Question

Find two rational and irrational numbers between $\sqrt{2}$ and $\sqrt{7}$.

Answer 1

Step 1: Find two rational numbers between the given numbers

Rational numbers are those numbers whose decimal representation is terminating or non terminating repeating.

We know that $\sqrt{2}=1.414....$ and $\sqrt{7}=2.645...$

So any two rational numbers between $1.414....$ and $2.645...$ are $1.6$ and $1.8$, decimal notations of both these numbers are terminating.

Step 2: Find two irrational numbers between the given numbers

Irrational numbers are those numbers whose decimal representation is non-terminating and non-repeating.

So any two irrational numbers between $\sqrt{2}$ and $\sqrt{7}$ are $\sqrt{3}=1.732050....$ and $\sqrt{5}=2.23606....$, decimal notations of both these numbers are non-terminating and non-repeating.

Hence the two rational numbers between $\sqrt{2}$ and $\sqrt{7}$ are $1.6$ and $1.8$ and two irrational numbers between $\sqrt{2}$ and $\sqrt{7}$ are $\sqrt{3}$ and $\sqrt{5}$.