 Hello and welcome to the session. In this session, we will learn about properties of modulus of complex numbers. Now we know that if a complex number z is equal to a plus b iota, then modulus of z is equal to square root of a square plus b square, where a is called the real part and b is called the imaginary part of the complex number and a and b are real numbers. Now let us discuss the first property of the modulus of complex numbers and that is the modulus of a complex number and the modulus of its conjugate are equal. That is if z is a complex number, then modulus of z is equal to the modulus of conjugate of z and also the modulus of a negative of a complex number is equal to modulus of a complex number. That is modulus of a complex number is equal to modulus of negative of the complex number. Therefore modulus of z is equal to modulus of conjugate of z is equal to modulus of negative of a complex number. Now let us prove this property for this let the complex number z is equal to a plus b iota, then modulus of z will be equal to square root of a square plus b square and the modulus of conjugate of z will be equal to now conjugate of z will be equal to Now here we will change the sign of the imaginary part in the complex number z so it will be a minus b iota therefore the modulus of the conjugate of z will be equal to square root of a square plus minus b square which is equal to square root of a square plus b square. Now minus z that is the negative of this complex number is equal to minus a minus b iota therefore the modulus of minus z will be equal to square root of minus a square plus minus b square which is equal to square root of a square plus b square. Hence from these three equations modulus of z is equal to modulus of conjugate of z is equal to modulus of minus z. Now let us start with the second property which is minus of modulus of z is less than equal to the real part of z is less than equal to modulus of z and minus of modulus of z is less than equal to the imaginary part of z is less than equal to the modulus of z where z is a complex number. Now let us prove this identity now here let z that is the complex number z is equal to a plus b iota then modulus of z will be equal to square root of a square plus b square. Now a square is less than equal to a square plus b square because b square is greater than equal to 0 which implies the square of the modulus of z that is a square plus b square will be equal to the square of modulus of z is greater than equal to a square which further represents a square root of a square plus b square. This implies modulus of z is greater than equal to plus minus a which further implies modulus of z is greater than equal to a or modulus of z is greater than equal to minus a. This implies modulus of z is greater than equal to a or minus modulus of z is less than equal to a which implies minus modulus of z is less than equal to a is less than equal to modulus of z. Now we know that in the complex number z which is equal to a plus b iota a is called the real part and b is called the imaginary part. So this implies minus modulus of z is less than equal to the real part of z is less than equal to the modulus of z. Similarly we can prove this that is similarly we can guess minus modulus of z is less than equal to b less than equal to modulus of z which implies minus modulus of z is less than equal to the imaginary part of z is less than equal to the modulus of z. Now let us discuss the next property which is the modulus of a complex number is 0 if and only if the complex number is 0. That means the modulus of a complex number that is z is equal to 0 if and only if the complex number is equal to 0 that is the real part of z is equal to the imaginary part of z is equal to 0. Now let us prove this property. Now let the complex number z is equal to a plus b iota then modulus of z is equal to 0 if and only if the modulus of z that is square root of a square plus b square is equal to 0. If and only if a square plus b square is equal to 0 if and only if a is equal to 0 and b is equal to 0. Now we know that a is called the real part of the complex number and b is called the imaginary part of the complex number. So a is equal to 0 and b is equal to 0 if and only if the real part of z is equal to 0 and the imaginary part of z is equal to 0. That means the complex number z is equal to 0 so this proves this identity that modulus of z is equal to 0 if and only if the complex number is equal to 0. Now next property is the product of a complex number with its conjugate is equal to the square of its modulus that is if z is the complex number then z into the conjugate of z is equal to the square of the modulus of z. Now let us prove this identity. Now let's the complex number z is equal to x plus y iota so conjugate of z will be equal to x minus y iota. Now z into the conjugate of z is equal to x plus y iota the whole into x minus y iota the whole. Now applying the formula of a plus b the whole into a minus b the whole which is equal to a square minus b square this is equal to x square minus iota square y square. Now iota square is minus 1 so this will be x square plus y square. Now modulus of z is equal to square root of x square plus y square therefore the square of the modulus of z will be equal to x square plus y square therefore this value will be equal to the square of the modulus of z. Now let us start with the next property and that is the modulus of the product of two complex numbers is equal to the product of their modulus that is if z1 and z2 two complex numbers then modulus of z1 into z2 is equal to modulus of z1 into modulus of z2. Now let us prove this property here that z1 is equal to a plus b iota z2 is equal to c plus b iota where a b c and d are real 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 a c minus b d the whole plus a d plus b iota the whole. Now modulus of z1 into z2 is equal to the square root of a c minus b d whole square plus a d plus b c whole square which is equal to square root of a square c square plus b square d square minus 2 a d plus b iota. Now a b c d plus a square d square plus b square c square plus 2 a b c d. Now these terms are cancelled each other and further this is equal to square root of now from these two terms taking a square common it will be a square into c square plus d square the whole plus from these two terms taking b square common it will be plus b square into c square plus d square the whole which is equal to square root of a square plus b square the whole into c square plus d square the whole. which is equal to square root of a square plus b square into square root of c square plus t square. modulus of z1 is equal to square root of a square plus b square and modulus of z2 is equal to square root of c square plus d square. So this is equal