 Hi and welcome to the session. Today we will learn about comparison of rational numbers. First of all let us see how to compare two positive rational numbers. Suppose we are given two rational numbers 1 upon 3 and 2 upon 7 and we need to compare these two rational numbers. For this first of all we will find out the LCM of the denominators of both the rational numbers that is 3 and 7. So LCM of 3 and 7 is equal to 21. Now we will find the equivalent rational numbers of these both rational numbers such that the denominator of these both rational numbers is 21. So to get the equivalent rational number of 1 by 3 we will multiply the numerator of the rational number 1 by 3 by 7 to get the denominator as 21. So we have 1 into 7 upon 3 into 7 which is equal to 7 upon 21. Now let's find out the equivalent rational number of 2 by 7. So let us multiply the numerator and denominator of 2 by 7 by 3. So we get 2 into 3 upon 7 into 3 which is equal to 6 upon 21. Now we will compare 7 upon 21 and 6 upon 21. Here it is very clear that the numerator 7 is greater than numerator 6. That means 7 upon 21 is greater than 6 upon 21 and this implies that 1 upon 3 is greater than 2 upon 7. So this is how we compare two positive rational numbers. Now let's see how to compare two negative rational numbers. Suppose we have two negative rational numbers minus 1 by 3 and 2 by minus 7 and we need to compare these two rational numbers. First of all let us express these two rational numbers with positive denominator. So here we need to compare minus 1 by 3 and minus 2 by 7. Now to compare two negative rational numbers we will compare them ignoring their negative sign and then we will reverse the order. So here we will compare 1 by 3 and 2 by 7. We have already compared 1 by 3 and 2 by 7 in the previous case and we get that 1 by 3 is greater than 2 by 7. So here we have 1 by 3 is greater than 2 by 7. Now let us consider their negative signs. So minus 1 by 3 and minus 2 by 7 for this we will just reverse the order. So that means minus 1 by 3 is less than minus 2 by 7. Next let us see how to compare a negative and a positive rational number. Suppose we have a negative rational number minus 1 by 3 and a positive rational number 2 by 7. Now we know that every positive rational number is greater than 0 and every negative rational number is less than 0. So a negative rational number will always be less than positive rational number. So from this we can say that minus 1 by 3 will be less than 2 by 7 as minus 1 by 3 is a negative rational number and 2 by 7 is a positive rational number. Now suppose we are given two rational numbers minus 1 by minus 3 and minus 2 by minus 7 and we need to compare these two rational numbers. Then first of all we will express these two rational numbers in their standard form. So minus 1 by minus 3 will be equal to 1 by 3 and minus 2 by minus 7 will be equal to 2 by 7. So now we need to compare 1 by 3 and 2 by 7. Now these two rational numbers are positive rational numbers and we already know how to compare two positive rational numbers. Thus to compare rational numbers first of all write them in their standard form and then compare them. Our next topic is rational numbers between two rational numbers. There are unlimited number of rational numbers between two rational numbers. Let's take an example. Let us find the rational numbers between minus 1 upon 5 and 2 upon 3. For this first of all we will express these two rational numbers with same denominator. So let us express these two rational numbers with denominator 15. So minus 1 upon 5 can be written as minus 1 into 3 upon 5 into 3 which will be equal to minus 3 upon 15. And 2 upon 3 can be written as 2 into 5 upon 3 into 5 which is equal to 10 upon 15. Now we need to find rational numbers between minus 3 upon 15 and 10 upon 15. So rational numbers between minus 3 upon 15 and 10 upon 15 will be minus 2 upon 15 minus 1 upon 15 0 1 upon 15 2 upon 15 and so on up to 9 upon 15. Thus these all are the rational numbers between minus 1 upon 5 and 2 upon 3. Now let us find few more rational numbers between minus 1 upon 5 and 2 upon 3. Now we will express these two rational numbers with denominator 30. So minus 1 upon 5 can be written as minus 1 into 6 upon 5 into 6. This will be equal to minus 6 upon 30 and 2 upon 3 can be written as 2 into 10 upon 3 into 10 which is equal to 20 upon 30. So now we need to find rational numbers between minus 6 upon 30 and 20 upon 30. So rational numbers between minus 6 upon 30 and 20 upon 30 will be minus 5 upon 30 minus 4 upon 30 minus 3 upon 30 and so on up to 18 upon 30 and 19 upon 30. So these are the rational numbers between minus 1 upon 5 and 2 upon 3. Now we can find more rational numbers between these two rational numbers by taking the denominator as 45, 60 and so on. So from this discussion we can see that we can get unlimited rational numbers between any two rational numbers. With this we finish this session. Hopefully you must have enjoyed it. But I take care and keep smiling.