Question

which numbers in the set listed below are irrational numbers? { -\frac{2}{5},0.005,3.2 \times 10^{-4},\pi,\sqrt{13} } select all correct options

Explanation:

Step1: Recall definition of rational and irrational numbers

A rational number can be written as a fraction $\frac{a}{b}$ where $a,b$ are integers and $b
eq0$. An irrational number cannot be written as a fraction and has a non - repeating, non - terminating decimal expansion.

Step2: Analyze $-\frac{2}{5}$

It is a fraction with $a = - 2$ and $b = 5$, so it is rational.

Step3: Analyze $0.005=\frac{5}{1000}=\frac{1}{200}$

It can be written as a fraction, so it is rational.

Step4: Analyze $3.2\times10^{-4}=\frac{3.2}{10^{4}}=\frac{32}{10^{5}}=\frac{8}{25000}$

It can be written as a fraction, so it is rational.

Step5: Analyze $\pi$

$\pi$ has a non - repeating, non - terminating decimal expansion ($3.1415926535\cdots$) and cannot be written as a fraction, so it is irrational.

Step6: Analyze $\sqrt{13}$

Since 13 is not a perfect square, $\sqrt{13}$ has a non - repeating, non - terminating decimal expansion and cannot be written as a fraction, so it is irrational.

Answer:

$\pi,\sqrt{13}$