 In this video, we are going to look at properties of division of integers. Let me just write integers here. So, whatever set that we are going to look at is the set of integers and we will perform division and we will come out with interesting properties. Very first property that I can think of is closure property. So, what is closure again? Closure is when you choose any two numbers from the given set and divide them. Do you get a number in the same set? Now, I will just quickly write the set of integers here. So, integer set I, some negative numbers that I need to write, then 0 and some positive numbers and then dot dot dot. So, this is how we can write integers and then we can choose any two numbers in C. Once we divide them, do we get a number from integer set? Let me choose 1 and minus 2. Let's say 1 divided by minus 2. We can write this as minus 1 by 2. Is this an integer? Not an integer. So, an integer set doesn't follow closure under division. I will just put this sign here. You could choose any other two numbers. Just remember that we cannot divide any number by 0. Whenever choose any two numbers, make sure that you should not divide by 0. So, you should not do something like this, 2 divided by 0. So, let's not do this. All right. Let's go to the next property. We already saw closure is not followed under integer set for division. And now we want to see whether the commutative property is followed. So, what is commutative property? So, whenever you choose any two numbers from integer set, say something like 10 divided by 2. Right? So, it's like saying 10th of ease being distributed among two people. So, everybody will get five toffees. But is it equal to 2 divided by 10? Is it? It is not. Because the meaning completely changes. It's like dividing two toffees among 10 people. And the answer would be different than what it would be when we divide 10 by 2. So, this is not equal. And therefore, whenever we have two numbers, a divided by b. If it is not equal to b divided by a, we say commutative property is not followed. And therefore, we can say division is not commutative for integers. Now, there is another property called associative property. This is just a kind of an extension for the commutative property when it comes to division. Let me just discuss that. So, in associative property, we choose three numbers. Let me just write it here. So, let's say we have three numbers. I'll just write associative property. So, we will have three numbers. For example, 10 divided by 5 divided by 2. Now, we use brackets like this. And the question is whether this is equal to 10 divided by 5 divided by 2 when the brackets are placed for another pair of numbers. Now, this is 2 divided by 2 because we will perform the operation in brackets first. And this will give us 1. But here, 10 divided by 5 divided by 2 is 2.5. So, this will give us 4. And so, these two things are not equal. And so, we can say that division is not associative for integers as well. Okay. Let's go ahead and see some special numbers for division. And these special numbers are called as 0 for division and identity for division. So, now what is a 0 for division? 0 for division is a special number dividing which will always give us 0. So, the fourth thing that we want to talk about is a 0 of division under integers. So, 0 for division is a number 0 itself. Why? Because once you divide 0 by any number, say 99, you will always get 0. 0 divided by 5 will be 0. 0 divided by minus 3 will be 0. So, every time you get the output 0 and that's why 0 of division is the number 0. Now, there is another special number and that's an identity for division under integers. And the identity is basically a number which you can use in the division operation, returns the same number. Identity in division for integers and this is going to be some number, some number a, so that you divide any given number say, let's say 65 divided by a gives you 65 itself. If you divide any number say minus 10 by a, it will give us minus 10. Can you think of such number? Yes. Yes, yes. It's 1, isn't it? If you replace a with 1, if you divide any number by 1, you are going to get the same number and therefore, this a or the identity in division is 1. That just means that once you divide any number by 1, you get the same number and these are few important properties for division. Let's just revise. First, we discussed closure property and we saw that division is not closed under integer set. So, a divided by b is not always an integer given that a and b are integers. Then we saw commutative property. Once you choose two integers a and b, a divided by b is not always equal to b divided by a. Then we came to associative property. So, if you choose three integers a, b and c, then in bracket a divided by b, bracket complete divided by c is not equal to a divided by in bracket b divided by c, bracket complete. Then we came to zero of division which is basically a number which when divided by any integer returns zero and such number is zero itself because zero divided by any integer is zero. Note that in any of these properties, the number with which you are dividing by cannot be zero, right? Lastly, we looked at identity in division which is a number by which when you divide any integer, you get the same integer and that is one.