Problem 6 Find the primitive 8 th roots of... [FREE SOLUTION]

91影视

Numbers, Sequences and Series

Keith Hirst

$Math Studyset 91影视 Explanations$ Math

1994 Edition

Chapter 6: Problem 6

Find the primitive 8 th roots of unity.

Short Answer

Expert verified

The primitive 8th roots of unity are $e^{\frac{\pi i}{4}}, e^{\frac{3\pi i}{4}}, e^{\frac{5\pi i}{4}}, e^{\frac{7\pi i}{4}}$.

Step by step solution

- Understand the Concept of Roots of Unity

Roots of unity are complex numbers that give 1 when raised to a specific positive integer power. The 8th roots of unity are the solutions to the equation $z^8 = 1$.

- General Form of the 8th Roots of Unity

The general form of the 8th roots of unity is given by $z = e^{2\pi i k / 8} $ where $k = 0, 1, 2, ..., 7$.

- Identify the Primitive Roots

A primitive 8th root of unity is a complex number that is an 8th root of unity but is not a root of unity for any lower order. To identify these, we need $k$ such that the fraction \frac{k}{8}\ is in its simplest form.

- List All Values of k

List all values of $k$ from 0 to 7: \{0, 1, 2, 3, 4, 5, 6, 7\}.

- Simplify the Fractions

Simplify \frac{k}{8}\ for each $k$: \frac{0}{8} = 0, \frac{1}{8}, \frac{2}{8} = \frac{1}{4}, \frac{3}{8}, \frac{4}{8} = \frac{1}{2}, \frac{5}{8}, \frac{6}{8} = \frac{3}{4}, \frac{7}{8}\.

- Select Primitive Roots

Primitive roots are those where \frac{k}{8}\ is in simplest form. Thus, the primitive 8th roots of unity are for $k = 1, 3, 5, 7$.

- Calculate the Primitive Roots

Substitute $k = 1, 3, 5, 7$ into $e^{2\pi i k / 8} $: $e^{\frac{2\pi i}{8}}, e^{\frac{6\pi i}{8}}, e^{\frac{10\pi i}{8}}, e^{\frac{14\pi i}{8}}$.

- Simplify Each Primitive Root

Simplify each expression: $e^{\frac{\pi i}{4}}, e^{\frac{3\pi i}{4}}, e^{\frac{5\pi i}{4}}, e^{\frac{7\pi i}{4}}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers

Complex numbers are a type of number that include a real part and an imaginary part. The imaginary unit is denoted as 'i', where ewline i^2 = -1...

Primitive Roots

Primitive roots are specific values within the roots of unity that are not roots of unity for any smaller order. For example, in our case...

Euler's Formula

Euler's formula is a key tool in understanding complex exponentiation. It states: ewline ewline e^{ix} = cos(x) + i*sin(x)...

Fraction Simplification

Simplifying fractions helps identify which roots are primitive. A fraction is considered simplest when both the numerator and the denominator are as small as possible...

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