The number system is the backbone of mathematics. Numbers are used in many different contexts and in many ways. In the number system, numbers are classified into whole numbers, odd and even numbers, negative and positive integers, rational and irrational numbers, real numbers, etc.
Again, real numbers are classified into rational numbers and irrational numbers. As of now, we will only discuss rational numbers.
Let us consider a small example to understand the concept of rational numbers.
John wants to buy five books, fourteen rupees each. His friend Joel wants to buy two similar books. So, they went to the stationery store. The shopkeeper said that a pack of five books cost ₹250. How much does each book cost? We can easily calculate the cost of each book by₹250/5.
Therefore, the numbers which are expressed in fractions, i.e., in the form of p/q, where p and q are integers and q=0 are called rational numbers.
Rational numbers are denoted by Q.
These are also called quotient numbers.
We can express any natural number like 10 as 10/1or20/2,………
We can express any whole number like 0 as 0/1or0/2,………
We can express any integers like – 4 as-4/1 or-8/2,...
So, we conclude that all natural numbers, whole numbers, and integers are rational numbers.
Hence, from the above discussion, we have understood the concept of rational numbers. Let's start finding the rational numbers between two rational numbers, which is our main concept.
The natural numbers between 7 and 2 are 6, 5, 4, and 3.
Let me ask you something. Are there any natural numbers between 4 and 5?
Now, the integers between – 2 and 4 are – 1, 0, 1, 2, and 3.
But, can we write the integers between – 6 and – 7?
The answer is that we cannot find any natural numbers or integers between two successive natural numbers or integers.
But, we can write rational numbers between any two successive integers. Wondering how? Let us explore.
Now, we will try to write the rational numbers between 4 and 5.
If a and b are any two rational numbers, thena+b/2 (mean of a and b) is also a rational number between them.
Therefore, 4+5/2=9/2is a rational number that lies between 4 and 5.
Thus, 4<9/2<5
Now, the rational numbers between 4 and 9/2 are
Thus,
In this way, we can go on finding numbers. In fact, there are infinite rational numbers between any two rational numbers.
Now, let’s learn a simple and easy way of finding rational numbers between any two rational numbers.
It's just that simple. In this way, we can easily find the required number of rational numbers.
NOTE:
1. It is easy to find rational numbers between two rational numbers with the same denominators.
EXAMPLE: Write any three rational numbers between 2/8and 9/8.
SOLUTION:3/8,4/8,5/8 are any three rational numbers between 2/8 and 9/8.
2. But in the case of different denominators, simply find the LCM of the denominators and equate them. After that process, apply the previous method.
EXAMPLE: Write any four rational between 2/7 and 4/3
SOLUTION: Since the denominators are different, let us find the LCM of 7 and 3, which is equal to 21. Now, equate the denominators such that the rational numbers will be 6/21 and 28/21.
Now, rational numbers between 2/7 and 4/3 are 7/21,8/21,9/21,10/21,…..26/21,27/21.
We can take any four rational numbers from the above rational numbers.