 Hello friends, I Welcome you all in this session on number system and this particular session is dedicated towards finding a Rational numbers between two given rational numbers. So you have learned about rational numbers We have given you the definition and the criteria of rationality Now in this session, we are going to understand if two rational numbers are given let us say one and Let us say five. So let us say I have one rational number one and Another rational number five both are rational numbers. Yes, they are why because both are of the form of p by q One can be written as one upon one q is not equal to zero and gcd of P and q is one. Is it that these are the three criteria for rationality a B and C, sorry. He's not equal to zero now. So one and five it is cake work for you You can always say two three four are in between 2.1 3.1 4.1 are also in between or 2.00579 is also in between one and five. So you can actually find infinite number of National numbers between two given national numbers, isn't it? But what is the formal way of? Doing it is there is some this is this is because you know that these are you know, you add a small Small number to the previous number you get a number between the two given numbers So is there in a formal way any formal way? So let us say if you have to find out One number between x and y. Okay. So idea is question is finding finding a Rational number so you have to find a rational number rational number between x and Y Let us say that number is Small q Okay, or for you know just to avoid confusion. Let us say that number is z So z is a rational number which is greater than x and less than y Okay, how to find such a number the idea is very simple. You know that The mean or if you if you don't know it so mean of two numbers always lie between the two numbers That means mean is nothing but average If you have studied about average so average of x plus y or sorry average of x and y is between is between x and y Always yeah, so what is the average of two numbers average of two numbers is so z can be written as x plus y by 2 right, so if you see This is a universal thing x is always less than x plus y by 2 and X plus y by 2 is less than Y right so hence if you have two numbers you just need to do You just need to find out x plus y by 2 and you get the You get one number between the two given numbers. Let us say I have been given as x is 3 and Let us say y is 7 So hence x plus y by 2 is nothing but 3 plus 7 by 2 which is 10 upon 2 which is nothing but 5 so hence 5 5 lies between 3 and 7 Let us take another example Another example, let us say I have to find out find One rational number One rational number Between Two and let us say 4.5. These are the two numbers given So what will you the solution so simply you add 2 plus 4.5? by 2 which is 6.5 by 2 or which is 3.25 so clearly 3.25 is between 2 and 4.5 another example could be if let's say they have given not in decimal form, but in fraction. Let us say find Now find a rational number between 1 by 2 and let us say 2 by 3 Okay, so what will be it so first first of all you must also ascertain that You know one is lesser another is greater any which way if you have two numbers one will be lesser than the other Now how to find this so now that number is nothing but 1 by 2 plus 2 by 3 sorry 2 by 3 upon 2 so you know how to calculate this this is nothing but LCM is 6 so if you take the LCM in the numerator it is 6 so 3 plus 4 by 2 which is nothing but 7 upon 12 Okay, so 7 upon 12 lies between half and 2 by 3 this is one one way of doing it the The process would not change if one of the numbers is my negative. Let us say find or both are negative find one rational number number Between Between minus 7 and minus 1.5 Okay, so you now know what is the process you just simply add them and Divide by 2 So how much is it? It is nothing but minus 8.5 by 2 which is nothing but minus 4.25 Which is which lies between the two numbers did you understand so hence if you have to find only one The idea is very simple add the two numbers and find out The average then you get the number between the two numbers now. Let us say the question is something like this find find two rational numbers two rational numbers between between Let us say one and 2.5 many times you will also see it is written as insert insert two rational numbers Insert two rational numbers. This is the language of the question between Between 1 and 2.5 in both the cases. It's only in the matter of language. Otherwise everything is same So how to do this again not to rational number So you now know you can find out the first rational number first first numbers will be nothing but simply 1 plus 2.5 by 2 so which is 3.5 by 2 which is 1.75 Okay, if you simplify 3.5 by 2 you'll get 1.75 now this next number could be you take any of the given numbers Let us say one and take the new number the first found out number as a second number in the case second case So second case second number will be you can simply pick any of these numbers Let us say I