Question

Insert the rational number between $3$ and $4$.

Answer 1

Hint: We can write a large number of rational numbers between two natural numbers or integers. If we have to write a number in between two numbers then we have to simply add the given two numbers and then divide the result by two, it gives an exact middle number between these two numbers. Similarly, again we can write a rational number between these two numbers and so on.

Complete step-by-step answer:
Given: we have to insert the rational numbers between $3$ and $4$.
Now, the first rational number between $3$ and $4$ is given by $\dfrac{{3 + 4}}{2} = \dfrac{7}{2}$.
The next rational number we can write between $3$ and $\dfrac{7}{2}$, and other between $\dfrac{7}{2}$ and $4$.
The rational number between $3$ and $\dfrac{7}{2}$ is given by $\dfrac{{3 + \dfrac{7}{2}}}{2} = \dfrac{{13}}{4}$.
The rational number between $\dfrac{7}{2}$ and $4$ is given by $\dfrac{{\dfrac{7}{2} + 4}}{2} = \dfrac{{15}}{4}$.
The next rational number we can write between $3$ and $\dfrac{{13}}{4}$, other between $\dfrac{{13}}{4}$ and $\dfrac{7}{2}$, other between $\dfrac{7}{2}$ and $\dfrac{{15}}{4}$. And other between $\dfrac{{15}}{4}$ and $4$.
Similarly, the rational number between $3$ and $\dfrac{{13}}{4}$ is given by $\dfrac{{3 + \dfrac{{13}}{4}}}{2} = \dfrac{{25}}{8}$.
The rational number between $\dfrac{{13}}{4}$ and $\dfrac{7}{2}$ is given by $\dfrac{{\dfrac{{13}}{4} + \dfrac{7}{2}}}{2} = \dfrac{{27}}{8}$.
The rational number between $\dfrac{7}{2}$ and $\dfrac{{15}}{4}$ is given by $\dfrac{{\dfrac{7}{2} + \dfrac{{15}}{4}}}{2} = \dfrac{{29}}{8}$.
The rational number between $\dfrac{{15}}{4}$ and $4$ is given by $\dfrac{{\dfrac{{15}}{4} + 4}}{2} = \dfrac{{31}}{8}$.
Thus, the rational numbers between two numbers can be written as so on.

Hence, the rational number $\dfrac{7}{2}$, $\dfrac{{13}}{4}$, $\dfrac{{15}}{4}$ and so on are between $3$ and $4$.

Note:
Rational number: Any number in the form of $\dfrac{p}{q}$ is said to be rational number if $p$ and $q$ are integers and $q$ is not equal to zero $\left( {q \ne 0} \right)$.