let z and y are two complect numbers such that:
z= a + bi
y= c+ di
Now let us verify:
sum of compelx numbers:
(a+ bi) + (c+di) = (a+c) + (b+d)i
Then , the sum is a complex number.
Now the difference:
(a+ bi) - (c+di) = (a-c) + (b-d)i
The difference is a complex number.
Now the product:
(a+bi)*(c+di) = (ac + bci + adi + bi*di)
= ac + (bc+ad)i - bd
= (ac-bd) + (bc+ad)i
The product is a complex number
None of them need be complex number
Sum of two compex compex numbers: 2+3i and 2-3i has a sum 4 which is not acomplex number.
The difference of two complex numbers need not be a acomplex number . Example : 5+3i - (3+3i) = 2 is not acomplex number.
Product of 2 complex number need not be a complex number.
example: 3+i and 3-i has the product (3+i)(3-i) = 3^2 -i^2 = 9- (-1) = 10 is not a complex number.
Let us take 2 complex numbers a+ bi and c+ di.
Now the sum of a+ bi and c+ di is (a+c) + (b+d)i , which is a complex number (as you can see a, b, c and d are different)
The difference of the two complex numbers a+ bi –( c+ di )= (a-c) + (b-d)i which is a complex number
The product of a+ bi and c+ di is (a+ bi)*(c+ di) = ac + (ad+cb)i –bd which is also a complex number
The inverse of a+bi = 1/(a+bi)= a-bi/(a+bi)(a-bi) = a-bi/(a^2 + b^2) which is also a complex number.
So all of the given options are complex numbers.