 Hello and welcome to the session. In this session, we will learn about operations with complex numbers. Let us discuss complex numbers. Now any number of the form a plus b are the where and b are real numbers of complex number is called the real part of the complex number and b is called the imaginary part of the complex number with complex numbers. Now we can perform all the four basic operations that is addition, subtraction, multiplication and division. All these operations when performed on two or more complex numbers the resultant is also a complex number addition. Now the addition complex numbers z1 is equal to a plus b iota and z2 is equal to c plus d iota is defined by the relation z1 plus z2 is equal to a plus b iota the whole plus c plus d iota the whole which is equal to a plus c the whole plus b plus t the whole into iota. Now here we know that z1 is a complex number and for the complex number z1, z2, a, b, c and d are real numbers therefore a plus c and b plus d are also real numbers so z1 plus z2 is of the form iota where p and q are real numbers so this is also a complex number so if z1 and z2 are two complex numbers then their addition that is z1 plus z2 is also a complex number. Now secondly let us discuss the negative of complex number now if the complex number z is equal to a plus b iota then negative of the complex number that is minus that minus z is equal to plus b iota the whole which is equal to minus a minus b iota that is minus z is equal to minus 5 minus 6 iota the whole which is equal to minus 5 plus the negative of a complex number is also a complex number. The complex number z1 is equal to a plus b iota and the complex number z2 is equal to c plus d iota where a, b, c and d are real numbers equal to a plus b iota the whole minus c c the whole plus d the whole into iota. Now z1 and z2 are two complex numbers and a, b, c, d are real numbers so a minus d and b minus d of the form x plus y iota where x and y are real numbers so this is a complex number therefore we can say if z1 and z2 are two complex numbers then z1 minus z2 is also a complex number that is the subtraction that is the subtraction of z1 and z2 is also a complex number. Now let us see one example now if the complex number z1 is equal to 2 plus 3 iota and the complex number z2 is equal to 5 plus equal to 2 plus 3 iota the whole minus 5 plus 6 iota the whole which is equal to 2 minus 5 the whole which is equal to minus 3 minus 3 iota. Now let us discuss one more operation that is the integer number which is equal to a plus b iota then z will be equal to m into a plus b iota which is equal to m into a plus m into b into iota into z that is 1 over m into a plus b iota is equal to 1 over m into a plus 1 over. Now let us see one example let the complex number z is equal to 2 is equal to 5 1 by m will be equal to z will be equal to into a into b that is 5 into 3 into iota so this is equal to 15 iota 1 by m into z will be equal to 1 by m that is 1 by 5 into a that is into 2 plus 1 by 5 into that is into 3 which is equal to now let us discuss the additive inverse now the complex number z2 the complex number z1 z2 is equal to 0 now let z1 is equal to a iota where a b c and d are real numbers plus b iota the whole plus d iota the whole which is z2 is equal to b plus d the whole into iota is equal to 0 equal to 0 v plus d is equal to 0 is equal to minus t and b is equal to minus t is equal to minus b therefore iota which is equal to c equal to as c is equal to minus 1 so it will be minus a and d is equal to minus b so it will be minus b which implies z2 is equal to minus a plus b iota the whole because b iota is a complex number where a and b are real numbers then iota the whole which is equal to minus a minus b iota plus b the whole into iota. The additive inverse of a complex number is again a complex number. Now let us discuss one example. Now if the complex number z1 is equal to 2 plus 3 iota then inverse is 3 iota that is the additive inverse of z1 will be equal to minus 2 which is equal to minus 2 minus 3 plus the multiplication of two complex numbers is equal to a plus b iota and the complex number z2 is equal to c plus d iota b c and d iota numbers then z1 into z2 is equal to a plus b iota the whole into c plus d iota the whole which is equal to iota square is minus 1 b c the whole into iota if b c minus b d also real numbers. So this product iota where b and q are real number x number. Now let us discuss one property related with the multiplication of two complex numbers and that is plus b iota the whole into a minus b iota the whole is equal to a square plus b square plus 0 which is equal to a square plus b square this is a real number with complex numbers iota the whole. Now let us discuss one example here if the complex number z1 is equal to 4 plus 2 iota and the complex number z2 is equal to 1 plus 3 iota then the product that is z1 into z2 is equal to 4 plus 3 iota the whole into 1 plus 3 iota the whole iota which is again a complex number the division of two complex numbers. Now if the complex number z1 is equal to a plus b iota z2 is equal to c plus d iota b c and d iota b c and d iota where number of percent z2 over z1 can be determined by multiplying the numerator and denominator by a minus b iota. So this is equal to now z2 is equal to c plus d iota over z1 is equal to a plus b iota so we will multiply the numerator and denominator the iota equal to c plus d iota the whole into a minus b iota the whole where upon a plus d iota the whole into a minus b iota the whole c a plus db as iota square is minus 1 the whole dn minus cb the whole into iota. Now a plus b iota the whole into a minus b iota the whole will be equal to a square plus b square as iota square is minus 1. So this can be written as a plus db whole upon a square plus b square plus d a minus cb whole upon a square plus b square into iota. There is also a real number c equals db is a real number and d a minus cb is also a real number. This means therefore if z1 and z2 are complex number complex number. Now if the complex number z is equal to a plus b iota where a and b are real numbers then its reciprocal that is 1 by z is equal to 1 by a plus b iota. Now multiply the numerator and denominator by a minus b iota. So this will be equal to minus b iota whole upon a square plus b square plus of minus b over a square plus b square into also of the complex number z is again a complex number. And the reciprocal the multiple pages involves complex number z into 1 by z is equal to 1. The complex number z is equal to a plus b iota then its multiple 1 by z will be equal to a minus b iota whole upon a square plus b of a rational number with a complex number. The complex number z is equal to a plus b iota and z that is p by q into z as multiplication of z by p and its division by q into a plus b iota the whole which is equal to over q into b over q into iota the whole is equal to 1. But operations whenever performed on complex numbers will result into a complex number. So in this session you have learnt about operations with complex numbers. So this completes our session. Hope you all have enjoyed the session.