 So let me welcome you to this course on basic analysis and this is called basic real analysis. So what is real and what is analysis? Let us understand the name of the course at least. So real because it concerns with the real numbers and analysis on the set of real numbers. So what does this mean? So we will try to understand what is the set of real numbers, what are the properties or threats of real numbers. So that is the object on which we will be doing our course. And then analysis basically means analyzing various aspects of real numbers and then functions on real numbers. You will find this is the sort of basic trend in most of the courses in mathematics as well as statistics. You have a basic object, you study that basic object, properties of the object and then you look at functions on that object. So that is a normal sort of way mathematics progresses and different topics progress. So what are real numbers? If I ask you that question, probably you may find it difficult to answer that. But think about what is the real number? Why did mathematicians invent real numbers? What is the need for that? When we start our journey in mathematics as childhood, we start counting. We get familiar with natural numbers 1, 2, 3, 4, so on. We come to next stage integers. That is because one wants to look at solve equations n plus m is equal to some k and that may not be always solved for a given n and m of k or from evolution point of view that might have been because of writing debt kind of thing. You borrowed some money, you returned some money, how much is left kind of a thing. So keeping account of those kind of things. So negative numbers must have origin, must have originating. And then 0 came much later, historically. Because when you say I have nothing, why should you write 0 for that? Why there should be a symbol for nothing? You can just write nothing. But it helps to have a symbol for that and gives us the credit that we discovered 0. Indians discovered 0. So let us take credit for that. Integers and then came evolution and then human beings evolved. One wanted to have fractions. Then you want to share things. Could have been anything like an old man is about to die, he has got 5 kids, wants to distribute his property, may be a piece of land. So how will he distribute? How much everybody gets kind of a fraction came into picture. So for a long time fractions were good enough. Till historically around 300 years before Christ, the Greek mathematicians, Greeks were the ones who were doing lot of mathematics before Christ, 300 years, Pythagoras, Euclid and so on. So one of the persons, they believed that any given any two magnitudes, so for them number it did not exist, it was only magnitudes, like 5 kg of something, 5 did not exist for them, it was 5 of something kind of magnitude. And they believed that given any two magnitudes, they are commensurable. What does commensurable mean? Means one is a multiple of the other. And two magnitudes, always one is a multiple of the other. And that in the modern language if you write, if A is one magnitude, B is another, then A by B is a number. That means what does that mean? A equal to some NB. So A by B is a number, that means it is a, what does it mean? It is a rational number. So they believed only in rational numbers. They did not believe that there is something beyond rational also. They thought every length should be a magnitude of something. And they were fascinated by geometry. So they wanted to represent numbers by geometric objects. So their idea was that we should be able to represent numbers on a geometric object that is line. So what they do? So they took the horizontal line and put a point. So I was saying that they believed that every number is a geometric object. So they took a horizontal line and marked a point called 0 and equal distance points and so on. Right? So all integers were, they were able to assign to every integer a point, a position on the horizontal line. So geometrically we were able to say that this point represents the magnitude 0. This represents 1. And similarly they were able to assign say for example this, right? So 3 by 2 for example, half of midpoint of that. So that way every fraction P by Q they were able to put on the line, horizontal line. And they were very happy and they thought that there is a 1 to 1 correspondence. I am slightly making it mathematical between the points on the line and the set of rational numbers, meaning what? Meaning that every rational number gets occupied by a point on the line, so geometric object and every geometric object that is a point on the line is represented by rational number. So that went on for quite some time till one person, I forget his name Eurosis I think he discovered that this is not always the case. And it is a very simple example that he gave. So let us take this line, this is 0, this is 1 and let us construct a square on this of length 1, right? And let us look at what is the length of the magnitude, that is the length, right? Geometric object. So if you like your, so that gives you a point. So OP, so OP is equal to OA, right? So he said that this point P cannot be represented by a fraction that was his claim, right? Because by Pythagoras theorem you know that this OA square that is OP square must be equal to 2 because this length is 1, this length is 1, so Pythagoras theorem Pythagoras mathematics was available to them. So they said this must be equal to 2. So mathematically this leads to a question, it led to a question, can