Question

18. Let z = a + bi represent a general complex number. As noted in the lesson, the conjugate of z, abbreviated conj(z) or conj(a + bi) is the complex number a-bi. Also, the modulus of z, modulus(z) is the "size" of z, or √(a2 + b2). Which of the following is true for all complex numbers?

A) All of the following
B) z*conj(z) = [modulus(z)]2
C) z + conj(z) = 2a
D) z - conj(z) = 2bi
E) None of the above
19. Using the definition of "size" of z from problem #18, which of the following is largest?
A) 3i
B) -7
C) 4
D) -2i
E) 4 + 4i
F) None of the above

Answer 1

Question (18):

Answer: (A) All of the following

(I am using
`z^*` in place of conj(z)

`z*z^*=(a+ib)(a-ib)=a^2-(ib)^2= a^2+b^2=\mbox{modulus(z)}^2`

`z+z^*=a+ib+a-ib=2a\\z-z^*=a+ib-a+ib=2bi`

Question (19):

Answer: (B) -7

modulus of -7 is sqrt(49)=7

the other choices are all < 7