 Hello friends, welcome to this session on real numbers and the first question we are going to answer today is, what are real numbers? So as you know, we are going to adopt a mechanism by which we will be answering one question in every session so that it becomes a good method for you to build your concepts. So in the first lecture we are going to discuss about what are real numbers. So the first question which comes to my mind the moment I encounter a phrase like this, so why are we calling real numbers real numbers? So numbers as we all know have a, you know, we have encountered numbers in every sphere of life. So whenever Virat Kohli hits a century, we know what this number is. So every time, you know, someone scores a goal, so I know, you know, let's say France defeated Brazil in a football World Cup match by let's say 4-2. So I know four goals were scored by France and two by Brazil. So everything, you know, is, you know, the numbers are helping us in our day-to-day life. And best example is our phone number, so 9-0-0 something like that. If we have a number, I know when and how to reach a person. So be it trade, be it sports, be it your score marks or whatever, you know, in your school everywhere we have numbers. And since our childhood we have been encountering different types of numbers. And the first type of number, if you recall, the first set of numbers you studied was nothing but natural numbers. If you remember natural numbers, so natural numbers are nothing but the counting numbers, counting numbers which we use like one, two, three, four, like that. So all these numbers are nothing but they are called natural numbers. But the problem was natural numbers. The problem with natural numbers was, and if you know, we represent this set by a letter N and for natural. Now the problem with this set was that if I had two equal numbers, let's say if I, when I didn't know or when the civilization, human civilization didn't know about zero, and you might be aware that it was an Indian mathematician called Ariveta who gave the world the symbol zero. So let's say before him, when people were not aware of how to represent nothingness, so this was a problem. The problem was how to find out what is two minus two, because two plus two is very easy and we know that two plus two is four, which also belongs to the same set of natural numbers. If you see, four is here. Right? But two minus two, there was no number to represent this. And then hence, humans coined a new symbol of zero and then including zero. So zero was the new symbol which was found out and then zero plus the set of natural number gave you the set of whole numbers. So whole numbers, as you now know, is a set of nothing but zero, one, two, three and so on and so forth. So the problem of two minus two was solved with the help of zero. But again, humans encountered a problem when they wanted to subtract, let's say, three from five. So till, let's say, your fourth and fifth grade when you are not taught integers, so you don't know what the answer would be. And hence, they formulated new set of numbers, they called it negative numbers and together with all the negative numbers, negative numbers like negative numbers, negative numbers were like minus one, minus two, minus three and so on and so forth, together with negative numbers and the whole numbers, a new set of numbers were formed and they were called the integers represented by z. So z was nothing but, if you see, lots of negative numbers minus three, minus two, minus one, zero, one, two, three and so on and so forth. Now again, the problem of subtracting a bigger number from a smaller number was solved by discovery of, let's say, integers, but what about division? So for example, 10 upon two was five. So everyone knew the answer, but the moment it was 10 upon three, there was no answer in the set of integers, isn't it? So hence what we had to do, we had to invent a new set of numbers and I'll write it here. So the new set of numbers were rational numbers. So we invented or discovered whatever word, whichever word you want to use, rational numbers. So you all know the rational numbers in your ninth grade, you might have studied that rational numbers are of the form of p by q, where p and q are integers. So p and q are integers and one vital information is q shouldn't be zero because division by zero is not allowed in mathematics and p and q are co-primes, p and q are co-primes. Co-primes are a pair of numbers whose highest or greatest common divisor is one. So these are the three criteria of defining rational numbers. So this was also done. But then again, humanity again hit a dead end when they were trying to find out solution to things like what is root of two? So what should be multiplied with itself to get two? Again there was no solution to such problems and hence because of this another set of numbers were discovered and they were called irrational numbers, irrational, irrational numbers. So those numbers which were not rational were called irrational numbers. So if I have to show it in a set notation or a Venn diagram form, which you will study anyways in tenth, sorry, eleventh grade, you will see that this is how we can represent, let's say all the numbers. So you can do this. So this is, let's say my, sorry, yeah, so what I'll do is, yeah. So let's say this is my set of natural numbers. So if you see this is set of natural numbers added to it is zero and you add to it a zero and you'll get a set of bigger set of numbers and we can call it as whole numbers, right? So a zero was added to this. Yeah, this is whole number. Now add it, add to it a set of all negative numbers, negative numbers and you will get another bigger set which is, which we call as integers, integers, so all negative numbers were added. So if you see the set of natural number is smaller than set of whole numbers, which is again smaller than natural, sorry, integers, now integers is another set within a bigger set and this set is called rational numbers, isn't it? So and rational numbers are, rational numbers are denoted by a letter Q, you all know this is Q, so