 Hello and welcome to the session. In this session we will discuss representation of rational numbers on the number line. We know that the rational numbers can be represented on a number line. As you can see this is a number line. We represent positive numbers to the right side of 0 and the negative numbers to the left side of 0. And this line extends indefinitely on both sides. Now let's see how can we represent a rational number on this number line. Consider a rational number 2 upon 3. Now we need to represent this rational number on this number line. First let's see what is the denominator of this rational number. It is 3. So this means that the first unit will be divided into 3 equal parts. That is the region between the numbers 0 and 1 would be divided into 3 equal parts. Now we have not taken the numbers on the left side of 0 since this rational number is a positive rational number. So we will divide the distance between 0 and 1 into 3 equal parts. This numerator of the rational number that is 2 tells us how many parts we consider out of the total parts that we make. So this says that we consider 2 parts out of the 3 equal parts that we have marked. And that 2 on the right side of the 0 since it's a positive rational number. We write this marking that is the first marking as 1 upon 3, the second one as 2 upon 3 and we can write 1 as 3 upon 3. Now our given rational number is 2 upon 3 which is this point. So this is the point which represents the rational number 2 upon 3 on the right side of the number line. Now let's discuss about the rational numbers between given to rational numbers. We have countless rational numbers between any 2 given rational numbers. We have 2 methods to find the rational numbers between given to rational numbers. Let's see what is the first method. Consider the rational numbers 2 upon 3 and 3 upon 2. Let's try and find out the rational numbers between these 2 given rational numbers. As you can see the denominators of these 2 rational numbers are different. So first we convert them to rational numbers with the same denominators. Now this 2 upon 3 can be written as 2 multiplied by 2 upon 3 multiplied by 2. This gives 4 upon 6. Then 3 upon 2 can be written as 3 multiplied by 3 upon 2 multiplied by 3 which gives us 9 upon 6. So we get 2 rational numbers 4 upon 6 and 9 upon 6 and the denominators are same. Now let's see what are the rational numbers between these 2 rational numbers. One is 5 upon 6, then 6 upon 6 or you can say 1, then we have 7 upon 6, then we have 8 upon 6. So these are the 4 rational numbers between the given to rational numbers 4 upon 6 and 9 upon 6 or you can say 2 upon 3 and 3 upon 2. Now let's see what is the second method to find out the rational numbers between the given to rational numbers. Let's consider the same rational numbers 2 upon 3 and 3 upon 2. In this method we will use the idea of mean to find the rational numbers between the given to rational numbers. If we have that a and b are any 2 rational numbers then a plus b upon 2 is a rational number between a and b. So to find the rational numbers between given to rational numbers we first find their mean. So the mean of 2 upon 3 and 3 upon 2 that is 2 upon 3 plus 3 upon 2 divided by 2 is equal to 13 upon 6 divided by 2 that is we have 13 upon 12. So 13 upon 12 is a rational number between 2 upon 3 and 3 upon 2. Next we find out the rational number between 2 upon 3 and 13 upon 12. So for this we will find their mean that is 2 upon 3 plus 13 upon 12 the whole divided by 2 this gives us 21 upon 12 divided by 2 which gives us 21 upon 24. So we have 21 upon 24 is a rational number between 2 upon 3 and 13 upon 12. So from here we get that 13 upon 12 is a rational number between 2 upon 3 and 3 upon 2 and then 21 upon 24 is a rational number between 2 upon 3 and 13 upon 12. Thus we get 2 upon 3 21 upon 24 13 upon 12 and 3 upon 2. So we have found out these two rational numbers between the given two rational numbers. So in this way we can find out as many rational numbers as we want between any two given rational numbers and we know that there are countless rational numbers between the given two rational numbers. This completes this session. Hope you have understood how we represent rational numbers on number lines and how we find the rational numbers between given two rational numbers.