 Hello friends, so welcome to this all-new session on number system We are going to deal with the multiple topics of number system in a series of such sessions In the first such session, we are going to today discuss number system review, right? So we are just going to recap whatever we have learned so far in the previous grade This particular session would be useful for all those 9th grade students who have completed 8th grade mathematics and So let's begin number system review. So guys, you know that you know Our entire surrounding our ecosystem all around us. We see a lot of utility of numbers, right? So your mobile number is a number your roll number in the class is a number So the number plate in your vehicle carries a number so you can you know when when you go to a Grocery shop or any any other shop when you exchange money Then there is a number your bank account number your parents bank account numbers. So are numbers So basically you see a lot of utility of numbers all around So if you really have to You know Understand the utility of number utility of the numbers, you know, there are three primary role numbers play And one is definitely to count. So let us say when in cricket you hit a six or a four So you count number of runs you are scoring, right? Similarly the marks achieved by you in any of the examination you will see there is a number attached to it So let us say there is a paper on mathematics which carries hundred full marks and you get hundred out of hundred So that's how you count how many marks you are getting or let us say how many pencils you have How many days in a week how many minutes in on in our and things like that. So basically numbers help in counting second Utility if you may notice for example, if you follow cricket, so You know Sachin Tendulkar who plays for India always used to wear a jersey Number ten, right? So hence you if you see there is another utility of Numbers and that is labeling correct So your flat number your house number your You know Let's say parking slot number and things like that or all where labeling Is used or labeling is done using numbers Another utility you can also think of is order Right ranking for example, someone comes first in the race. So there is a first Rank then there is a second rank and so on and so forth in exams like let's say Iitj there is all India rank. So there also there is an order. You must be you know given some merit rank in your class So that's an order. So all these are utility of numbers Now we have also learned different types of numbers so far So if you see the basic counting numbers are natural numbers natural numbers, you know So natural numbers is nothing but all those numbers which are used for counting Example one two three and so on and so forth These are the first things you learn when you take up arithmetic in your childhood natural numbers But then we sooner, you know graduated into another set of numbers which is called whole numbers and whole numbers Whole numbers are nothing but numbers starting from zero all counting numbers plus zero Three four five and so on and so forth are all whole numbers Now the set of natural numbers, please notice. I am saying set set of Natural numbers right natural numbers is denoted by a letter n Okay, so n for natural so n similarly set of whole numbers is nothing but It's denoted by a letter w so you can clearly see W the set of w has one extra member that is this zero Then the natural numbers, isn't it? So hence if I have to pictorially represent the two sets I can say if this is my Natural number set then if I the whole number set will will constitute all the natural numbers Plus zero and hence this will be called a whole number set Now natural numbers set have Numbers like one two three so on and so forth right now Why did we require whole numbers in the first place? Why was zero needed? So if you know natural numbers if you add two apples with three apples, you'll get five apples Right, so two plus three was five right. This was very much easy and if you if you let's say do five plus Five plus another number seven so you get twelve so if you see if you add two natural numbers if you add two natural Numbers you get we get Another natural number isn't it another natural number this particular Property is Not that simple the what what I what we what I mean is If you add two natural numbers, you always get another natural number This particular property is called closure property right so we say closure Property why closure property because to find the solution of a sum like three plus seven You will get the solution in the set of natural numbers only that means if you add a natural number to another natural number You will get another natural number right so hence We don't require any other new number Outside the set of natural numbers to find the solution of any addition operation, right? You cannot you know you take any number of examples Let us say seven plus twenty one is twenty eight again If you see this is one natural number Added to another natural number you get another natural number right this particular thing is called closure property and we say that set of set of natural numbers set of natural numbers is closed What do I what am I saying closed for addition? This is how we describe it closed for addition means you don't require any intervention external intervention To find the solution of some of two natural numbers, but my friends. Let us do another operation You have studied subtraction multiplication division, etc. So let us now say If I find out let us say five minus three, you know the answer is two right So if you see one natural number Subtracted to another natural number or subtracted from another natural number gives you another natural number But there is a problem. What is the problem if I subtract five? from five You now know that the answer is zero But if someone who has studied only natural numbers or whose knowledge comprised of let's say the Set of natural numbers then he will be confused What is the solution? so hence If you know the history Indians first first, you know Indians are the first people who came out with the concept of zero All right, so five minus five was now Expressed as zero, but then zero this zero doesn't belong to doesn't belong to Counting numbers or natural numbers belong to natural numbers because we can't you don't really count nothing, right? Natural numbers so hence hence when zero was added to the set of natural numbers We got a new set of numbers Majority of which was already known which was called natural numbers now. We are getting another set of Another set and now that set is called