 Hello friends, so welcome again to this session on number systems now in the last session We studied you know some basics about number system. We saw a few types of numbers as well We saw natural numbers We saw whole numbers and we also saw integers and we saw that all these numbers the set of numbers had to be Formulated because there was some kind of limitation to the set of number system till that point, right? So for example, we discussed that the set of numbers natural numbers was not closed for subtraction That means if you would have Subtracted a bigger number from a smaller number you would not get a solution in the same set of natural numbers So we had to go with let's say a set of whole numbers and that's and then Subsequently with a set of integers. So now you you have you would You also understood last time that set of natural numbers is represented by letter n And what were natural numbers 0 1 2 3 counting numbers, right? Not not 0. I'm sorry It's 1 2 3 4 and so on and so forth then 0 was added to the set of natural numbers and Whole numbers were found out. So and then We also found out set of integers, which is represented by Letter Z and we understood the logic behind all these letters and symbols as well now going forward We will be you know discussing some more types of numbers before that we also learned the concept of number line that means the set of integers which is like Universal set so far where all the other set sets for example sets of Whole numbers and natural numbers are within the set of integers itself So integer set of integers is the largest in set. We are saying largest in terms of all the others are included in the Integer set as of now now we represented numbers integers on number line. We had a infinitely long number line and We started from any point. Let us say we Mentioned this as zero and then we distributed or divided the line into equal intervals in equal parts so and then we Mentioned or we Indicated those points by our integers one two three four on the right hand right hand side of zero and Negative integers on the left hand side of zero and in both direction it is increasing infinitely Okay, so we also learned few properties natural numbers are closed for addition. What does this mean? You add two natural numbers. You will get a natural number only But natural numbers are not closed for subtraction Why because if you it's not necessary that when you subtract two natural numbers You will get a natural number example three minus five is not a natural number. Isn't it not a Natural number this we learned last class last session whole numbers are nothing but natural number plus zero Whole numbers are closed for addition. That means like natural numbers if you add two whole numbers You will get a whole number But like natural numbers whole numbers are also not closed for subtraction. That means if you subtract Zero minus five. It's not a natural or a whole number Right, then we invented something called integers and integers is a set of all negative zero and positive numbers Integers are closed both for addition as well as subtraction Why because now you know three minus five is negative two minus two which is also an integer So integer is a set of Numbers which is you know close for both addition as well Subtraction now moving ahead will today Understand the concept of or let's say a new type of number, which is called rational rational numbers What are rational numbers and why are they called so? Right, so first we'll define rational numbers and before that also we will see why rational numbers were required So for example, let us say you were you know owner of a field where let's say 10 farmers are working on one thing on your field then farmers then farmers are working on your field and you are giving them as A salary One loaf of bread to each one of them one loaf of bread, right? One day, you know one guy didn't turn up So only nine farmers were there nine farmers were there and now you have to distribute So for ten farmers you had how many loaves of bread ten loaves, right? And now only nine has turned up and you have ten to distribute How would you distribute this? Now it would be you know very intuitive for you guys because you have already, you know learned this But during that time when there was no concept of Fractions, it was very difficult to you know Do these things so hence but later on They came up with this that okay, we can devise a new number now So each one of them got one loaf each and The last one was divided into ten parts. I'm sorry nine parts and Each of them each part it was then distributed to each one of the farmers now This is a new concept right until that time there was no concept of a number Which is a part of a number, right? So for example one upon nine is the ninth part of one ninth part of One so one bread was divided into nine parts now each one was a part of a hole, right? So part which was less than the whole itself So one part of the loaf was less than the whole bread. This is nothing. This is something New so hence now you are saying that on the number line if you had zero and then next is one You are trying to find out a number between zero and one which was not known earlier now These type of numbers which were like which are also called fractions Fractions are now called rational numbers rational numbers why the word rational and The the word rational gets its justification from the word irrational because it was very difficult to and define the significance of irrational numbers and Hence these numbers came to be known as rational numbers, right? Also, you can see like this this rational