to modulus of z1 into modulus of z2. Hence we have proved the given identity. Now in channel we can write modulus of z1 into z2 and so on up to zn is equal to modulus of z1 into modulus of z2 into so on up to modulus of zn. That is the modulus of the product of m complex numbers is the product of the module of z1 and if z1 is equal to z2 is equal to z then the modulus of the square of z is equal to the modulus of z into the modulus of z which is equal to the square of modulus of z. That is the modulus of the square of a complex number is equal to square of the modulus of that complex number. Similarly if z1 is equal to z2 is equal to so on is equal to zn which is equal to z then the modulus of z raised to power n is equal to the modulus of z raised to power n where n belongs to the set of natural numbers. Now let us discuss the next property that is z inverse that is z raised to power minus 1 is equal to the conjugate of z over the square of modulus of z. Now let us prove this identity. Now we know that z into the conjugate of z is equal to the square of the modulus of z. Now this property we have proved earlier therefore z into the conjugate of z over the square of the modulus of z is equal to 1 which implies the conjugate of z over the square of the modulus of z is equal to 1 over z which is equal to that raised to power minus 1. Hence the given property is proved. Now let us discuss the next property that is the modulus of z1 plus z2 is less than equal to modulus of z1 plus modulus of z2. Now let us prove this identity. Now the square of the modulus of z1 plus z2 is equal to z1 plus z to the whole into the conjugate of z1 plus z to the whole because z into the conjugate of z is equal to the square of the modulus of z. This property that we have proved earlier. So this will be equal to z1 plus z to the whole into now the conjugate of z1 plus z2 is equal to conjugate of z1 plus conjugate of z2 the whole. Now multiplying we get z1 into the conjugate of z1 plus z1 into the conjugate of z2 plus z2 into conjugate of z1 plus z2 into conjugate of z2. Now using this property z1 into the conjugate of z1 can be written as the square of the modulus of z1 and this term can be written as plus the square of the modulus of z2 plus z1 into the conjugate of z2 plus z2 into the conjugate of z1 the whole. Further this is equal to the square of the modulus of z1 plus the square of the modulus of z2 plus z1 into conjugate of z2 plus now this term can be written as the conjugate of z1 into the conjugate of z2 the whole. Now we know that the conjugate of a conjugate of a complex number is equal to the complex number itself. So we can write this as the conjugate of z1 into the conjugate of conjugate of z2 so this will become the complex number z2 so in the resultant we get the conjugate of z1 into z2. Now this is equal to the square of modulus of z1 plus the square of the modulus of z2 plus 2 into real part of z1 into conjugate of z2 because z plus the conjugate of z is equal to 2 real part of z since a is less than equal to square root of a square plus b square therefore the real part of a complex number that is if z is equal to a plus b iota then a is the real part of the complex number can never exceed its absolute value that is the absolute value or the modulus of z that means the real part of z is less than equal to the modulus of z so here the real part of z1 into conjugate of z2 will also be less than equal to the modulus of z1 into conjugate of z2 that is it is less than equal to the square of the modulus of z1 plus the square of the modulus of z2 plus 2 into the modulus of z1 into conjugate of z2 the whole. Now this is less than equal to the square of the modulus of z1 plus the square of the modulus of z2 plus 2 into modulus of z1 into the modulus of conjugate of z2 now we know that the modulus of the conjugate of z is equal to the modulus of z therefore the modulus of the conjugate of z2 will be equal to the modulus of z2 therefore the square of the modulus of z1 plus z2 is less than equal to now this whole expression will become modulus of z1 plus modulus of z2 whole square now taking the positive square root this implies modulus of z1 plus z2 is less than equal to modulus of z1 plus modulus of z2 and this is called the triangle inequality now let us discuss the next property and that is the modulus of z1 minus z2 is greater than equal to modulus of the modulus of z1 minus modulus of z2 now let us prove this property now let the modulus of z1 is greater than modulus of z2 now z1 can be written as z1 minus z2 the whole plus z2 which implies modulus of z1 is equal to modulus of z1 minus z2 the whole plus z2 now this further implies now using the triangle inequality modulus of z1 is less than equal to modulus of z1 minus z2 plus modulus of z2 which further implies modulus of z1 minus modulus of z2 is less than equal to modulus of z1 minus z2 which implies modulus of z1 minus z2 is greater than equal to modulus of z1 minus modulus of z2 and if the modulus of z2 is greater than modulus of z1 then modulus of z2 is equal to modulus of z2 minus z1 the whole plus z1 implies modulus of z2 is less than equal to modulus of z2 minus z1 the whole plus modulus of z1 that is further triangle inequality which implies modulus of z2 minus modulus of z1 is less than equal to modulus of z2 minus z1 which implies modulus of Z2 minus modulus of Z1 is less than equal to modulus of Z1 minus Z2 which further implies modulus of Z1 minus Z2 is greater than equal to modulus of Z2 minus modulus of Z1. Now let us see equation number 1 and this be equation number 2. So from 1 we get modulus of Z1 minus Z2 is greater than equal to modulus of the modulus of Z1 minus modulus of the Z2. Now let us discuss the next property and that is the modulus of quotient of two complex numbers is the quotient of their modulus that is the modulus of Z1 over Z2 is equal to the modulus of Z1 over the modulus of Z2 where Z1 and Z2 are two complex numbers. Now let us prove this property. Now we know that the square of the modulus of Z is equal to Z into the conjugate of Z. Therefore the square of the modulus of Z1 over Z2 is equal to Z1 over Z2 into the conjugate of Z1 over Z2 which is further equal to Z1 over Z2 into the conjugate of Z1 over the conjugate of Z2. Now Z into the conjugate of Z is equal to the square of the modulus of Z. So here this will be equal to the square of the modulus of Z1 over the square of the modulus of Z2. Thus the square of the modulus of Z1 over Z2 is equal to the square of the modulus of Z1 over square of the modulus of Z2 which implies the modulus of Z1 over Z2 is equal to modulus of Z1 over modulus of Z2. So in this session you have learnt about the properties of modulus of a complex number. So this completes our session. Hope you all have enjoyed the session.