Question

Find six rational number between $2$ and $3$

Answer 1

Find the rational number between the two numbers

Rational number is a type of real numbers, which is in the form of $\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.$ where $q$ not equal to $0$. Any fraction with non-zero denominators is a rational number.

Let $x$ be a rational number such that $x$ is less than ${\frac{1}{6}}^{th}$of the difference between the two number.

$$\begin{gathered} \therefore x<\frac{1}{6}(3-2) \\ \Rightarrow x<\frac{1}{6} \end{gathered}$$

Say $x=\frac{1}{7}$

From the property of rational number we know that sum of two rational number is a rational number.

So we can say the rational in between $2$ and $3$ are of form $2+n\frac{1}{7}$ where $n=1,2,...,6$

$\therefore$the rational numbers are $2+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,2+\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,2+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,2+\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,2+\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,2+\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$7$}\right.$

$=\frac{15}{7},\frac{16}{7},\frac{17}{7},\frac{18}{7},\frac{19}{7},\frac{20}{7}$

Hence, the six rational numbers between $2$ and $3$ are $\frac{15}{7},\frac{16}{7},\frac{17}{7},\frac{18}{7},\frac{19}{7},\frac{20}{7}$.