Start with two arbitrary complex numbers and perform the operations indicated. Then find the restrictions the real and the imaginary parts should satisfy.
Example Solution: 4+2i and 4-2i
Practice makes perfect
Let's recall the standard form of a complex number.
a+ bi
In the expression above a and b are real numbers, and i=sqrt(-1). In this form a is the real part of the complex number, while b is the imaginary part. Recall that to add (or subtract) two complex numbers, we add (or subtract) their real parts and their imaginary parts separately.
(a+bi) + (c+di ) = (a+c) + (b+d)iTherefore, if we want to get rid of the imaginary part we need that b+d=0. Or equivalently, d=- b. Let's now see what is needed for their product to have just real parts. Again we will consider two arbitrary imaginary numbers, (a+bi) and (c+di).
As we can see, if we want the imaginary part to be zero we have the condition ad+bc = 0. This implies ad = - bc. We can apply the first condition d =- b to the result we just found to simply this a bit more.
We have found a second condition, a=c. Therefore, considering both conditions found, d = - b and c = a, the second arbitrary complex number c+di takes the form shown below.
c + di ⇒ a + ( - b)i=a-bi
With this in mind, we can write an example of two complex numbers whose sum and product is a real number. For example, 4+2i and 4-2i. Let's check this. We will first see if the sum is a real number.
