Question

Find five rational numbers between -1 and 1.

Answer 1

Hint: The given question is related to rational numbers and their representation on the number line. Using the definition of a rational number and equivalent fraction of -1 and 1, find the rational numbers between -1 and 1.

Complete step-by-step answer:

Before moving to the options, let us talk about the definitions of rational numbers followed by irrational numbers. Rational numbers are those real numbers that can be written in the form of ${\displaystyle \frac{p}{q}}$ such that both p and q are integers and $q\ne 0$ . In other words, we can say that the numbers which are either terminating or recurring when converted to decimal form are called rational numbers. All the integers fall under this category.

Now, moving to irrational numbers. Those real numbers which are non-terminating and non-recurring are termed as irrational numbers. The roots of the numbers which are not perfect squares fall under the category of irrational numbers. $\pi \text{ and }e$ are also the standard examples of irrational numbers.

Now, coming to the question, we have to find five rational numbers between -1 and 1. We know, we can write -1 as ${\displaystyle \frac{-1}{1}}$ and 1 as ${\displaystyle \frac{1}{1}}$ .

Now, we will multiply 5 in the numerator and denominator of both the fractions. So, we can write -1 as ${\displaystyle \frac{-1\times 5}{1\times 5}}={\displaystyle \frac{-5}{5}}$ . Also, we can write 1 as ${\displaystyle \frac{1\times 5}{1\times 5}}={\displaystyle \frac{5}{5}}$ . Now, we can easily find five rational numbers between ${\displaystyle \frac{-5}{5}}$ and ${\displaystyle \frac{5}{5}}$ . The five rational numbers between ${\displaystyle \frac{-5}{5}}$ and ${\displaystyle \frac{5}{5}}$ are ${\displaystyle \frac{-3}{5}},{\displaystyle \frac{-1}{5}},{\displaystyle \frac{1}{5}},{\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{4}{5}}$ .

Note: By using the above stated method, we can find infinite numbers of rational numbers between any two rational numbers. Instead of multiplying by 5, we can multiply by any other number as per the number of rational numbers required between the two numbers.