Please do question 22 part k and please show each step Proposition 4.2.21. Given z and w are complex numbers, then z+ w =z+ w.PROOF. We may write z as a + bi and w as c+ di. ThenZ+ w = a + bi +c+di= (a – bi) + (c – di)= (a + c) – (b+ d)i= (a + c) + (b+ d)iby definition of conjugatecommutative, associativeby definition of conjugate%3D= z + wby definition of complex additionExercise 4.2.22. Prove each of the following propositions (follow the styleof Proposition 4.2.21).(a) (z) = z(g) |z|3 = |2°| (*Hint*)(b) z. w = zW(h) z-1(*Hint*)|z|2(c) If a is real, then az = az(d) |z| = |7|(i) ]z-1| :1(*Hint*)|z|(e) zz = |z|2(j) (z)-1 = =1(f) |zw| = |z||w|(k) (zw)-1 = w-1z-1

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Answered: (k) (zw)-1 = w-!z-1

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(k) (zw)-1 = w-!z-1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.4: Complex And Rational Zeros Of Polynomials

Problem 36E

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Question

Please do question 22 part k and please show each step

Proposition 4.2.21. Given z and w are complex numbers, then z+ w =
z+ w.
PROOF. We may write z as a + bi and w as c+ di. Then
Z+ w = a + bi +c+di
= (a – bi) + (c – di)
= (a + c) – (b+ d)i
= (a + c) + (b+ d)i
by definition of conjugate
commutative, associative
by definition of conjugate
%3D
= z + w
by definition of complex addition
Exercise 4.2.22. Prove each of the following propositions (follow the style
of Proposition 4.2.21).
(a) (z) = z
(g) |z|3 = |2°| (*Hint*)
(b) z. w = zW
(h) z-1
(*Hint*)
|z|2
(c) If a is real, then az = az
(d) |z| = |7|
(i) ]z-1| :
1
(*Hint*)
|z|
(e) zz = |z|2
(j) (z)-1 = =1
(f) |zw| = |z||w|
(k) (zw)-1 = w-1z-1

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