Question

List three rational numbers between -2 and -1.

Answer 1

Hint: Keeping in mind the definition of rational numbers, we will try to find rational numbers between -2 and -1. Any rational number between -2 and -1 must be greater than -2 and smaller than -1.

Complete step-by-step answer:
Whenever we want to find rational numbers between two numbers, it is necessary that the two numbers always form a pair of like fractions. Since, the given numbers can be written as $\dfrac{-2}{1}$ and $\dfrac{-1}{1}$. So, here it is clear that the denominators of both the numbers are equal so they are like fractions.
Now, we have to find rational numbers that are less than -1 and greater than -2.
We know that a rational number is always of the form $\dfrac{p}{q}$ where p and q are integers and q is not equal to zero. So, any rational number less than -1 can be given as $-1-\dfrac{p}{q}$.
Let us take $\dfrac{p}{q}=\dfrac{1}{2}$.
So, $-1-\dfrac{1}{2}=\dfrac{-2-1}{2}=\dfrac{-3}{2}$
If, $\dfrac{p}{q}=\dfrac{1}{3}$, then:
$-1-\dfrac{1}{3}=\dfrac{-4}{3}$
Similarly, a rational number greater than -2 is given as:
$-2+\dfrac{p}{q}$
Let us take $\dfrac{p}{q}=\dfrac{1}{3}$
So, $-2+\dfrac{1}{3}=\dfrac{-6+1}{3}=\dfrac{-5}{3}$
Therefore, we have got three rational numbers between -2 and -1 as $\dfrac{-3}{2},\dfrac{-4}{3}\,and\,\dfrac{-5}{3}$.

Note: It should be noted here that any rational number less than -1 is given by $-1-\dfrac{p}{q}$ because in case of negative integers, we not only see the magnitude but also have to be careful about the sign of the number. The same concept applies when we have to find numbers greater than -2. We should also keep in mind that the greater the magnitude the greater will be the number.