 Hello and welcome to the session. In this session we will learn about properties of addition with complex numbers. Now we know that any number is a form plus v iota where a and b are real numbers is called a complex number. And here a is called the real part and b is called the imaginary part of the complex number. Now let us start with the properties of addition with complex numbers. Here is the closure property. Now the search of complex numbers is closed under addition. If we perform addition operation on two or more complex numbers the resultant will also be a complex number. Two complex numbers z1 and z2. Now let the complex number z1 is equal to a plus b iota and the complex number z2 is equal to c plus d iota where dcd two numbers z1 plus z2 will be equal to a plus c the whole plus v plus d the whole into a iota which is of the form xq a iota. a, b, c and d are real numbers then a plus c and b plus d are also real numbers. That means z1 plus z2 is of the form p plus q iota where p and q are real numbers. That means z1 plus z2 is also a complex number. Thus we can say that a set of complex numbers is closed under addition. The second property that is the commutative property now addition is commutative that is let z1 is equal to a plus b iota plus d iota where a, b, c and d are real numbers then is equal to plus b iota plus d iota the whole which is equal to a plus c the whole plus d the whole into a iota. Now addition is commutative and the set of real numbers c plus a the whole into a iota. a iota the whole is equal to z2 plus therefore equal to z2 plus z1. There is z1 and z2 belongs to the set of complex numbers. Thus we can say addition is commutative in the set of complex numbers and that z1 plus z2 is equal to z2 plus z1. Now let us discuss the associated property now the addition in the set of complex numbers to the whole plus z3 is equal to z1 plus z3 the whole z2 z3 complex numbers. Now let us view this property now let z1 is equal to a plus b iota z2 is equal to c plus d iota and z3 a, b, c, d and now first of all z1 is z2 the whole plus z3 Now putting the values of z1, z2 and z3 this will be equal to plus b iota plus c plus d iota the whole equal to iota the whole. Now by the definition of addition this can be further written as c the whole plus e the whole plus d the whole plus f the whole into iota. Now a, c, e, b, g, f are real numbers and this can be further written as the definition of addition this can be written as b iota the whole into iota the whole. Now we are equal to a plus b iota the whole plus d iota the whole plus e plus f iota the whole c plus d iota is equal to z2 the whole. z1 plus z2 the whole plus z3 is equal to z1 plus z2 plus z3 the whole where z1, z2 and z3 belongs to the set of complex numbers. Hence addition is associated in the set of complex numbers plus the existence of additive identity. Now the complex number 0 plus 0 iota which is equal to 0 is the additive identity in the set of complex numbers c that is if set is a complex number 0 is equal to z is equal to 0 plus z where z belongs to the set of complex numbers. Now let us prove this let z is equal to a plus b iota where a, b are real numbers and is equal to a plus b iota the whole plus 0 plus 0 iota the whole by the definition of addition this can be further written as a plus 0 the whole plus b plus 0 the whole into iota which is further equal to a plus b iota and a plus b iota is equal to 0 plus z is equal to 0 plus 0 iota the whole plus a plus b iota the whole which can be further definition of addition 0 plus a the whole plus 0 plus b the whole into iota which is equal to a plus a plus b iota is equal to the complex number thus from these two is equal to z and z plus z is also equal to f belongs to the set of complex numbers. Now let us discuss the existence of additive inverse. Now every element in the set of complex numbers has an additive inverse. Flex number x number minus z minus z is equal to 0 is equal to minus z the whole. Now let us prove it the complex number z is equal to a plus b iota where a and b are real numbers the minus z will be equal to minus a where minus z is also a complex number. Now plus or minus z is equal to a plus b iota the whole which is equal to minus a the whole b plus or minus b the whole into iota which is equal to 0 plus 0 iota which is equal to 0 the whole plus z is equal to 0 therefore z plus of minus z is equal to 0 is equal to minus z the whole. Flex number z a complex number minus z such that equal to 0 is equal to minus z the whole plus z therefore tip inverse learnt about the properties of addition with complex numbers. So this completes our session hope you all have enjoyed the session.