Final answer:
In the complex number a + bi, a and b are always real numbers, which can be demonstrated by multiplying the complex number by its conjugate, resulting in the sum of the squares of a and b.
Step-by-step explanation:
In the complex number a + bi, where i is the imaginary unit, the values a and b are always real numbers. A complex number is defined by a real part a and an imaginary part b, where b is multiplied by the imaginary unit i. This structure allows for complex numbers to encapsulate both real and imaginary components in a single number system.
When you multiply a complex number by its conjugate, for example, A * A' = (a + ib) * (a - ib), you end up with a² + b². This result clearly shows that the complex parts 'vanish' because the product of i and -i is -1, which when multiplied by b² gives a real number. Hence, the product of a complex number and its conjugate results in a real number, specifically the sum of the squares of the real and imaginary parts of the original complex number.