Chapter 15: Chapter 15

Problem 431

Find all of the nth roots of 1 . That is, find the values of $z$ which satisfy the equation $$ z^{\mathrm{n}}=1, z \neq 0 . $$

Short Answer

Expert verified

The nth roots of 1 can be expressed as the set: \[ z = \left\{ e^{i\frac{m\pi}{n}} \,|\, m = 0, 1, 2, ..., n-1 \right\} \]

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write the equation in polar form

First, we need to write the equation $z^n = 1$ in polar form. The polar form of a complex number z is given by $z = re^{i\theta}$, where r is the modulus of z and $\theta$ is the argument of z. Since the modulus of the number 1 is 1, and its argument can be any multiple of $2\pi$, we can write the polar form of 1 as $1 = 1 \cdot e^{(2k\pi)i}$, where k is an integer. Now, let's denote the polar form of z as $z = e^{i\alpha}$, substituting this into the given equation, we get: \[e^{in\alpha} = 1 \cdot e^{(2k\pi)i}\]

Step 2: Use De Moivre's theorem

Next, we will use De Moivre's theorem, which states that for any complex number z = r(cos$\theta$ + i sin$\theta$) and any integer n, the nth power of z can be expressed as: $(z^n)^{1/n} = r(\cos(\frac{\theta + 2\pi k'}{n}) + i\sin(\frac{\theta + 2\pi k'}{n}))$, where $k'$ is an integer. Comparing both sides of the equation in polar form, we have, \[\cos(n\alpha) + i\sin(n\alpha) = \cos(2k\pi) + i\sin(2k\pi)\]

Step 3: Equate the real and imaginary parts

To solve for $\alpha$, we can equate the real parts and the imaginary parts: \[\cos(n\alpha) = \cos(2k\pi)\] \[\sin(n\alpha) = \sin(2k\pi)\] Since sin(2kπ) = 0, this implies that nα is a multiple of π, \[n\alpha = m\pi\] where m is an integer. Now, dividing by n, we get \[\alpha = \frac{m\pi}{n}\]

Step 4: Determine the nth roots of unity

Finally, to find all the possible values for z, we substitute $\alpha = \frac{m\pi}{n}$ back into the polar form of z, $z = e^{i\alpha}$: \[z = e^{i\frac{m\pi}{n}}\] To get n distinct values, we take m = 0, 1, 2, ..., n-1. Thus, the nth roots of 1 can be expressed as the set: \[ z = \left\{ e^{i\frac{m\pi}{n}} \,|\, m = 0, 1, 2, ..., n-1 \right\} \]

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Chapter 15: Chapter 15

Find all of the nth roots of 1 . That is, find the values of \(z\) which satisfy the equation $$ z^{\mathrm{n}}=1, z \neq 0 . $$

Short Answer

Step by step solution

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Step 1: Write the equation in polar form

Step 2: Use De Moivre's theorem

Step 3: Equate the real and imaginary parts

Step 4: Determine the nth roots of unity

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Chapter 3

Chapter 4

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Chapter 7

Chapter 8

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Chapter 11

Chapter 12

Chapter 13

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