 Hello and welcome to the session. In this session we will learn about properties of multiplication with complex numbers. First of all let us discuss the closure property. Now the search of complex numbers is closed under multiplication. That is if we perform multiplication operation on two or more complex numbers the result will also be a complex number. Now let us prove this. Let the complex number z1 is equal to a plus b iota and the complex number z2 is equal to c plus d iota where c, d are real numbers and z1 and z2 belongs to the set of complex numbers c then z1 into z2 is equal to a plus b iota the whole into c plus d iota the whole is equal to a c minus b d the whole as iota square is minus 1 plus b c the whole into iota. Now as a, b, c and d are real numbers therefore a, c, v, d, a, d and b, c will be also real numbers and the region and subtraction will be also the real numbers that means a, c minus v, d is also a real number and a, d plus b, c is also a real number which is of the form q iota where p and q are real numbers therefore z1 into z2 is also a complex number thus if we perform multiplication operation this numbers z1 and z2 then the product that is the result of complex number thus we can say that the set of complex numbers is closed under multiplication thus the next property that is is commutative in the set of complex numbers that is if z1 belongs to the set of complex numbers then z1 into z2 is equal to z2 into z1. Now let us prove this property let z1 is equal to a plus b iota and z2 is equal to c plus d iota where a, b, c are now z1 into z2 is equal to a plus b iota the whole into the whole plus b d the whole as iota square is minus 1 plus bc the whole into iota. Now let us find z2 into z1 which is equal to c plus d iota the whole iota the whole which is equal to db the whole into iota now as a, b, c and d are real numbers and we know that multiplication is commutative in the set of real numbers therefore yes db will be equal to bd the whole plus into bc plus dn will be equal to ad the whole into iota as multiplication is commutative in real numbers further this is equal to ac minus bd the whole we know that addition is commutative in real numbers b, c, a and d are real numbers so bc will be also a real number and ad will also be a real number so bc plus ad can be written as ad plus bc the whole into iota from these two equations z1 into z2 is equal to z2 into z1 thus we can say that multiplication is commutative in the set of complex numbers because the next property which is multiplicative in the set of complex numbers c that is z1, z2 and z3 belong to the set of complex numbers c then z1 into z2 the whole into z3 is equal to z1 into z2 into z3 the whole now let us prove this property all this z1 that is the complex number z1 is equal to a plus b iota z2 is equal to c plus d iota and z3, c, d, a and f are real numbers z1 into z2 the whole into z3 so this will be equal to b iota the whole into c plus d iota the whole this whole into further this is equal to this bd the whole d plus bc the whole into iota this whole into into the whole now in sum of this it will be equal to this bd e minus adf minus bcf the whole plus minus bdf plus ad vce the whole into iota now from these two terms taking a common it will be equal to a into ce minus df the whole and from these two terms taking minus b common it will be minus b into de plus cf the whole now from these two terms taking a column it will be a into cf plus de the whole and from these two terms taking b column it will be plus b into this df the whole this whole into iota further it can be written as a plus b iota the whole into this df the whole plus iota into cf plus de the whole now z1 is equal to a plus b iota this complete term will be equal to z2 into z3 so this is equal to z1 into z2 into z3 the whole therefore we are getting z1 into z2 the whole into z3 is equal to z1 into z2 into z3 the whole thus multiplication is associated in the set of complex numbers now let us discuss the existence of multiplicative identity in the set of complex numbers c now the complex number one which is equal to one is zero iota is the multiplicative identity of complex numbers x number z we have z into one is equal to z is equal to one into z now let z is equal to a plus b iota that is the complex number z is equal to a plus b iota where a and b are real numbers then z into one that is a plus b iota the whole into one plus zero iota will be equal and here b iota into zero iota will be equal to b into zero into iota square and iota square is minus one so it will be minus b into zero the whole a into zero plus one into v the whole into iota now here these terms will be equal to zero so in the result we will get a plus b iota and a plus b iota is equal to the complex number z one into z will also be equal to the complex number z thus from these two equations z into one is equal to z is equal to one into z one which is equal to one plus zero iota is the multiplicative identity in the set of complex numbers now let us discuss the existence of multiplicative inverse in the set of complex numbers now every non zero complex number has a multiplicative inverse that is every non zero complex number z one there exists is equal to one is equal to z two into z one now let us z one that is the complex number z one is equal to a plus b iota the complex