 Hello and welcome to this session. In this session we will learn about square root of a complex number, cube roots of unity and properties of cube roots of unity. First of all let us discuss square root of a complex number. Now let the square root of a complex number a plus b iota plus y iota that is square root of a plus b iota is equal to x plus y iota where x and y belong to the set of real numbers. Now squaring both sides this implies a plus b iota is equal to x plus y iota whole square which is equal to x square minus y square as iota square is minus 1 plus 2 x y iota. Now equating the real and imaginary dots on both sides we get now equating the real dot we get a is equal to x square minus y square and equating the imaginary part we get b is equal to 2 x y. Now let this be equation number 1 and this 2 is the equation number 2. Now x square plus y square whole square can be written as x square minus y square whole square plus 4 x square y square which implies x square plus y square is equal to square root of x square minus y square whole square plus 4 x square y For some price, x square plus y square is equal to square root of, now a is equal to x square minus y square, so x square minus y square whole square will be equal to a square plus g is equal to 2 whole square, so whole x square y square which can be written as 2 x y whole square will be equal to b square. Now let this be equation number 3 and this is the equation number 3, so solving 1 and 3 we get a is equal to into a plus plus b square the whole, we get y square is equal to 1 by 2 into square root of a square plus b square minus a the whole, which implies x is equal to a root of 1 by 2 into a root plus square root of a square plus b square the whole plus minus square root of 1 by 2 into square root of a square plus b square minus a the whole. Now this is the equation number 2, y is equal to b, now if b is greater than 0 then negative but if b is less than 0 equal to 0 then from this equation we can say that x and y will have either positive signs or negative signs such if b is less than 0 then a square root of a plus b i is equal to obtaining the values of x and y that is the required into square root of a square plus b square is equal to minus square root of 1 by 2 into square root of a square plus b square the whole minus a root of 1 by 2 into square root of a square plus b square b is less than 0. The x and y will have the opposite side. Now let us discuss some of the important points as well as of a complex number example. In this, let the nth unit of the complex number a plus b over tau is equal to s where a, b are real numbers and b is equal to 0. Then tau is equal to x raise to the power n which is equal to real number plus to the power n the square root of a plus b iota is equal to x plus y iota where a, b, x and y are real numbers then iota is equal to this y iota. Now the cube roots of unity, my left is equal to 1 raise to the power 1 by 3 1 is equal to c minus 1 the whole into x square plus x plus 1 the whole this is equal to 0 and this implies and here this is a quadratic equation in x. So by using the formula we can find this b that is minus 1 plus minus square root of b square that is 1 square which is 1 minus 4 you see that is 4 that is 2 into 1 which is 2. This whole into minus 1 minus 3 whole upon 2 this is whole equal to root 3 whole upon 2 tau minus 1 is equal to iota root 3 whole upon 2 therefore of unity root 3 whole upon 2 minus 1 minus iota root 3 whole upon 2 root is real that is 1 and the other two roots are the complex numbers. Let us discuss the properties of cube roots of unity. Now the first property complex of unity is the square of other. Now we know that the complex cube roots of unity minus 1 plus iota root 3 whole upon 2 the complex roots of unity minus 1 plus iota root 3 whole upon 2 whole square is equal to plus of minus 3 as iota square is minus 1 minus 2 iota root 3 which is equal to minus 2 minus 2 into iota root 3 which is further equal to minus iota root 3 cube root of minus 1 minus iota root 3 whole upon 2 then this will be equal to 1 plus iota root 3 x cube root of unity cube root of unity is denoted by omega then cube root would be omega square plus cube root of unity is the square of other. So if we are taking one complex cube root as omega then the other would be omega square the three cube roots of unity omega omega square. Now let us discuss the second property that is of unity is 0 1 plus omega plus omega square is equal to 0. Now let us prove this omega square will be equal to 1 3 whole upon 2 the whole plus minus 1 minus iota root 3 whole upon 2 the whole iota root 3 minus 1 minus iota root 2 this is equal to 0 by 2 which is equal to 0 therefore the sum of three cube roots of unity. Now let us discuss the next property the product of unity 1 into omega into omega square is equal to 1. Now let us prove this now 1 into omega into omega square will be equal to 1 into whole upon 2 the whole into minus on solving this is equal to minus 1 whole square minus iota root 3 whole square minus 1 so this is equal to 4 over 4 which is equal to 1 therefore the product of three cube roots of unity is 1. So 1 into omega into omega square is equal to 1 implies omega cube is equal to 1. Now let us discuss the next property which is the conjugate of omega is equal to omega square and the conjugate of omega square is equal to omega. Now let us prove this of omega is equal to the conjugate of minus 1 plus i whole upon minus 1 minus iota omega square the conjugate of omega square is equal to the conjugate of minus 1 minus iota which is equal to iota root 3 whole upon 2 which is each complex cube root of unity the reciprocal now let iota root 1 plus iota root 3 whole upon 2 and beta is equal to minus iota root 3 whole upon 2. Now alpha into beta is equal to minus 1 plus iota root 3 whole upon 2 the whole into minus 1 minus iota root 3 whole upon 2 the whole which is equal to minus 1 whole square minus iota root 3 whole square which is equal to 1 plus 3 whole upon 4 which is equal to 4 over 4 implies alpha is equal to 1 over beta beta is equal to 1 over alpha. We can say that each complex cube root of unity is the reciprocal of other. Now let us discuss the powers of omega of 3 cube roots of unity is 1 and from that we have got the result as is equal to 1 raised to power 4 can be written as omega raised to power 3 into omega which is equal to now omega cube is 1 so this is equal to 1 into omega which is equal to now omega raised to power 5 can be written as omega raised to power 3 into omega square now omega cube that is omega raised to power 3 is equal to 1 so it will be equal to 1 into omega square which is equal to omega square now omega raised to power 6 can be written as omega raised to power 3 whole raised to power 2 1 square which is equal to 1 now omega raised to power 3 n can be written as omega raised to power 3 whole raised to power now this is equal to now omega raised to power 3 is equal to 1 so this is equal to 1 raised to power n which is equal to 1 also omega raised to power and written as omega raise to power 3 m into omega which is further equal to omega raise to power 3 4 raise to power n into omega. Which is equal to 1 raise to power n into omega which is equal to 1 into omega, which is equal to omega. Now omega raise to power, minus is 3m is equal to 1 by the omega raise to power 3m which can be written as 1 over omega raise to power 3 that is omega cube raise to power is further equal to 1 by the 1 raise to power m which is equal to 1 over 1 which is equal to 1. In terms of omega, in terms of we have learnt about complex number, cube groups of unity or cube groups of unity. So this completes our session. Hope you all have enjoyed the session.