Explanation Formula used: i) Strategy: Proof by Generalized Induction 1. Basic Step: The statement is verified for n = a . 2. Induction Hypothesis: The statement is assumed true for n = k , where k ≥ a . 3. Inductive Step: With this assumption made, the statement is then proved to be true for n = k + 1 . ii) Multiplication in complex number can be defined as, For ( a , b ) , ( c , d ) ∈ ℂ , ( a , b ) ( c , d ) = ( a c − b d , a d + b c ) Proof: To prove this result by using, mathematical induction. The statement is, “ z 1 ⋅ z 2 ⋅ … ⋅ z n ¯ = z 1 ¯ ⋅ z 2 ¯ ⋅ … ⋅ z n ¯ for complex numbers z 1 , z 2 , … , z n . .” Now, for n = 2 Let z 1 , z 2 ∈ ℂ Thus, z 1 = a + b i , z 2 = c + d i with a , b , c , d ∈ ℝ By using the definition of complex conjugate, z 1 ¯ = a + b i ¯ = a − b i , z 2 ¯ = c + d i ¯ = c − d i …... (1) Consider z 1 ⋅ z 2 = ( a + b i ) ⋅ ( c + d i ) By using the definition of multiplication in a complex number, ⇒ z 1 ⋅ z 2 = ( a c − b d ) + ( a d + b c ) i ⇒ z 1 ⋅ z 2 ¯ = ( a c − b d ) + ( a d + b c ) i ¯ By using the definition of a complex conjugate, ⇒ z 1 ⋅ z 2 ¯ = ( a c − b d ) − ( a d + b c ) i ⇒ z 1 ⋅ z 2 ¯ = a c − b d − a d i − b c i By using the addition in a complex number is commutative, ⇒ z 1 ⋅ z 2 ¯ = a c − a d i − b d

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 8.4, Problem 24E

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To prove: z1⋅z2⋅ … ⋅ zn¯ = z1¯ ⋅ z2¯ ⋅ … ⋅ zn¯ for complex numbers z1, z2, …, zn..

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Chapter 8.4, Problem 23E

Chapter 8.4, Problem 25E

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Chapter 8 Solutions

Elements Of Modern Algebra

Ch. 8.1 - True or False Label each of the following...Ch. 8.1 - Prob. 2TFE Ch. 8.1 - Prob. 3TFE Ch. 8.1 - Prob. 4TFE Ch. 8.1 - Prob. 5TFE Ch. 8.1 - Prob. 6TFE Ch. 8.1 - Prob. 7TFE Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E

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To prove: z 1 ⋅ z 2 ⋅ … ⋅ z n ¯ = z 1 ¯ ⋅ z 2 ¯ ⋅ … ⋅ z n ¯ for complex numbers z 1 , z 2 , … , z n . .

Solution Summary: The author explains how to prove a complex number by using mathematical induction.

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To prove: z1⋅z2⋅ … ⋅ zn¯ = z1¯ ⋅ z2¯ ⋅ … ⋅ zn¯ for complex numbers z1, z2, …, zn..

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Chapter 8 Solutions