 Hello and welcome to the session. In this session we will discuss about complex numbers. A number of the form a plus i b is a complex number in which we have a and b are the real numbers and the value for this i is square root minus 1 that is we have i square equal to minus 1. For a complex number z equal to a plus i b we have this a is the real part of the complex number and it is denoted by Rez and b is the imaginary part of the complex number z and it is denoted by Imz. Consider a complex number z equal to 2 plus 3i here this 2 is the real part of the complex number z and 3 is the imaginary part of the complex number z. Next we discuss algebra of complex numbers. First we have addition of two complex numbers. Consider a complex number z1 equal to a plus i b and a complex number z2 equal to c plus id. Then the sum of these two complex numbers given by z1 plus z2 is equal to a plus c plus i into b plus d. The addition of complex number satisfies the following properties. The first one is the closure law according to which we have that for any two complex numbers z1 and z2 their sum that is z1 plus z2 is also a complex number. The next property is the commutative law according to which we have for any two complex numbers z1 and z2 z1 plus z2 is equal to z2 plus z1. Next is the associative law which says that for any three complex numbers z1 z2 z3 z1 plus z2 plus z3 is equal to z1 plus z2 plus z3. The next property is existence of additive identity according to which we have that there exists a complex number 0 plus i0 which is denoted by 0. This is called the additive identity or we can say 0 complex number such that for every complex number z we have z plus 0 is equal to z. Then we have existence of additive inverse that is to every complex number z equal to a plus ib we have a complex number minus z that is minus a plus i into minus b which is called the additive inverse or we can also say negative of the complex number z and we have z plus minus z is equal to the additive identity which is 0. Next we discuss difference of two complex numbers given any two complex numbers z1 and z2 the difference that is z1 minus z2 is equal to z1 plus minus z2. Then next is multiplication of two complex numbers consider a complex number z1 equal to a plus ib and a complex number z2 equal to c plus id then their product that is z1 multiplied by z2 is equal to ac minus bd plus i into ad plus bc. Now multiplication of complex numbers presents the following properties the first one is the closure law according to which we have that for any two complex numbers z1 and z2 their product that is z1 multiplied by z2 is also a complex number. Next is the commutative law which says for complex numbers z1 and z2 z1 multiplied by z2 is equal to z2 multiplied by z1. Then next property is the associative law which says that for any three complex numbers z1 z2 z3 z1 multiplied by z2 the whole multiplied by z3 is equal to z1 multiplied by z2 into z3. Next property is existence of multiplicative identity that is there exists a complex number 1 plus i0 which is denoted as 1 this is called the multiplicative identity such that we have the complex number z multiplied by 1 is equal to z. Then the next property is existence of multiplicative inverse according to which we have that for any non-zero complex number z equal to a plus ib where we have a is not equal to 0 and b is not equal to 0 we have a complex number a upon a square plus b square plus i into minus b upon a square plus b square which is denoted by 1 upon z or z inverse. This is the multiplicative inverse of the complex number z such that we have that z multiplied by 1 upon z is equal to 1. Now the next property is the distributive law that is for three complex numbers z1 z2 z3 we have z1 into z2 plus z3 is equal to z1 into z2 plus z1 into z3 and also z1 plus z2 into z3 is equal to z1 into z3 plus z2 into z3. Now we have division of two complex numbers given any two complex numbers z1 and z2 where we have z2 is not equal to 0 the question z1 upon z2 is defined by z1 multiplied by 1 upon z2. Let's try and find out the product 2 plus 3i multiplied by 4 minus 5i. This is equal to 2 into 4 that is 8 minus 3 into minus 5 that is minus 15 plus i into 2 into minus 5 that is minus 10 plus 3 into 4 that is 12 which is equal to 8 plus 15 plus i into 2 that is we have 23 plus 2i. Next we discuss power of i that is for any integer k we have i to the power 4k is equal to 1 then i to the power 4k plus 1 is equal to i, i to the power 4k plus 2 is equal to minus 1 and i to the power 4k plus 3 is equal to minus i. Now let's try and evaluate i to the power 7 this can be written as i to the power 4 multiplied by i to the power 3 that is we have i to the power 7 is equal to i to the power 4 plus 3 now this is of the form i to the power 4k plus 3 now we know that i to the power 4k plus 3 is equal to minus i so i to the power 7 is equal to minus i. Now we have square roots of a negative real number we know that for all positive real numbers a and b we have square root a multiplied by square root b is equal to square root a b and if we have a and b are negative real numbers then square root a multiplied by square root b is not equal to square root a b. Now we shall discuss some identities the first one is z1 plus z2 the whole square is equal to z1 square plus z2 square plus 2 z1 z2 where we have z1 and z2 are the complex numbers then the next is z1 minus z2 the whole square is equal to z1 square minus 2 z1 z2 plus z2 square next is z1 plus z2 the whole cube is equal