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Exercise 4.4.5. In this exercise you will give a different proof that there are exactly 4 4th roots of unity, by showing that any complex apart from 1, -1, i, or -i cannot possibly be a 4th root of unity. First we suppose that w is a complex number such that w g {1,-1, i, -i}. (a) Show that (w – 1)(w+1)(w – i)(w+ i) # 0. (*Hint*) (b) Show that this implies that w is not a 4th root of unity. (*Hint*)

Exercise 4.4.5. In this exercise you will give a different proof that there are exactly 4 4th roots of unity, by showing that any complex apart from 1, -1, i, or -i cannot possibly be a 4th root of unity. First we suppose that w is a complex number such that w g {1,-1, i, -i}. (a) Show that (w – 1)(w+1)(w – i)(w+ i) # 0. (*Hint*) (b) Show that this implies that w is not a 4th root of unity. (*Hint*)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 4.4.5. In this exercise you will give a different proof that there are exactly 4 4th roots of unity, by showing that any complex apart from 1, -1, i, or -i cannot possibly be a 4th root of unity. First we suppose that w is a complex number such that w ¢ {1, –1, i, –i}. (a) Show that (w – 1)(w+ 1)(w – i)(w + i) # 0. (*Hint*) (b) Show that this implies that w is not a 4th root of unity. (*Hint*)
Transcribed Image Text:Exercise 4.4.5. In this exercise you will give a different proof that there are exactly 4 4th roots of unity, by showing that any complex apart from 1, -1, i, or -i cannot possibly be a 4th root of unity. First we suppose that w is a complex number such that w ¢ {1, –1, i, –i}. (a) Show that (w – 1)(w+ 1)(w – i)(w + i) # 0. (*Hint*) (b) Show that this implies that w is not a 4th root of unity. (*Hint*)
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