Final answer:
The necessary and sufficient condition for two complex numbers to have a real sum and a real product is that the numbers are either both real or they are complex conjugates of each other.
Step-by-step explanation:
To determine the necessary and sufficient condition for two complex numbers, let's say a and b, to have a real sum and a real product, we must delve into the fundamental properties of complex numbers. A complex number is generally represented as a + bi, where a is the real part and bi is the imaginary part.
Now, the sum of two complex numbers is real if and only if their imaginary parts cancel each other out. This happens in two scenarios: (a) Both a and b are real numbers with no imaginary part, or (b) a and b are complex conjugates of each other, meaning if a is x + yi, then b is x - yi. The sum in both cases will be a real number as the imaginary parts will cancel out.
Similarly, the product of two complex numbers is real if their imaginary parts cancel out in the multiplication process. This also occurs when the two numbers are either both real or complex conjugates. Using the identity A* A = (a + ib) (a − ib) = a² + b², we can see that multiplying a complex number by its complex conjugate yields a real product, as the imaginary parts effectively cancel out.
Therefore, the correct answer to the question is: d) (a) and (b) are real, or complex conjugates of each other. This is because when complex numbers are either both real or complex conjugates, their sum and product will necessarily be real.