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Problem 431

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

Expert verified
The nth roots of 1 can be expressed as the set: $z = \left\{ e^{i\frac{m\pi}{n}} \,|\, m = 0, 1, 2, ..., n-1 \right\}$
See the step by step solution

## 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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