number also contains a word ratio Isn't it so hence if you can see this is kind of a you know ratio of two numbers to integers You know to be precise, but the thing is that's just a justification It's not that it's ratio and hence it is called rational number rational So if you if you know what does rationality means rationality means objective Objectivity where you can explain stuff Right, we can we can reason out you can justify but irrationality means where you can you know the reasons are not known or it is Unjustified right so hence now this was this was originally because of this something like this that if there is a square fence and Each side is of one unit Let us say the square is of side unit one then the diagonal length the diagonal length is something which was not known So how to find out the length of diagonal now today, you know, it's nothing but the root two, isn't it? But root two was appearing to be irrational or you know, it was it would have it was having no significance as such When irrational numbers were you know discovered or invented whichever way you want to say it Hence the other set of numbers became rational to the set of numbers which were called irrational numbers will discuss rationally rational in a bit In more detail sometime later, but just understand there were few numbers whose justification was not known How to calculate root two was not known. Basically. What is root two? What should be a number x which when multiplied by itself should get? To this was what the definition of square root was right but Unfortunately, they could not find that one number and hence they were little Beford little that how can there exist a number whose choir or which when multiplied by itself gets you too So they could never find that number, you know One number they so there were methods by which they can they did a lot of trial and error to get a right Right x value, but they could not do it. Hence. They said these are something called irrational numbers and Everything else which could be justified, for example, if there is a loaf of bread and if you divide into two parts Right, so one part can be written as one upon two. So this so this was coined as a Rational number or for example again another bread which is divided into three parts. Let us say one two and three So one part was one upon three again This was reasonable it can you know, it makes sense But you know, you were not able to find out square root of two was something which is unreasonable at that point in time Hence these numbers were called irrational numbers. I Hope you understood it. So now let's you know define rational numbers. So what are rational numbers? What are rational numbers? So you now know rational numbers must be some kind of a fraction. So you say a number Number, this is a formal definition of a rational number a number in the form of form of P upon q. Okay, where where? P and q are Integers Integers so you now know integers now we are going beyond integers and that is p upon q this is a kind of a fraction and q is not equal to 0 why because division by a division by Zero is not defined in mathematics is not defined We don't know what would be the quotient like right and we also have another criteria So you can write a this is a criteria a criteria B and Criteria C is HCF Then which in technical terms we also called GCD GCD or this is highest Highest Common factor you must have must have studied which is also called greatest common divisor greatest Common divisor, this is these are the same terms But in technical language GCD is used more often and it is denoted by GCD and then two numbers that is p and q So GCD of P and q must be equal to 1 that means P and q must not have common factors If there are common factors, they must be reduced by eliminating those common factors. So example Example 5 upon 6 is a rational number. Why because both 5 and 6 are integers our integers Six is clearly not equal to zero and if you see GCD of 5 and 6 that is the HCF of 5 and 6 is also one there is no common factor between 5 and 6 similarly minus 2 by 3 right both minus 2 and 3 are integers integers clearly 3 is not equal to 0 and GCD of There is no common factor between minus 2 and 3 Yep, so these are examples of Rational numbers or for that matter all integers you must notice all integers are Rational numbers Rational Numbers why because any integer for example, it says 7 can be written as 7 upon 1 and then you can see it satisfies all the three conditions of rationality correct similarly minus 5 is nothing but minus 5 upon 1 but all rational numbers All rational numbers Need not be Need not be integers Right for example 1 upon 2 is a rational number, but it is not an integer not an integer Isn't it all rational numbers need not be integers? So I hope you understood the concept of natural numbers. There are three criteria for rational numbers It must be in the form of p by q where p and q are integers q must not be equal to 0 and GCD of The two numbers p and q must be 1. Please remember this Definition one other thing is Let us say if there is a number like or there's a fraction like 6 upon 8 Is it a rational number in this form? It is not but it can be reduced to a rational number like 3 by For now 3 by 4 is the correct rational representation of 6 by 8. Okay Understood similarly 12 upon 15 can be represented as or written as What 4 upon 5, right? So this is now a Rational number. I hope you understood the concept of rational number and the three criteria Please bear in mind these three criteria are going to be used multiple number of times for various concepts and problem solving Thank you