to z1 cube plus 3 z1 square z2 plus 3 z1 z2 square plus z2 cube next identity is z1 z2 the whole cube is equal to z1 cube minus 3 z1 square z2 plus 3 z1 z2 square minus z2 cube and next is z1 square minus z2 square is equal to z1 plus z2 multiplied by z1 minus z2. Next we discuss modulus and the conjugate of a complex number consider a complex number z equal to a plus ib then modulus of z is denoted by this and this is equal to square root of a square plus b square which is a non-negative real number then the conjugate of the complex number z is denoted by z bar and this is equal to a minus ib now we have some results regarding the conjugate and the modulus like z multiplied by z bar is equal to modulus z square then modulus z1 z2 is equal to modulus z1 multiplied by modulus z2 next is modulus of z1 upon z2 is equal to modulus z1 upon modulus z2 provided that modulus z2 is not equal to 0 next we have z1 z2 whole bar is equal to z1 bar multiplied by z2 bar then we have z1 plus minus z2 whole bar is equal to z1 bar plus minus z2 bar next we have z1 upon z2 the whole bar is equal to z1 bar upon z2 bar provided that z2 is not equal to 0 consider a complex number z equal to 2 plus 5 i now modulus z is equal to square root 2 square plus 5 square which is equal to square root 4 plus 25 thus we get modulus z is equal to square root 29 then conjugate of the complex number z is given by z bar and this is equal to 2 minus 5 i next is the argon plane a plane having a complex number assigned to each of its point is called the complex plane or the argon plane a complex number z equal to x plus i y corresponds to the ordered pair x y which can be represented geometrically as a unique point P with coordinates x and y in the x y plane so in the argon plane the modulus of a complex number that is modulus z equal to square root x square plus y square is this distance that is the distance between point P and the origin the x axis in the argon plane is called the real axis and the y axis in the argon plane is called the imaginary axis point P is the representation of the complex number z and point P has coordinates x and y now conjugate of the complex number z that is z bar equal to x minus i y is represented by the point Q with coordinates x and minus y this point Q is the mirror image of the point P on the real axis now we discuss polar representation of a complex number let point P represent the complex number z given by x plus i y let the directed line segment OP be of length r and theta be the angle which OP makes with the positive direction of x axis we may note that the point P is uniquely determined by the ordered pair of real numbers r and theta that is we have the point P has coordinates r theta r theta is the polar coordinates of the point P we have a complex number z equal to x plus i y now in this when you take x equal to r cos theta and y equal to r sin theta we get z equal to r into cos theta plus i sin theta this is the polar form of the complex number z here we have modulus of z is given by r that is equal to square root x square plus y square and this theta is called the argument of z or we can also say amplitude of z which is denoted by this value of theta such that theta is greater than minus pi and less than equal to pi is called the principal argument of z and it is denoted by this consider a complex number z equal to minus 2 i let's represent this in the polar form that is we have the complex number z equal to 0 minus 2 i now in the polar form the complex number is of the form z equal to r cos theta plus i sin theta so when we compare these two we get that r cos theta is equal to 0 and r sin theta is equal to minus 2 now from these two that is on squaring and adding these two we get r equal to 2 conventionally we take r greater than 0 when we put r equal to 2 in both these equations we get cos theta equal to 0 and sin theta equal to minus 1 now from these two we get theta equal to minus pi by 2 so now when we put r equal to 2 in this and theta equal to minus pi by 2 in this equation we get z equal to 2 into cos of minus pi by 2 plus i into sin of minus pi by 2 this is the polar form of the given complex number z equal to minus 2 i next we have quadratic equations we already know how to solve the quadratic equations when the discriminant is non-negative that is it is greater than or equal to 0 now for a quadratic equation a x square plus b x plus c equal to 0 where we have a b and c are real coefficients and we have a is not equal to 0 we assume that b square minus 4ac that is the discriminant is negative that is less than 0 so now the solution to this quadratic equation is available in the set of complex number and it is given by x equal to minus b plus minus square root b square minus 4ac upon 2a now since b square minus 4ac is negative so we get x is equal to minus b plus minus square root 4ac minus b square i upon 2a to check that how many roots does an equation have we have a fundamental theorem of algebra according to which we have that a polynomial equation has at least one root this is the fundamental theorem of algebra there is a consequence of this theorem we have a result which is of immense importance which says that a polynomial equation of degree n has n roots consider the equation x square plus 5 equal to 0 let's try and solve this we have x square equal to minus 5 that is we get x equal to plus minus square root minus 5 or we can say that x is equal to plus minus square root 5i this completes the session hope you understood the concept of complex numbers