5. 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.

 

 

She wondered whether the same would happen in case of rational numbers too?

 

She took two rational numbers $\frac{-3}{2}$ 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 ($2$,$3$) $= 6$.

 

Convert both the numbers with denominator as $6$.

 

$\frac{-3\times 3}{2\times 3}=\frac{-9}{6}$ and $\frac{-2\times 2}{3\times 2}=\frac{-4}{6}$

 

We have,

 

 

(or)

 

 

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

 

She doubts that are there only $4$ rational numbers between $\frac{-3}{2}$. She gets to know there are more than $4$ rational numbers between $\frac{-3}{2}$ because if we find the multiples of denominators, then many more rational numbers can be inserted between $\frac{-3}{2}$.

 

For example,

 

$\frac{-3}{2}=\frac{-3\times 4}{2\times 4}=\frac{-12}{8}=\frac{-3\times 10}{2\times 10}=\frac{-30}{20}$ and $\frac{-2}{3}=\frac{-2\times 4}{3\times 4}=\frac{-8}{12}=\frac{-2\times 10}{3\times 10}=\frac{-20}{30}$

 

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