 Hi and welcome to the session. Today we will learn about multiplication of integers. Let us start with multiplication of a positive and a negative integer. Suppose we have two positive integers a and b and we want to multiply a with minus b that is a positive integer and a negative integer. Then first of all we will multiply a and b as whole numbers and then we will put a minus sign before the product. So from this we will get a negative integer as an answer. Similarly if we want to multiply minus a with b then this will be equal to minus of a into b. Let us take an example for this. Let us find 2 into minus 6 this will be equal to minus of 2 into 6 which will be equal to minus 12. Thus we can say that product of a positive and a negative integer is a negative integer. Next we have multiplication of two negative integers. Suppose we have two positive integers a and b and we want to multiply minus a with minus b. Then first of all we will multiply a and b as whole numbers and then we will put a positive sign before the product. Or we can simply write it as a into b. Thus product of two negative integers is a positive integer. Here is an example for this. Suppose we want to multiply minus 2 and minus 6 then this will be equal to 2 into 6 which will be equal to 12. Now let us move on to product of three or more negative integers. If the number of negative integers in a product is even then the product will be a positive integer. Here is an example for this. Suppose we want to find out minus 3 into minus 5 into minus 2 into minus 1. Now here we want to find out the product of negative integers. So first of all we will count the number of negative integers which is 4 over here and 4 is an even number. That means the product will be a positive integer. So this will be equal to plus 3 into 5 into 2 into 1 which will be equal to 30. Now if the number of negative integers in a product is odd then the product will be a negative integer. Here we want to multiply minus 3 by minus 5 by minus 2. Now here we have three negative integers and three is an odd number. So that means the product will be a negative integer. So this will be equal to minus of 3 into 5 into 2 which will be equal to minus 30. I hope the topic of multiplication of integers must be clear to you. Now let us move on to our next topic. Properties of multiplication of integers. First of all we have closure under multiplication. This property states that integers are closed under multiplication. That is for any integers a and b, a into b is an integer. For example let us take two integers minus 5 and 2. So minus 5 into 2 is equal to minus 10 which is also an integer. Thus we can say that integers are closed under multiplication. Now let's see commutativity of multiplication. This property states that multiplication is commutative for integers. That is for any two integers a and b, a into b is equal to b into a. For example let us take two integers minus 5 and 2. So minus 5 into 2 is equal to 2 into minus 5 this is equal to minus 10. Now let's move on to multiplication by 0. According to this property product of any integer by 0 is 0. That is for any integer a, a into 0 is equal to 0 into a is equal to 0. For example if we have an integer minus 5 then minus 5 into 0 is equal to 0 into minus 5 is equal to 0. Now let's see the multiplicative identity for integers. Here 1 is the multiplicative identity for integers. That is for any integer a, a into 1 is equal to 1 into a is equal to a itself. For example if we have an integer minus 31 then minus 31 into 1 is equal to 1 into minus 31 is equal to minus 31 itself. Now suppose we have an integer a and we multiply a with minus 1 then this will be equal to minus 1 into a. And this will be equal to minus a. So from this we can say that minus 1 is not the multiplicative identity for integers. Rather when we multiply our integer by minus 1 then we get the additive inverse of that integer. So now let us find out the additive inverse 20. So for this we will multiply 20 by minus 1 and we will get minus 20. So minus 20 is the additive inverse of 20. Similarly the additive inverse of minus 11 will be given by minus 11 into minus 1 which is equal to 11. So that means 11 is the additive inverse of minus 11. Next property is associativity for multiplication. According to this property if we have three integers a, b and c then a into b the whole into c will be equal to a into b into c the whole. Let us take an example. Suppose we have three integers minus 5 minus 2 and 3 then minus 5 into minus 2 the whole into 3 will be equal to minus 5 into minus 2 into 3 the whole which will be equal to 30. Now last property is distributive property. According to this property for any three integers a, b and c a into b plus c the whole will be equal to a into b plus a into c. This is known as distributivity of multiplication over addition. For example if we have three integers 5, 2 and 3 then 5 into 2 plus 3 will be equal to 5 into 2 plus 5 into 3 which will be equal to 25. Now for integers we also have a into b minus c is equal to a into b minus a into c. For example let us take the integers 5, 2 and 3 so 5 into 2 minus 3 is equal to 5 into 2 minus 5 into 3 which is equal to minus 5. Now let us see how we can make our multiplications easier. The above properties that is commutativity, associativity and distributivity of integers helps us to make our multiplications simpler and easier. Let us see how. Suppose we want to multiply minus 50 by 17 by 2 then first of all we will rewrite it as minus 50 into 2 the whole into 17 using commutative property. So this will be equal to minus 100 into 17 which will be equal to minus 1700. Thus in this session we have learnt about multiplication of integers and properties of multiplication of integers. With this we finish this session. Hope you must have understood all the concepts. Good bye, take care and keep smiling.