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Chapter 15: Problem 1

List in ascending order the fractions that appear in the Farey sequences \(F_{7}\) and \(F_{8}\).

Short Answer

Expert verified
0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1.

Step by step solution

01

Understanding Farey Sequences

Farey sequences, denoted by \(F_n\), are sequences of completely reduced fractions between 0 and 1, arranged in increasing order, such that the denominator of these fractions is less than or equal to \(n\). We will first list the fractions for \(F_7\) and \(F_8\).
02

Identify Fractions in \(F_7\)

Fractions in \(F_7\) are 0/1, 1/7, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 6/7, and 1/1. These fractions have denominators 7 or less.
03

Identify Fractions in \(F_8\)

Fractions in \(F_8\) are 0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 3/4, 4/5, 5/6, 7/8, and 1/1. These are reduced fractions where denominators are 8 or less.
04

Combine and Sort the Fractions

Combine the fractions from both \(F_7\) and \(F_8\), while ensuring to only include unique fractions. Then, sort them into ascending order: 0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1.
05

Verify Sorted List

Check that each fraction appears once and they are indeed sorted in ascending order from 0/1 to 1/1. Each fraction corresponds to one from either \(F_7\) or \(F_8\) with denominators preserved.

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Key Concepts

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

Reduced Fractions
In mathematics, a reduced fraction is a fraction in which the numerator and denominator share no common factors other than 1. The process of reducing a fraction involves simplifying it to its most basic form.
For example, the fraction \( \frac{4}{8} \) can be reduced to \( \frac{1}{2} \) because the greatest common divisor (GCD) of 4 and 8 is 4. Dividing both the numerator and the denominator by 4 simplifies the fraction.
  • To check if a fraction is reduced, find the GCD of the numerator and denominator. If the GCD is 1, the fraction is already reduced.
  • In reduced fractions, the denominator is important as it determines the size of the pieces the whole is divided into.
Reducing fractions helps in comparing fractions easily and in number theory problems, like Farey sequences, where only reduced fractions are listed in a sequence. This simplification process is crucial for clear mathematical expressions.
Number Theory
Number theory is a fascinating branch of mathematics focused on the properties and relationships of numbers, especially integers. It deals with various elements such as prime numbers, greatest common divisors, and fractions.
A key concept in number theory relevant to Farey sequences is the idea of integer relation through fractions.
  • Integers in number theory help describe mathematical structures and patterns, like the denominators in Farey sequences.
  • Farey sequences use number theory to list fractions that show simple integer relationships, all in reduced form.
Understanding number theory can unravel interesting properties, such as how fractions relate to each other in Farey sequences by minimizing the denominator and creating a list of unique fractions. It's all about uncovering hidden patterns in numbers.
Denominators
The denominator is the bottom part of a fraction, indicating how many equal parts the whole is divided into. In the theory of Farey sequences, the denominator plays a crucial role as it determines which fractions are included in the sequence.
When constructing Farey sequences like \(F_7\) or \(F_8\), the primary rule is that the denominators should be less than or equal to the given number \(n\).
  • In \(F_7\), all denominators are 7 or less, meaning fractions like \( \frac{1}{8} \) would not be included since 8 exceeds 7.
  • Denominators are essential because they guide the order and uniqueness of fractions within Farey sequences.
In essence, denominators help to control and filter the pool of fractions, ensuring that only those that meet specific criteria are examined. This makes it easier to list fractions systematically, validating the structure and organization of the Farey sequences.

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Most popular questions from this chapter

Prove that if the irrational number \(x>1\) is represented by the infinite continued fraction \(\left[a_{0} ; a_{1}, a_{2}, \ldots\right]\), then \(1 / x\) has the expansion \(\left[0 ; a_{0}, a_{1}, a_{2}, \ldots\right] .\) Use this fact to find the value of \([0 ; 1,1,1, \ldots]=[0 ; \overline{1}]\).

Find the simple continued fraction representation of \(3.1416\), and that of \(3.14159 .\)

Express each of the rational numbers below as finite simple continued fractions: (a) \(-19 / 51\). (b) \(187 / 57\). (c) \(71 / 55\). (d) \(118 / 303\).

Compute the convergents of the following simple continued fractions: (a) \([1 ; 2,3,3,2,1]\). (b) \([-3 ; 1,1,1,1,3]\). (c) \([0 ; 2,4,1,8,2]\).

Obtain the immediate successor to the fraction \(\frac{5}{8}\) in the Farey sequence \(F_{11}\). [Hint: Use the initial part of Theorem 15.10.]
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