Question

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

Answer 1

Hint: First, before proceeding for this, we must know the values of the numbers that is $\sqrt{2}$and $\sqrt{3}$in the decimal form to easily get the rational number between them. Then, the rational number is the number which is written in the form of $\dfrac{p}{q}$and $q\ne 0$. Then, we can clearly see that number 1.5 exists between 1.414 and 1.732. Then, we get $\dfrac{3}{2}$ as a rational number which is in the form of $\dfrac{p}{q}$ and denominator of the number is not zero which proves that 1.5 is the correct rational number between $\sqrt{2}$and $\sqrt{3}$.

Complete step by step answer:
In this question, we are supposed to find the rational number between $\sqrt{2}$ and $\sqrt{3}$.
So, before proceeding for this, we must know the values of the numbers that are $\sqrt{2}$and $\sqrt{3}$in the decimal form to easily get the rational number between them.
Then, we get the value of $\sqrt{2}$and $\sqrt{3}$as:
$\begin{align}
  & \sqrt{2}=1.414 \\
 & \sqrt{3}=1.732 \\
\end{align}$
So, we need to find the rational number in between 1.414 and 1.732.
So, before finding the rational number, we must know what is a rational number?
Then, the rational number is the number which is written in the form of $\dfrac{p}{q}$and $q\ne 0$.
Now, we can clearly see that number 1.5 exists between 1.414 and 1.732.
Now, just we need to check whether 1.5 is a rational number or not.
So, for that we need to convert it into fraction form as:
$1.5=\dfrac{15}{10}$
Now, by taking to the lowest form, we get:
$\dfrac{15}{10}=\dfrac{3}{2}$
So, we get $\dfrac{3}{2}$ as a rational number which is in the form of $\dfrac{p}{q}$ and denominator of the number is not zero which proves that 1.5 is the correct rational number between $\sqrt{2}$and $\sqrt{3}$.
Hence, we get the rational number between $\sqrt{2}$and $\sqrt{3}$ as 1.5.

Note:
Now, to solve these type of the questions we can also find any other rational number between $\sqrt{2}$and $\sqrt{3}$as we can also take the rational numbers as 1.48, 1.49, 1.6, etc. as all of these numbers gives the rational form by converting them into fraction. But $\sqrt{2}$and $\sqrt{3}$gives a non terminating decimal value which is not a rational number.