and non - terminating. And $2\sqrt{10}$ also has a non - repeating and non - terminating decimal expansion, so it cannot be expressed as a fraction of two integers. Thus, $\sqrt{40}$ is an irrational number.

Answer:

$\boldsymbol{\sqrt{40}}$