number z two is equal to c plus d iota where a b c and d are real numbers now we have two is equal to one plus b iota the whole plus c plus d iota the whole is equal to now one condition as one plus zero iota which further implies bd the whole now it is ac minus bd as iota square is minus one plus bc the whole into iota is equal to one plus zero iota now comparing this is bd is equal to one that is comparing the real parts and comparing the imaginary parts we have 80 plus bc is equal to zero now let this be equation number one and this be equation number two now solving equation number one and two we were over a square plus b square this value in one we get d is equal to minus b over a square plus b plus b square is not equal to zero is not equal to zero that is we have taken this for a non-zero complex number a plus b iota taking z one into z two is equal to one we are getting the values of c and d z two is equal to this d iota which will be equal to square plus b square the whole of minus b b square the whole into iota thus we can say that x number z one which is equal to a plus b iota there exists a complex number z two which is equal to c plus d iota and we have obtained the value of z two and this is known as the multiplicative inverse of z one and the multiplicative inverse z one which is equal to z two is denoted by z one inverse or one over z one and next let us discuss that the multiplication is distributed over addition in the set of complex numbers c that is if z one z two and z three belong to the set of complex number c then z one into z two plus z three the whole is equal to z one into z two plus z one into z three which is called the left distributive the right distributive property now let us prove the left distributive property that is we will find z one into z two plus z three the whole consider these values of z one z two and z three will be equal to a plus b iota the whole into c plus d iota the whole plus a f iota plus b c iota iota square is minus one which is further equal to c minus b d the whole while combining these two terms combining these two terms and taking iota common it will be plus iota into a d b c the whole plus now combining these two terms it will be a e minus b f the whole and combining these terms it will be plus b iota whole that z one and z two will give this value and the product of z one and z three will give this value so this is equal to z one into z two plus z one into z three so we have proved the left distributive property similarly we can prove the right distributive property now let us discuss some identities z one plus z two whole square is equal to z one square plus two z one z two plus z two square where z one and z two are the complex numbers now let us prove this identity now z one plus z two whole square is equal to z one plus z two the whole into z one plus z two the whole addition in the set of complex numbers so we can write it as z one into z one plus z two the whole plus z two into z one plus z two the whole is equal to z one square plus z one z two and z two into z one will be equal to z one into z two as multiplication is commutated in c that is a set of complex numbers so this will be plus z one into z two plus z two square which is further equal to z one square plus two z one z two plus z two square now let us move to the second identity that is z one minus z two whole square is equal to z one square minus two z one z two plus z two square now let us prove z one minus z two whole square can be written as z one minus z two the whole into z one minus z two the whole which is further equal to z one into z one minus z two the whole minus z two into z one minus z two the whole this is further equal to z one square minus z one z two will be equal to minus z1 z2 as multiplication is commutative we will see plus z2 square so this will be equal to z1 square minus 2 z1 z2 plus z2 square. Now let us see the third entity that is z1 plus z2 whole cube is equal to z1 cube z1 square z2 plus 3 z1 z2 square plus 3 z1 square plus z2 cube. Now let us prove the third entity. Now z1 plus z2 whole cube commutative as z1 plus z2 whole cube into z1 plus z2 whole square. Now from the first entity we have z1 plus z2 whole square is equal to z1 square plus 2 z1 z2 plus z2 square. Now multiply z1 cube z1 square z2 plus 3 z1 z2 square plus z2 cube. Now the next entity is z1 minus z2 whole cube is equal to z1 cube minus 3 z1 square z2 z2 square minus z2 cube. z1 minus z2 whole cube can be written as z1 minus z2 whole into z1 minus z2 whole square which is further equal to z1 minus z2 whole into now z1 minus z2 whole square from the second entity is z1 square minus 2 z1 z2 plus z2 square whole. Now multiply z1 square z2 plus 3 z1 z2 square minus z2 cube. Now the next entity is z1 square minus z2 square is equal to z1 plus z2 whole into z1 minus z2 whole. Now let us start with its proof. Now we will start with the right hand side z1 plus z2 whole into z1 minus z2 whole z1 into z1 minus z2 whole plus z2 into z1 minus z2 whole. Now this is equal to z1 square minus z1 z2 plus z2 z1 so it can be written as z1 z multiplication is commutative in C so it will be plus z1 z3 minus these terms are cancelled with each other so this is equal to z1 square minus z2 square. So in this session we have learnt about multiplication with complex numbers and I have enjoyed the session.