Whole numbers right so now I I assume that you have understood the Concept of natural numbers and whole numbers now now Does this mean that? Does this mean that? in a set of set of set of whole numbers whole numbers set of whole numbers is closed is closed for Subtraction is this statement true What does it mean? It means that if you choose a whole number a and Let another whole number be and you subtract them will you get see which is see which is Which is always always a whole number always a whole number Really not Right because you can take an example for example It was very comfortable if you subtract a smaller number from a greater number like five minus three and you know The answer very easily. It's true, but what happens if I reverse these two Numbers right that means if I want to find out three minus five Does this answer? Belong to the set of whole numbers that means will you get? will you get a solution a solution in Whole number set in the set of whole numbers will you get? whole number set But a set is nothing but the collection of all whole numbers right so will you get a solution in the set of whole numbers? That means will you find a whole number which will fetch you three minus five actually no No, so hence another set of numbers now were required and hence now that you know that answer is minus two But this is minus two doesn't belong to Doesn't belong to Doesn't belong to what set of set of whole numbers Set of whole numbers. So hence this is a this number is an outsider. Isn't it outsider? outsider to the set of whole numbers Right and hence we say friends that set of whole numbers set of whole numbers whole numbers is Not is Not not closed closed for subtraction It is definitely closed for Addition but it is not closed for subtraction meaning what you will never you will it's not necessary that you will get a whole number when you take the difference of two whole numbers correct hence to remove this You know Problem or let's say to remove this in capability of finding the difference of Higher number from a smaller number We in we find another set of numbers and now those are called integers Integers right. So what are integers now integers are nothing but a set of Whole numbers Whole numbers set of whole numbers that is zero one two and all that plus set of set of negative Counting numbers negative counting Numbers and those are minus one two Minus three minus four and so on and So forth guys, right? So this is nothing but we have now found out a new set of numbers and those are called Integers now the set of integers is denoted by Z Z and Z is for Xalan this word Xalan, right? It's a German word Xalan that means numbers Okay, so hence Hence integers are a new set of numbers which in comprise of not only The whole numbers but negative counting numbers so pictorially you can you can represent You can represent integers and the entire set of numbers that you know now is like this How do I represent pictorially so hence if this is my set of natural numbers? Then you add zero you will get the whole number set Then you add negative numbers to it. You'll get Integers Isn't it so hence natural number belongs to the set of whole numbers and the whole numbers now belong to the set of Integers right another way of pictorially representing it is called the concept of number line number Line so if you draw a line Infiniti long line Let us say this is my line and it is going in both the directions and let us take any other point as let's say zero That's starting point as zero now divide the line into Equal intervals. Let's say one two three so that the difference between the two units or two points is always same Correct, so I'm doing it with rough hand what you can do is you can take a ruler and draw neatly So this point is represented as one so this is exactly what is this distance? This distance is one unit isn't it and this is to this is three likewise four five six and this goes on Similarly on this side, you'll see minus one minus two minus three minus four minus five and all that Okay, so if you see All the integers can be represented on a number line which is extending infinitely in both the direction You have to just put one starting point that is zero We have put it here and on both the sides on the right hand side and the left hand side You will see, you know infinitely many numbers which can be represented on the number line It's just the constraint on the space. Hence, we are not able to draw longer number lines But philosophically speaking there is there extends an infinitely long number line where all the integers could be Accommodated and represented such that the difference between consecutive integer is always the same correct, this is a very important concept now Now what is the learning of learning so far? So we learned these points first natural numbers natural numbers are closed for Closed for what for addition? We know that two natural numbers added will fetch you another natural number We also learned that natural numbers are Not closed for closed for Subtraction not close for Subtraction third we also learned that whole numbers whole numbers is Nothing, but w which is equal to n plus a element zero, right? So natural number added to added to an element or zero added to that set of natural numbers you'll get whole numbers and We also learned same as natural number whole numbers are closed for closed for addition addition but again whole numbers all numbers are Not not closed for closed for subtraction subtraction hence and hence We read it. We needed some external intervention and we formulated new set of numbers which are called integers Integers now which are nothing but what are integers it denoted by letter Z for xalan and It is nothing but an element comprising of Infinitely many numbers minus three minus two minus one zero one two Three and so on and so forth, right? And then we learned integers integers integers are closed closed both both for addition addition as well as as well as subtraction Subtraction isn't it? So if you take any two integers and you try to find out either some or the difference you will always get a Integer so very important property of integers and then third and then one more thing is number line We understood that all the integers Whole numbers and negative in negative in accounting numbers can be represented in a on a Number line isn't it? This is what we learned so far. So in the in the in the next few Sessions, we'll understand why Did we require any other set of Numbers or what happens if we try to divide and multiply two integers? What should be the result? So let's meet in the next session. Thank you