I find a number x belonging to rationals? Say that x square is equal to 2, right? They only thought of geometric problem that whether this point on the horizontal line is representable by a fraction or not, right? So now mathematically if you can frame this as a question, does there exist a number x which is a rational number and 1 square is equal to 2? And most of you have gone through a proof of that, this is not so, right? So there does not exist x belonging to Q such that x square is equal to 2, right? Probably it is very interesting to go through a proof of this which is normally found in textbooks. So the proof is by contradiction, proof is by contradiction. So that means what? So assume there exists x equal to p by Q such that x square is equal to 2, right? So this is, I am just quoting the proof that most of the textbooks have, I think the most famous book on analysis is mathematical analysis by Rodin, principles of mathematical analysis. And you will find a proof of that or if you look at NCRT standard 10th textbook, you will find a proof there also of school level. So that means, so this implies p by Q square is equal to 2. So that implies p square is equal to 2 of Q square, right? So that implies what? It implies that p square is even, right? Because there is a multiple. So we assume proof is by contradiction. That is another technique given by the Greek mathematicians the proof by contradiction, they were the first one to own. So it says p square is even and that says p is even. I am quoting the proof word by word either given in Rodin or in textbooks of school level. So implies p is equal to 2 K implies, so put it back here. So 2 K square is equal to 2 Q square. So that implies 4 K square is equal to 2 Q square and that implies 2 K square is equal to Q square. So that obviously implies Q square is even and that implies Q is even. So here are two things namely p is even, so here is 1 and here is Q is even. So that means 2 must be a factor of p and Q both but we could have been a bit smarter, could have started where p and Q have nothing in common, right? Even common factors are there in p by Q would have cancelled it, right? In the beginning. So that is called an equivalent form of a rational number. So that would have, this would lead to a contradiction. So that proves that, that gives a proof. Now what is my worry is, so how does this happen? If p square is even, why should be p even? That fact is not supported in school books at all. I do not think that is supported even in textbooks, in mathematics, even higher level. That means a proof and that is not proved in any one of the classes, right? It can be proved very easily, right? One line says if you take a odd number, if square is always odd. That is all. A one line has to be added in the bracket because square of an odd number is, but nobody bothers to mention that fact. And school kids and even teachers, they assume, they remember this, everybody remembers this proof. Nobody tries to understand this proof. So that is an unfortunate thing, right? So one could add a line and be happy about it, okay. I want to give you an alternate proof of this. Proof is same essentially, but it is interesting. So let us assume, so here is proof to assume p by Q square by Q square by Q square by Q square is equal to 2 and that implies p square is equal to 2 Q square, right? Now what I am going to do is, 2 is a prime, right? And 2 occurs on the right hand side. So if I look at the prime decomposition of the number 2 Q square right hand side, every number can be written as a product of primes. So look at the prime factorization of the number 2 Q square. How many times the number 2 can appear in that? 2 is already there. For Q square if a prime occurs in the prime factorization of Q square, it must occur even number of times, right? So total number of times 2 can appear as a prime in the prime factorization of right hand side is odd because 2 is already sitting there. So look at the left hand side, how many times 2 can appear? Contradiction, proof is over. If I look at the prime factorization, the prime appearing in the prime factorization of left hand side, it can appear even number of times. In the right hand side, it should appear in odd number of times. So that is a contradiction. That is all, right? So look at 2 in the prime that leads to a contradiction. Proof is over. You do not have to do anything. And advantage of this, that is how mathematics progress is. In this, can I replace 2 by 3? Same proof. If I replace 2 by 3 and repeat the arguments, what does it prove? It will prove 3 is not, there is no fraction P by Q whose square is equal to 3, y3. I can put any prime number. There is no rational whose square is a prime. Same proof works without any change at all. Only I have to replace 2 by 3, 3 by P. However, y prime, I can put P by Q square cannot be, whole square cannot be a number 5 which is a perfect square. That will also work, right? So that is the advantage anyway. So this is what, so Q has a problem, right? The rational numbers and the interesting thing is the person who discovered this in the Greek mathematics around 200 years before Christ probably, you know what was the reward he got? He was put on a boat without food, without any help and left in the deep sea to die because the Greek philosophers did not want to accept that they have a problem in their mathematics. They wanted to hush up this discovery and as a result, discovery of this fact was delayed by 2000 years approximately. So it