Question

which of the following are irrational numbers? 3, √10, -4, π

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 its decimal expansion is non - repeating and non - terminating.

Step2: Analyze each number

• For the number $3$: We can write $3=\frac{3}{1}$, where $p = 3$ and $q = 1$ are integers with $q

eq0$. So, $3$ is a rational number.

• For the number $\sqrt{10}$: Since $10$ is not a perfect square, $\sqrt{10}$ has a non - repeating and non - terminating decimal expansion. It cannot be written as a fraction of two integers. So, $\sqrt{10}$ is an irrational number.
• For the number $- 4$: We can write $-4=\frac{-4}{1}$, where $p=-4$ and $q = 1$ are integers with $q

eq0$. So, $-4$ is a rational number.

• For the number $\pi$: The decimal expansion of $\pi$ is $3.1415926535\cdots$ which is non - repeating and non - terminating. It cannot be expressed as a fraction of two integers. So, $\pi$ is an irrational number.

Answer:

The irrational numbers among the given numbers are $\boldsymbol{\sqrt{10}}$ and $\boldsymbol{\pi}$. So the correct options (the ones that should be checked) are: $\sqrt{10}$ (the second option) and $\pi$ (the fourth option). The numbers $3$ (first option) and $- 4$ (third option) are rational numbers and should not be checked.