 Hello friends. So welcome again to another session on number systems. So, so far we have studied Rational numbers and before that we also learned different types of numbers like natural numbers We talked about natural numbers. We talked about whole numbers We also talked about integers and Then we saw rational numbers Rational numbers isn't it Rational numbers and and we also understood one thing that all these numbers were developed one after the other and every time there was a need need of a New set of new set of numbers new set of numbers Isn't it so we knew that a smaller number cannot subtract a bigger number Hence we had The use of integers and then we also knew that two equal numbers two equal natural numbers if they are subtracted We needed a new number because Five minus five for example in natural numbers didn't have a solution But hence the moment we had set of whole numbers that is once zero was included Then we could do five minus five six minus six and so on and so forth Similarly three minus five was not defined in the set of whole numbers So then we had to formulate a new set of numbers called integers to define such operations Then we also saw that things like five upon nine seven upon eight These are not defined in terms of integers. That means Division was not allowed in integers for all given integers. So hence We figured out another set of numbers called rational numbers and here We observed two types of rational numbers whose decimal representation was either terminating So we say that decimal representation was either terminating. Isn't it if you recall? terminating as in let us say one by two was nothing but 0.5 and The other other other example or other type was non-terminating Non-terminating non-terminating but we had repeating repeating or recurring decimal representation for example one by three was given as 0.3333 so on and so forth and formally rational numbers were defined as P upon Q the numbers which can be expressed in P icon in the form of P upon Q P and Q belongs to the set of integers, right and Q was not equal to zero and then another one was GCD of P and Q GCD or else HCF of P and Q was One these were the criteria for defining rational numbers today. We are going to understand what are irrational numbers, right? So, you know What if what if the what if the decimal representation? Do not terminate as well as do not represent repeat, okay? So, let us understand such kind of number before that. Let us take a problem statement Let me say I have a I have a square plot square plot and I have square plot of unit one as side Okay, so the side length is one now. The question is I want to fence the diagonal. I Want to fence the diagonal, okay? So hence I go to a hardware shop and I ask the hardware shopkeeper to give me a wire of Wire such that I could fence the diagonal of my field Then the shopkeeper asks me to give me the length of the diagonal. Then I said, okay Let me calculate the diagonal. So hence I know that by Pythagoras theorem I have let us say this is X So hence X square will be one square plus one square that is two So there must be a number X such that when it multiplies when I multiply itself with When I multiply the number with itself, I should get two. That's what my challenge was Now I thought what could be This type of number how to find out this what is the value of X? What is the value of X which when multiplied by itself? I will get two So to answer this what I did was I started doing some trial and error. So I knew one So I multiplied one by one to get result one But one is less than two. So that means I can clearly say that one is definitely not the square root Then I took two Right two as value of X But when two is multiplied by two itself it gets it gives you four So that means the number X lies between One and two clearly why because one square is one And two square is four. So hence X square lies between one and four Now I also knew rational numbers, isn't it? So let me just try to find out What if I take X equals to 1.5 Okay, so X equals to 1.5. So X square X square X square. I'm writing it here if X is equal to 1.5. So X square will be 2.25 Okay, which is greater than two. So hence X is not 1.5 So let me try X is equal to 1.4 So X square will be 1.96 Which is less than two that means again the square lies between These two numbers. So let me take X is equal to 1.45 Now if you see X equals to 1.45 the square is 2.1025. So if you see X square is 2.1025, which is again greater than two So X equals to 1.45 is also not the solution So let me take now let me take a value which is less than slightly less than this. So let me take 1.42 So if you take X equals to 1.42 right now I have to find out the square of 1.42 So if you see X square in this case will be 2.0164 But unfortunately this is also greater than two. So we have still not been able to find out the exact square root of two. So we can keep on the journey And hence let us say 1.41. Let us try this now So if you see 1.41 square is 1.9881 my god now again it it went down Right went down to uh below two. So hence again we have the other Value of X was 1.42. This is 1.41 So let us take let me take our middle value again 1.415 So if you see 1.415 square is Beautiful if you see this is 2.00225 But again it is closer to do but still it is more than two That means The square root of X is less than A square root of two is less than 1.415. So hence what else? So let us say I am taking 1.414 Okay, so if I do that then the square root is One or sorry square is 1.9993 96 Right again unfortunately, we are not able to strike the real square root So which is again, you know, it is less than two. So if you keep on doing the process Actually, it was later on discovered that this process never ends. So however close You try to find out the square root of two You will never be able to you will never be able to find exactly Uh the number which when multiplied by itself will give you two. So A simple calculator You know square rooting will give you Root of two which is now expressed as root two root two is nothing but The value of X which when multiplied by itself will give you two Is being found out to be 1.414213562 And so on and so forth. Okay So the observation here is If you if you notice carefully None of the digits are repeating. There's no pattern. Right. So what do we conclude? There is no pattern or repetition No pattern No repetition, isn't it? No repetition So this is the this is a new new thing which we have now discovered Still so far we were seeing there were two types of digital symbol representation of any number One was so what was it? One was uh terminating the symbols, right terminating terminating the symbols and then the other one was The other one was non terminating. So it was not terminating for sure non terminating but it was Repeating the digits used to repeat after a point in after point for example 1.313131 like that So on and so forth right 1.3333 it is nothing but if you see this is Um, so this goes on isn't it? So this is nothing but 4 upon 3 Right 4.3 is like that. But now we are getting a new case which is non terminating non terminating as well as as well as As well as non repeating non repeating Non repeating so these kind of numbers were later on termed as irrational So this is these time these kind of numbers whose decimal representation are non terminating and non repeating Are called irrational numbers Irrational numbers why irrational because During that point in history when people were trying people were trying to find out root 2 They could never find a number which when multiplied by itself would get you too So people were little, you know startled. They were confused that how can there be a number They are not able to find a number which when multiplied to itself will fetch you too So it was it was appearing to be a unreasonable thing to them Hence they called it irrational Numbers call them irrational numbers. So they were not trying they were not able to find out To a reasonable extent how you know how to calculate such numbers So hence the the term given to those type of numbers was irrational So there was no reason why such numbers existed. So hence the term irrational numbers Okay, another examples of examples of irrational numbers are if you take a prime and take a square root of it You will get any rational number like root 3 root 4 and anything sorry not root 4. I'm sorry root 5 and anything which is not a perfect square like root 6 root 8 root 10 all are All are irrational numbers all are irrational numbers If you now try to find out one number which when multiplied to itself would which fetch you 3 5 6 8 and 10 etc You will never find one right. So as such numbers are called irrational numbers Okay, and another definition or in other words, you can call irrational number like this. So those those numbers numbers numbers Which are which are real. So we will we'll discuss in in some time. What are real numbers? So Numbers which are real but not rational but not rational are called Irrational Understood number which are not real numbers which are not rational numbers are called irrational numbers Irrational numbers right. So now we define another set of numbers which is called real numbers. So real numbers are Real numbers are Numbers numbers Which are either Which are either rational rational or Irrational so set of set of real numbers is denoted by R And r Is nothing but set is a sum of set of this is equal to i that is irrational plus q which is Rational right. This is our definition of real numbers