was only in 1878 and by a mathematician called Richard Dedekin and a mathematician called George Kenter in 1871, they said rational numbers are not complete in the sense that much kind of equations do not have solutions. So we should discover, we should add more numbers to rational numbers so that it becomes complete. So that construction is a non-trivial construction, both did independently. Both approaches are different but both lead to a common object which is called a complete ordered field. So nowadays either it is done in undergraduate course, the construction of real number system or it is just assumed. So because we have a prescribed labels to finish, so we will start with that we are given the object called real number system. We will say what are the properties given of it and go ahead with it. So let us start with the real number system. So this is the first part of our course. So real number system it is a set first of all. What is that set? So real number is a set. What are the objects in that set we do not specify because you actually have to construct them using rational numbers and the process is a bit long. So we assume that we have a set called the set of real numbers and is denoted by this funny symbol. Normally you will find such symbols quite common. This is called script R. So that is real numbers. It has two operations on it. It is a set with two binary operations. One is addition, other is multiplication. So here are the algebraic properties. There is operation of addition. There is operation of multiplication with the usual properties and what are the usual properties. So let me just start. I think all of you know these properties basically saying addition is commutative, addition is associative, multiplication is commutative, multiplication is associative. How does addition and multiplication interact with each other? These are two operations on the same set. So they interact in a way that is called the distributive property. It is like saying two human beings living in the same room. How do they should interact? What are the rules for interacting? Otherwise both are independent kind of a thing. So this is the interaction between them. Then there is a unique element denoted by this symbol. There is unique element denoted by this symbol which has that this plus x is equal to x and this multiplied by x is equal to x for every x not equal to 0. These are called additive identities and multiplicative identities. This one is read as 0 and this one is read as 1. So multiplicative identity is given the name 1, additive identity is given the name 0 and more than that for every x there is another real number in that set denoted by minus x whether when you add you get 0 and similarly for multiplication x not equal to 0 x into there is a number called x minus 1 which gives you 1. So essentially it says that under addition and multiplication the two form a abelian group or a group if you know that word if you do not know forget about it then it will matter you should remember all these things. Then there is something called an order on real numbers very important. That means given two in that set of real numbers you can compare them. And the comparison says that given any two so this is the properties namely given any two numbers we should be calling as numbers at present given any two objects in that set R you can compare given x and y either x will be less than y or y will be less than x or the two will be same. This is called the law of trichotomy that given three at least one and only one of them will hold. And this is very important the second one if x and y now there is a order right there is a 0, 0 is the additive identity given any x if x is bigger than 0 y is bigger than 0 then their sum n product should be bigger than 0. So sum n product of what we call as positive objects should be positive this is the rule and once this is true you say that you have got a set R with addition with multiplication with order with all these properties is called an ordered field. Such an object is called a field in algebra right where group is there and under addition it is a abelian group under multiplication it is a abelian group the two interact distributive property that thing is called a group sorry that thing is called a field. On field there is an order and that order respects addition and multiplication both it interacts nicely in what way if x and y are bigger than 0 then x plus y is bigger than 0 x into y is bigger than 0 right. So this is called an ordered field so till now what I have said is that R is an ordered field. There is one more crucial property of this which is called the completeness property which requires a bit of more discussion to state also basically if you look at the rational numbers you can add rational numbers you can multiply rational numbers you can also compare rational numbers one rational number bigger and they all have all the properties that till now we have stated for reals. So rationales also form an ordered field what is the difference between the rationales and the reals right. So we want to describe that because that is going to be crucial for our future discussions but anyway there are some things you can try to prove as far as order is concerned these are some properties obvious properties you can try to prove right x is less than y and z is bigger than 0 then x z should be we all assume this kind of things right in our arithmetic but what I am saying is given that R is an ordered field using only the axioms