 Hey friends, welcome again to another session on number system and in this topic We are going to discuss finding an irrational number between two numbers. Let us say if there are two numbers given a and b and You are asked to find out an irrational number Between them then how to go about it So if you recall what did we do when a if you had to find out a rational number between two numbers What am I saying a rational number between two numbers? Then we knew that a rational number rational number Rational number between between a and b a rational number between a and b is nothing but the average or arithmetic mean of them So a plus b by 2 So is a rational number lying between a and b and b if a and b are not equal Okay, so this is for the rational number. What about rational number? So similarly an Irrational number an irrational number Irrational number lying between an irrational number lying between a and b is given by root of a and b a b if a b is not a perfect Perfect square Okay, so if not a perfect if a b is not a perfect square then root of a b is a Irrational number lying between a and b let us take few examples. So let us say I have to find out find out question is find find one Irrational number Irrational number between 2 and 3 2 and 3 so clearly what is it? It is nothing but under root 2 into 3. So is equal to root of 6 Okay, so root 6 is a Is an irrational number lying between 2 and 3 another example. Let us say I have to find out irrational between 2 and 2.5 2 and 2.5 any rational number between 2 and 2.5 will be nothing but 2 into 2.5 which is root of 5 Right root of 5. So and actually if you use a calculator, you will see that root of 5 is between 2 and 2.5 another example, let us say I have root of 3 and root of let us say 5 I have to find out an irrational number between root 3 and root 5 how to do it Nothing but under root multiply both root 3 into root 5 So hence it is under root root 15 Right root 15, which is nothing but 15 to the power 1 by 4 so fourth root of 15 is between root 3 and root 5 Right, so let us say another example if we have 1 by 2 and 2 by 3 So what do you need to do simply? a number Irrational between rational number between 1 by 2 and 2 by 3 is under root 1 by 2 into 2 by 3 is equal to under root 1 by 3 Okay, so all these are examples of irrational number between two numbers a and b if a and b are If a and b is perfect square then what happens then if a into b is perfect square then root of a b will be another Irrational number then you repeat this process of finding The irrational number, what do I mean? I say let us say a b is equal to let us say c square. Okay, a b is a perfect square. So root of a b will be c Right, and we are dealing when a b is a when a b is a perfect square a b is a perfect Square so a b can be expressed as c square. So root a b is equal to c. So hence now an irrational number Irrational number between between a and b will be nothing but an irrational number Irrational number between a and c is it it Right, so on a number line if I have to represent if this is a this is let us say b And somewhere this is c. So if you if you find I find out a rational number here Which is nothing but root of ac then your job is done. Yeah So this is all about how to find out any rational number between two given numbers And why is it working? By the way is simply this let us say a is less than b So clearly a square is less than b square Also, if you see a is less than b that means a b is less than b square Multiplying both sides by b you'll get a b is less than b square similarly when a is less than b Then a square is less than a b Multiplying both sides by a right. So hence Uh, and we are considering here a to b a and b to be positive, right? So hence if you if you add this inequality or let us say from inequality one and two you'll get what a square is less than a b is less than b square hence taking square root all across you'll get a is less than root a b is less than b So hence if a b is not a perfect square, then definitely root a b is an irrational number between Irrational number between irrational number between a and b Okay, let us say if this method doesn't appeal you then you can always have this methodology Let us say you had to find out An irrational number between two and three. So, you know, what is an irrational number? It must have Non-terminating Non-terminating as well as non repeating Non repeating decimal Decimal decimal representation Decimal representation Isn't it? So hence you can always say This type of number will be here rational between two and three which type of number. Let us say 2.01 0 0 1 0 0 0 1 Then 4 0 1 Then 5 0 1 Right 5 0 1 then 6 0 1 And so on and so forth. Isn't it? This is a if you see this is non-terminating non repeating decimal representation So this is definitely greater than 2 And definitely less than 3 Okay, this is another way of finding an irrational number between 2 and 3 you can have done any other Such arrangement like 2 point 3 1 3 3 1 3 3 3 1 3 3 3 3 1 and so on and so forth. Yeah, again non-terminating Non repeating so it is more than 2 but less than 3 This is another way of finding any rational number between 2 numbers Now if they say find more than one irrational number between 2 numbers, then you know what to do. You can repeat the process So let us say if I have to find out 2 between a and b. So I'm finding out two irrational numbers Numbers Between A and b so you first find out c so a B root root ab will be the first number And second number will be nothing but under root a into root ab Likewise, and you can repeat the process. Isn't it example? So if 2 and 3 I have to find out Two irrational number between 2 and 3. So what do I do first is nothing but 2 into 3 which is root 6 Then I find out either one between these two or between these two Right. So the next number will be nothing but 2 into root 6 under root So which is nothing but if you simplify, this is 4 6 a 24 root 24 is another such number then again if I want one more So you can use these these two these two and you can say under root 3 root 6 is another such number which is nothing but 9 6 of 54 So if root 54, so if you see root 24 root 6 and root 54 or all Less than I'm sorry greater than 2 but less than I'm sorry Yeah, yes, so this is nothing but oh, this is nothing but root of root of 54. I'm sorry This is root of root of 54 And this is root of root of 24 right now. So hence if you see this is nothing but Yes, root of 24, which is nothing but 24 fourth power This is 54 fourth power. Is it it so hence 2 is less than 24 fourth power fourth root Then root 6 then 54 fourth power Like that. Yeah, is less than 3 okay So this is what It is all about finding irrational numbers between two given numbers