 Hello and welcome to the session. In this session we will learn about conjugate and properties of conjugates of complex numbers. First of all let us discuss conjugates of a complex number. If a complex number z is equal to a plus b iota then the number this b iota is called the complex conjugate simply conjugate plus b iota denoted by conjugate of z that is conjugate of z is equal to a minus b iota. The conjugate of a complex number by simply changing the sign of the imaginary part in the given complex number. Now let us see one example here z is equal to 5 plus 7 iota then conjugate of z is equal to the conjugate of 5 plus 7 iota. Now we will change the sign of the imaginary part so we will change the sign and it will be 5 minus 7 iota. Now let us discuss the second example let the complex number is equal to minus 2 minus 3 iota then the conjugate of z is equal to the conjugate of minus 2 iota and here again we will change the sign of the imaginary part will be equal to minus 2 plus 3 iota. Therefore we can see that the conjugate of a complex number by changing the sign of the imaginary part of the given complex number. Now let us discuss this geometrically here the point p x y represents the number that is the complex number z which is equal to x plus y iota which q is the conjugate number that is the conjugate of z which is equal to x minus y iota and the coordinates of the point q are x minus y reflection or image of the point p in the real axis. Minus y represents the complex number minus that which is equal to minus x minus y iota the reflection of the point q in the imaginary axis. Therefore we can see that is the reflection of the conjugate of z in the imaginary axis. Now let us discuss properties of conjugates of complex numbers. Now let us discuss the first property that is the conjugate of the conjugate of a complex number is the complex number itself. That is if z is a complex number then of the conjugate of a complex number is equal to the complex number itself. Now let us prove this. Now let z is equal to x plus y iota then the conjugate of z is equal to now here we will change the sign of the imaginary part. So this will be equal to x minus y iota. The conjugate of complex number is equal to the conjugate minus y iota. We will change the sign of the imaginary part. So this will be equal to x plus y iota which is equal to the complex number itself. Now we have proved the first property. Now let us discuss the second property. Now the second property is a complex number is purely real. That means now let us take z as the complex number. So a complex number is equal to its conjugate that is z is equal to the conjugate of z that is z is purely this property. Now here let z is equal to x plus y iota is equal to the conjugate of this complex number that is z is equal to the conjugate of z which is equal to x plus y iota is equal to the conjugate of z that is the conjugate of x plus is equal to the conjugate of this number plus y iota plus y iota is equal to 0 is equal to 0 as these terms are with each other y is equal to 0. Now y is the imaginary number x plus y is the imaginary part and if y is equal to 0 then that means purely complex number is equal to 0 if and only the complex number purely real. Now let us discuss the third property and the third property is the conjugate of the sum of two complex numbers is the sum of z is if z1 and z2 plus numbers that means the conjugate of z1 plus z2 is equal to two conjugates. That means this is equal to the conjugate of z1 plus the conjugate of z2. Now let us fill this property. Let us take the complex number z1 is equal to x1 plus y1 the complex number z1 is equal to the conjugate of this number which is equal to x1 minus y1 iota and the conjugate of z2 will be equal to the conjugate of this number which is equal to x2 minus now conjugate of z1 plus conjugate of z2 is equal to minus y1 iota the whole number equal to x1 plus x to the minus to the whole into i1 plus z2 which will be equal to the conjugate plus y1 iota can be obtained by changing the sign of the imaginary part so this will be equal to x to the whole i to the whole as equation number one this is equation number two now the right hand side of both of these equations are seen therefore to get the conjugate of z1 plus z2 is equal to conjugate of z1 plus conjugate of z2. Now let us define into which the conjugate of the difference of two complex numbers is equal to the difference of their conjugates that is if z1 of z1 minus z2 is equal to conjugate of z1 minus conjugate of z2 plus number z1 is equal to x1 and the complex number z2 is equal to x iota now the conjugate of z1 minus z2 is equal to the conjugate of the whole minus which can be now the conjugate of this number can be obtained by changing the sign of the imaginary part so this will be equal to x1 minus x to the whole minus y1 minus y to the whole into iota further as y2 iota the whole now the conjugate of z1 is equal to x1 minus y1 iota z2 is equal to x2 minus y2 iota the conjugate of z1 minus is the conjugate of z2 therefore conjugate of the difference of two is the product of two complex numbers is the product that is if z1 and z2 plus numbers into z2 is equal to conjugate of z1 into conjugate of z1 is equal to x1 plus y1 iota that is the conjugate of z1 and conjugate of z1 now iota the whole into obtain conjugate of x1 minus as iota square is minus 1 so it will be minus y1 y2 the whole plus x2 y1 the whole now here also we can obtain the conjugate of this number by changing the sign of the imaginary so this will be equal to minus y1 y to the whole minus now z1 into conjugate of z2 will be equal to iota the whole conjugate of z1 into z2 is equal to conjugate of z1 into conjugate of z2 z1 into z2 into z3 and is equal to conjugate of z1 into conjugate of z2 into conjugate of z3 and so on to conjugate of z n. Now let us discuss the next property which is the conjugate of the quotient of two complex numbers, non-zero denominators z 1 where z 1 and z 2 d two complex numbers is equal to the conjugate of z 1 over conjugate of z 1 over z 2 is equal to y 1 iota y 2 the whole. I am using the denominator by multiplying the numerator and denominator by the conjugate of the denominator. Now the conjugate of the denominator is x 2 minus y 2 iota. So this will be equal to the conjugate of y 2 iota the whole to the whole over iota the whole plus y 1 y 2 the whole the whole into iota by multiplying equal to minus 1 this is equal to plus y 1 y 2 the whole to the sign of the measuring part. So this will be minus minus x 1 y 2 the whole into iota still is the value of conjugate of z 1 over z 2 the whole. Now let us see equation. Now let us find conjugate of z 1 over conjugate of z 2. Now this is equal to now conjugate of z 1 is equal to conjugate of z 2 is x 2 minus y 2 iota. Now on rationalizing the denominator this will be equal to this y 1 iota the whole this y 2 iota the whole is y 2 iota. This implies conjugate of z 1 over conjugate of z 2 is equal to 2 the whole i to the whole into iota. Now from a and b z 1 over z 2 the whole is equal to conjugate of z 1 over conjugate of z 2. For the equation a and b conjugate of a complex number is equal to the conjugate of the square of that complex number that is the complex number then the square of the conjugate of z is equal to now let the complex number z is equal to then the conjugate of z is equal to x minus y as the conjugate of z. Then is y iota whole square now applying the formula of a minus b whole square this will be equal to z square is minus 1 so it will be x square minus y square the whole minus 2 x y iota that square will be equal to x plus y iota whole square. Now applying the formula of a plus b whole square which is this y square the whole of the square of z will be equal to the conjugate of this number which will be minus y square the whole now we are changing the sign of the imaginary part so this will be minus 2 x y iota. Now let this be equation a and this be equation b of z is equal to the conjugate of black's numbers. So in this session we have learnt about conjugate of a complex number and properties of conjugates of complex numbers. You all have enjoyed the session.