Question

Find a rational number between $\sqrt{2}$ and $\sqrt{3}$.

Answer 1

Hint: We will be using the concept of rational and irrational numbers. As we know that there are infinitely many rational numbers between any two-irrational number. we will also be using the approximation to find the desired answer.

Complete step-by-step solution -
No, we have to find a rational number between $\sqrt{2}$ and $\sqrt{3}$. If we write the approximate value of $\sqrt{2}$ that is $\sqrt{2}$=1.414…. where … represents that the expansion goes on to infinite times.
Similarly, $\sqrt{3}$ it can be written as $\sqrt{3}$=1.732…. where … represents that the expansion goes on till infinity.
Now we know that there are infinite possible rational numbers between two irrational numbers. So, the number which we are finding will be one of them.
Now, let us suppose x be a rational number that lies between $\sqrt{2}$ and $\sqrt{3}$. This implies that
$\sqrt{2}$Now dividing and multiplying by 10 in (1) we get
$\dfrac{10}{10}\sqrt{2}<\dfrac{10}{10}x<\dfrac{10}{10}\sqrt{3}$ ……. (2)
Now, taking 10 inside the square root in (2) we get
$\dfrac{\sqrt{200}}{10}Now we have to find a perfect square which lies between 200 and 300. We can easily see that 256 is square of 16 and lies between 200 and 300. Now we can substitute x as
$\dfrac{\sqrt{200}}{10}<\dfrac{\sqrt{256}}{10}<\dfrac{\sqrt{300}}{10}$ …. (4)
We can see that 256 lies between 200 and 300, in the denominator 10 is common in all terms in (4).
So x=$\dfrac{\sqrt{256}}{10}$ is a rational number which lies between $\sqrt{2}$ and $\sqrt{3}$.
We can further simplify our answer as follows:
x=$\dfrac{\sqrt{256}}{10}$=$\dfrac{16}{10}=\dfrac{8}{5}$
Hence $\dfrac{8}{5}$is the required rational number.

Note: There are infinitely many rational numbers between two irrational numbers so the answer to this question can be infinitely many. For example, we can see that $\dfrac{3}{2}$ is also a rationale that lies between $\sqrt{2}$ and $\sqrt{3}$. Similarly, there are more such numbers.