Question

How many fractions lie between $0$ and $1$ ?

Answer 1

Hint: In this question one thing is sure that whatever be the number of fractions between $0$ and $1$ , the integer in numerator part will always be less than the integer in denominator part because it lies between $0$ and $1$ . Also, we know that there are an infinite number of integers we have.

Complete step by step solution:
In the given question,
Between any two whole numbers there is a fraction. Between $0$ and $1$ there is $\dfrac{1}{2}$ . In fact, there are infinitely many fractions between any two whole numbers.
Between $0$ and $1$ there are also $\dfrac{1}{3},\,\,\dfrac{1}{4},\,\,\dfrac{1}{5}$ and any number that can be written as $\dfrac{1}{n}$ where n is some whole number.
In addition, there are fractions like $\dfrac{2}{3},\,\dfrac{3}{4},\,\dfrac{4}{5}$, and so on. More generally, if m and n are positive whole numbers and m is smaller than n, then $\dfrac{m}{n}$ is a fraction that lies between $0$ and $1$ and we know that there are infinitely many values of m and n.
Therefore, there are infinite fractions between $0$ and $1$ .
In a similar way, there are also infinitely many fractions in between any other pair of whole numbers.
But we can go even further than this. If you give me any two rational numbers x and y on the number line, then no matter how close together they are, we can always find infinitely many other rational numbers that lie between them.

Note: Whatever two rational numbers x and y you pick on the number line, we can always find infinitely many other rational numbers that lie between them. Also, any sum or product of two rational numbers is always itself a rational number, since when you add or multiply two fractions you always get another fraction.