Question

Find 12 rational numbers between -1 and 2.

Answer 1

Hint: Define the rational numbers first and try to find out any fractions in the form of \[\dfrac{p}{q}\], where $q$ is not equal to 0, that are considered as rational numbers. So, between -1 and 2 there are an infinite number of rational numbers. Choose any out of them.

Complete step-by-step solution -
We have been asked in the question to find twelve rational numbers that lie between -1 and 2. So, before we start with that, we need to understand what rational numbers are. In mathematics, a rational number is defined as a number that can be expressed as the quotient or a fraction $\dfrac{p}{q}$ of two integers, where numerator is $p$ and the denominator $q$ is non-zero. Since $q$ may be equal to 1, every integer is a rational number. The set of all rational numbers are often referred to as ‘the rationals’. The field of rational numbers is usually denoted by a bold Q. It was thus denoted by Giuseppe Peano in 1895, after the Italian word, ‘quoziente’ meaning quotient. For example: $\dfrac{7}{3},2,1,\dfrac{5}{7}$ etc. The decimal expansion of a rational number either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over again. Moreover, any repeating or terminating decimal represents a rational number. For example: 0.25,0.5,0.8333 etc.
As we have been asked to find 12 rational numbers between -1 and 2, thus the numbers that can be considered as any of those 12 rational numbers could be, $\dfrac{-1}{9},\dfrac{-1}{8},\dfrac{-1}{7},\dfrac{-1}{6},\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},0,\dfrac{1}{6},1,\dfrac{3}{2},\dfrac{5}{2}$as they lie between -1 and 2 as a fraction in the form of $\dfrac{p}{q}$ where $q$ is not equal to 0.
Hence, the 12 rational numbers between -1 and 2 are, $\dfrac{-1}{6},\dfrac{-1}{7},\dfrac{-1}{8},\dfrac{-1}{9},0,\dfrac{1}{6},\dfrac{1}{4},\dfrac{1}{3},\dfrac{1}{2},1,\dfrac{3}{2},\dfrac{5}{2}$.

Note: Generally, when students solve questions based on rational numbers from books, they get confused when their answers do not match with the answer given in the book. This is because there are an infinite number of rational numbers between any two numbers on a number line. So, they can choose any numbers from amongst them.