am taking one So out of one and 2.5. I took one and then the next number instead of 2.5 you take 1.75 Right, and then repeat the process. So what is it? It is nothing but 2.75 by 2 which is nothing but 1.3 7 5 Okay, so hence 1.3 7 5 and 1.75 lie between between Which numbers 1 and 2.5 so you can go on Repeating this process So you take the first number and the newly found number or any of the two given numbers or any of the you know, whatever You could have also done this you could have taken 2.5 and 1.75 and then you would have divided by divided some by two you would get another such number Right, so hence you can continue this process Let us say the question said instead of to find three know the process One is you repeat the process three times and you get Right, so what will you do? You will first find out The first number is any ways 1.75 then second number is 1.3 75 Then you can always take one and one point newly found number 1.3 75 by 2 You'll get another number likewise you can do it, right? So this is one way of finding as many natural rational numbers between two given numbers But let us say if you you know, is there any other method of doing it? Yes, there is so let us see the alternate alternate method alternate method What is alternate method? Let us say you have to find out again same numbers will take one and one and 2.5 you have to find out let us say find find 10 rational numbers rational numbers Now it will be too cumbersome if you go by the first method which we discussed you keep on adding You know and then dividing by 2 and all that instead of that do we have a better number? Yes a better method Yes, how method is this first of all first of all convert Convert all First of all convert all the given numbers into fraction numbers into fraction into Fraction how to do that? so it is in this case, so it is 1 and 2.5 is nothing but 25 by 10, isn't it? Correct now what so hence basically 2.5 and 2.5 1 and 2.25 by 10 now second step is equate the denominators equate the Denominators, what does it mean denominators? So somehow I want to find I want to you know, what is the denominator with one? Is one right can I have n as the denominator in case of one also? Yes, I can then write it is 10 upon 10 isn't it? Then upon 10 n the other number is 25 upon 10 Isn't it now easily if you see there are Various numbers already there which is so you can now find out lots of numbers between 10 upon 10 and 25 upon 10 so numbers like 11 upon 10 12 upon 10 13 upon 10 14 upon 10 15 upon 10 all these numbers till let us say 20 upon 10 all are all these numbers are greater than 1 that is 10 upon 10 and Less than 25 bond 25 upon 10 Isn't it? This is all you have to you can do it without much of an hassle much of an hassle now Well, now let us take another example another example is let us say insert insert five rational numbers five rational numbers between between 1 by 2 and 2 by 3 So if you see it's already Fraction form is it it so what to do first equate the denominator? How so one way is so I have to convert 2 and 3 into something common So let us say you have to basically take the LCM of 2 and 3 So LCM you take of 2 and 3 is nothing but 6 so this will be the denominator So hence if 6 is the denominator in the first one So hence 1 by 2 can be written as 3 by 6 and second one can be written as what 4 by 6 isn't it so now 3 and 3 by 6 and 4 by 6 are my Numbers again, what did I do? I found out the LCM of the denominators and then keeping the denominator is constant I converted the given fractions into Another fraction right so 3 by 6 and 4 by 6 but can I insert 5 rational numbers between 3 by 6 and 4 by 6? Looks difficult. So what I do is add 1 to 5 Okay, adding 1 to 5 you'll get 6 Now you multiply 6 to the numerator and denominator in both the fractions so hence 3 by 6 can be written as 3 by 6 Right, and it is 4 by 6 into 6 by 6 Correct multiply and divide the numerator and the denominator You know multiply 6 to both numerator and denominator in both the fractions Right, so one by one in case of 3 by 6 another in case of 4 by 6 So what will you get? You'll get 18 upon 36 and you'll get 24 upon 36 now clearly you can see lots of so you know the fraction denominators are same. So the numbers are now 19 upon 36 20 upon 36 21 upon 36 22 upon 36 and 23 upon 36 Right, so hence it is important to add one add one to the number of the desired number of Rational numbers which are to be inserted right then you will get extra Values to pick from and hence you will see these are the five natural oh sorry fractions or Rational numbers between Which two values it was between 18 by 36 and 24 by 36, but 18 by 36 is as good as 3 by 6 and 24 by 36 is as good as 4 by 6 which is as good as saying between 1 by 2 and 2 by 3 if you see 18 by 36 is nothing but 1 by 2 and 24 by 36 is nothing but 2 by 3 correct So hence we could find out find out these many Rational numbers between the two given rational numbers You will get more clarity when you go through the problem solving sessions Thank you