Answer 21-9
When we want to multiply one complex number by another, we treat both factors as binomi-
als, keeping in mind the fact that j^2 =−1. Therefore,
(a 1 +jb 1 )(a 2 +jb 2 )=a 1 a 2 +ja 1 b 2 +jb 1 a 2 +j^2 b 1 b 2
=a 1 a 2 +ja 1 b 2 +jb 1 a 2 −b 1 b 2
=a 1 a 2 −b 1 b 2 +ja 1 b 2 +jb 1 a 2
= (a 1 a 2 −b 1 b 2 )+j(a 1 b 2 +b 1 a 2 )
Question 21-10
What is the conjugate of a complex number a+jb, where a and b are real numbers? What
happens when we add a complex number to its conjugate? What happens when we multiply
a complex number by its conjugate?
Answer 21-10
We can get the conjugate of any complex number a+jb by reversing the sign of the imagi-
nary part. Therefore, a+jb and a−jb are conjugates of each other. When we add a complex
number to its conjugate using the rule from Answer 21-8, we get
(a+jb)+ (a−jb)= (a+a)+ (jb−jb)= 2 a+j 0
= 2 a
When we multiply a complex number by its conjugate using the rule from Answer 21-9,
we get
(a+jb)(a−jb)=a^2 −jab+jba−j^2 b^2
=a^2 −jab+jab+b^2
=a^2 +b^2
Chapter 22
Question 22-1
What is the polynomial standard form for a quadratic equation in the variable x?
Answer 22-1
When a quadratic equation in x is written in polynomial standard form, it’s formatted like this:
ax^2 +bx+c= 0Part Three 501