of an ordered field you can prove all these properties. For example you can prove one is bigger than 0 multiplicative identity there is additive identity you can prove one should be bigger than 0 these are two objects in R and there is an order so 1 and 0 should be comparable with each other we know that they cannot be equal because multiplicative identity cannot be equal to additive. So which is bigger than what? So one can prove only using these axioms that 1 has to be bigger than 0 all this is nice very nice to prove these things so try to prove yourself we will not prove it we will assume these things okay. So here is something you can identify what are natural numbers if we are calling R as a set of real numbers then where is our familiar object natural numbers in this right. So here is the familiar object let us call a set to be inductive S is a subset of R is R we call it as a inductive set what is the property you want one should belong to that it is a non-empty set one should belong and whenever a number n belongs whenever an object n belongs right then there is one additive identity I can add 1 to n right so x plus 1 or n plus 1 for every number real number x look at x plus 1 that should also belong right. So let us we can call it as additive successor of n number x plus 1 as a additive successor of x so whenever x belongs additive successor is also inside it that is set right that is second property. So for example can you give some examples of such sets for example look at all x bigger than or equal to 1 we have got set of real numbers right it is a ordered field in with that look at all x so that x is bigger than or equal to 1 so 1 belongs to it right x plus 1 is going to be bigger than 1 so it is going to belong so that is a inductive set right but what we want is we want the smallest inductive subset of reals we have defined what is the inductive set one belongs successor belongs I want to construct a set which is inductive which is a subset of R and it should be smallest. So take the intersection of all of them given two inductive sets if you intersect that again is a inductive set because one belongs to both so one belongs to intersection if x belongs to intersection x belongs to A x belongs x plus 1 belongs to A x plus 1 belongs to B so it belongs to intersection also. So intersection of inductive sets is again a inductive set if I look at intersection of all inductive subsets of R that is also a inductive set and it is a smallest that we denote as by n that is a set we denote by n and you can see clearly here what does inductive means is a induction that is your applying what you call as mathematical induction if one belongs n belongs n plus 1 belong and the smallest with that property this is what is mathematical induction also so that also holds in our new setup and this set n one belongs to it so its successor must belong 1 plus 1 should belong right you can call that you can give a name you can give a symbol to it called as 2. So you can define 2 to be 1 plus 1 3 to be successor of 2 so one can prove that n is nothing but 1 the multiplicative identity its successor its successor its successor and so on that is all that intersection is nothing but this the smallest set and that is the definition normally we take of natural numbers. So we have identified natural numbers as part of our new setup of real numbers which in abstract object we do not know what that is but we are trying to identify familiar things once you have integers or once you have the natural numbers you can define what are integers we have addition we have 0 additive identity for every x there is minus x so collect minus x for every x in n along with 0 call that as set of integers right once you have set of integers you can have fractions the rational numbers so that is n multiplied by m inverse where n is a natural number and m is a integer and n is a natural number normally you will find in books it is written n by m where n and m are integers and m is not equal to 0 there is no need to do that you can just write n over m where n is integer m is a natural number because you never write denominator to be a negative anywhere right because fraction means what 1 by 2 means what divide 1 into 2 parts and look at one each one of them that is 1 by 2 what is 1 by minus 2 if you look at fractions as we understand it from our primary sections what is minus 2 parts of 1 can you divide so it does not so the logical kind of difficulty in understanding so take at minus 1 as an object and divide it into 2 parts so and anyway when you do arithmetic of rational numbers you always take signs on the numerator and then take denominators and LCM and whatever it is right so anyway so this is a better way of writing fractions as so integers natural numbers integers fractions we have identified these objects as part of our set R now there are things which are not part of yo for example those objects like X say that X square is equal to 2 there is no such number so those we call as a set theory compliment of Q in R we already have a bigger set now R so take compliment the numbers objects which are not rational in R they are called irrational numbers so that is our definition of irrational and one can do many things one can go to decimal representations and prove all those things the decimal representation of a rational either repeats or terminates and all those things we will not do that we will assume all this but this is logically one can start with a ordered field and do everything right.