1. Rational numbers between two rational numbers

Asha wanted to count the numbers between $-9$ and $12$ . She found that totally $20$ integers are present, with $8$ negative integers, $zero$ and $11$ positive integers present between $-9$ and $12$ (excluding $-9$ and $12$). She also gets to know that there are no integers present between any two consecutive integers, that is, for example, between $8$ and $9$ there is no integer. So, between any two consecutive integers the number of integers is $0$.

 

She doubts whether the same will happen in case of rational numbers too?

 

She took two rational numbers $\frac{-3}{2}\mathit{and}\frac{-2}{3}$

 

She converted them to rational numbers with same denominators, for which the denominator of both the numbers should be converted into LCM.

 

She found the LCM of ($2$,$3$) $= 6$.

 

Converting both the numbers with denominator as $6$,

 

 

We have,

 

 

(or)

 

 

She could find rational numbers $\frac{-8}{6},\frac{-7}{6},\frac{-6}{6},\frac{-5}{6}$ between $\frac{-3}{2}\mathit{and}\frac{-2}{3}$.

 

She doubts are there only $4$  rational numbers between $\frac{-3}{2}\mathit{and}\frac{-2}{3}$. She gets to know there are

more than $4$ rational numbers between $\frac{-3}{2}\mathit{and}\frac{-2}{3}$ because if we find the multiples of denominators, then many more rational numbers can be inserted between $\frac{-3}{2}\mathit{and}\frac{-2}{3}$.

 

For example,

 

 

Now, between $\frac{-30}{20}\mathit{and}\frac{-20}{30}$ there are $9$ rational numbers present.