SOLUTION: 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
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Question 178761: 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 SQRT(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
FNone of the above
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
#18.
 = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - (b^2)(i^2) = a^2 - (b^2)(-1)= a^2 + b^2 = \[modulus(z)\]^2)
, hence B true.
 = (a + bi) + (a - bi) = 2a + 0bi = 2a)
, hence C true.
 = (a + bi) - (a - bi) = 0a + 2bi = 2bi)
, hence D true.
I have an objection to selecting answer A, however. Answer A says "All of the following. Since Answer E follows Answer A, the word "All" in Answer A includes Answer E. But... You get the idea. It is sort of like trying to determine the truth value of the sentence: "This statement is false."
=============================================
#19.
For the first four givens, either a or b in 
is zero, so the 'size' of the complex number reduces to the absolute value of the remaining part. The first 4 answers have a size of 3, 7, 4, and 2 respectively. The only thing you now have to determine is whether the size of Answer E is smaller than 7 or not.
Since 
, 
, hence 7 is the largest magnitude of the given numbers. Answer B.
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