Question

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

Answer 1

Hint: Before solving this question we must know the value of square roots of 2 and 3 which will help us to solve the question very easily. Also, one must know what a rational number is as it will let us choose the appropriate options.

Complete step-by-step solution:
Now, before solving the question let us talk about numbers which are called rational numbers.
In the world of mathematics, the numbers which are called rational numbers are those numbers that can be expressed as the quotient or fraction that is $\dfrac{p}{q}$ of two numbers where q cannot be equals to 0. Now, if q is equal to 1, then it gives an integer value. So, every integer is a rational number.
In mathematics, all those real numbers which are not rational numbers are called irrational numbers.
Now, in question, it is given that we have to find a number which is a rational and irrational number which lies between $\sqrt{2}$ and $\sqrt{3}$.
Now, we know that $\sqrt{2}$ is approximately equivalent to 1.414 and $\sqrt{2}$ is approximate equals to 1.732.
To find a rational number between $\sqrt{2}$ and $\sqrt{3}$ so what we can do is we can take p and q such that we get a terminating decimal from expression $\dfrac{p}{q}$ which lies between $\sqrt{2}$ and $\sqrt{3}$.
And, to get an irrational number between $\sqrt{2}$ and $\sqrt{3}$, we can take surds which lies between $\sqrt{2}$ and $\sqrt{3}$, where surds are those numbers irrational numbers which cannot be represented accurately in the form of a fraction.
Let us consider a rational number $\dfrac{3}{2}$ and an irrational number ${{(6)}^{\dfrac{1}{4}}}$.
Now, on seeing the values of $\dfrac{3}{2}=1.5$ and ${{(6)}^{\dfrac{1}{4}}}=1.5650.....$ and comparing them with values of $\sqrt{2}=1.414$ and $\sqrt{3}=1.732$, we can say that $\sqrt{2}< \dfrac{3}{2}< \sqrt{3}$ and $\sqrt{2}< {{(6)}^{\dfrac{1}{4}}}< \sqrt{3}$
Hence, $\dfrac{3}{2}$ and ${{(6)}^{\dfrac{1}{4}}}$ are the rational number an irrational number lying between $\sqrt{2}$ and $\sqrt{3}$, respectively.

Note: One must know the difference between rational and irrational number so that he can discard the options on the basis of definition and also one must know the values of $\sqrt{2}$ and $\sqrt{3}$ as it makes one solve the question easily and very fast.