Q is rational numbers and then, then we have added to it irrational numbers and you will get a bigger set still and this bigger set is called, this bigger set is called real numbers. So that's what we saw Q add and add to it irrational numbers I, so I was added to it and then this bigger set is real numbers. So if you see, it is a bigger set which encompasses all the different types of numbers which we have studied so far, right? So natural number we say in technical terms is a set subset of W, W is a subset of integers, integers, set of integers is a subset of rational numbers and rational numbers and irrational numbers put together comprise of real numbers, okay? This is what in, let's say in diagram notation, now you must be wondering why do we call real numbers real? So what is the reason we call them real? Now the story goes back to 17th century so don't think that we are using this term real numbers for this particular set of numbers prior to, we were not using it prior to 17th century when, you know, a famous French scientist, not scientist he was basically a philosopher but he had contributed a lot in the field of mathematics and physics for French philosopher and his name was philosopher and his name was René Descartes, okay? And this is the same philosopher on whose name we have today something called Cartesian coordinate system, Cartesian coordinate, coordinate system. So you might have studied in history that this was a time when René Descartes was going on in Europe and René Descartes was one of the, you know, philosophers who inspired several other philosophers and scientists. So the concept of real number was first used by French philosopher René Descartes. Now why? Why did he use this concept of real numbers? So what he was doing was he was trying to solve polynomial equations, polynomial equations. So polynomial equations you have already encountered to an extent and equations, right? So few of them, some examples of them are like linear equations, you have already studied linear equations in one and two variables, linear equations and then you have also studied or you might have studied quadratic equations as well, correct? So many a times if you see you don't find a solution to a particular equation. For example, if you have studied quadratic equation then good enough you might have encountered this otherwise even if you have not. So let's say if I have equation like x square plus 3x plus 1 equals 0. So I do not have any real value of x. So you can't find any value of x within this set of real numbers. There is no value of x within this set of real numbers which will make this polynomial equal to 0, right? So hence when René Descartes was solving such equations he said that okay there are no solution in the set of real numbers but then he coined something called some, you know, so again mathematicians hit a dead end so how to solve such equations and hence something called imaginary numbers. Imaginary numbers were discovered which came to rescue and now we have solution to these type of equations also in a set of imaginary numbers which is again part of a complex number which we'll study in grade 11th. This thing in grade 11th you'll be studying. So this is what he was trying to do and to differentiate between let's say real and real and imaginary. So let's say he was trying to differentiate it from imaginary numbers, imaginary solutions to equation like this and he coined the term real. And since then we are using it in our, you know, mathematics courses and it has become a wide area of study. So in higher grades you'll be studying things like real analysis, real algebra, etc. So this is the introduction to let's say our real numbers. Now one important property or most important characteristic of real number is we can express or represent them in a number line. So what one good thing about real number is we can express them in a number line. So this is very important. Number line and hence every number, every real number, every real number, real number can be expressed or represented, let's use the word represented on a number line. Let us see it how on a number line, number line. So whether it is 1, whether it is 0, whether it is 0.5, whether it is 0.33 bar or whether it is root 2 or 3 root 7, you name it. If it is real number it can be expressed in a number line. So let me show you how. So if you see here, there is a number line already drawn, right? So if you can see it starts with, you know, let me zoom a bit in and then you see there is 0 here, right? So there are equal intervals between the numbers 0, 1, 2, 3, 4 on the right hand side minus 1, 2, 3. So if you can see, we have order here, order in numbers. All numbers are ordered. So minus 1 comes before 0 and 0 comes before 1 and so on and so forth. This particular property is called ordinality. So all the numbers, all the real numbers have order, you know, can be arranged in an order. Now you can find out any number, you know, let's say where is 3.5. So you have to go to 3.5C, 3.5 is here. I can also go to 3.56. How? So if I zoom in, so this is 3.6. So let's 3.56 I have to find out. So if you see 3.55, 3.56 is here, right? So you can go in and in and you can get all the numbers represented here. So if you, as you keep zooming in, you will see all the numbers coming there. So you name it and the number will be there between any two points in the number line. So that's the unique characteristic of real numbers, anything. So any number can always or you can always express it in a number line. So I think we could answer in this session, what did we learn? We learned what are real numbers? What are real numbers? So you now know what are real numbers and we also understood the concept. Why are they called real numbers? Why are they called real numbers, real numbers? And thirdly, we saw representation, representation, representation of real numbers, real numbers on a number line, on a number line. I hope you enjoyed the session. So for next question or next set of sessions, please subscribe to our channel and you will be getting regular updates on these sessions. Thanks a lot.