Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- 1 The fundamental theorem of arithmetic
- 2 Modular addition and Euler's ɸ function
- 3 Modular multiplication
- 4 Quadratic residues
- 5 The equation xn + yn = zn, for n = 2, 3, 4
- 6 Sums of squares
- 7 Partitions
- 8 Quadratic forms
- 9 Geometry of numbers
- 10 Continued fractions
- 11 Approximation of irrationals by rationals
- Bibliography
- Index
11 - Approximation of irrationals by rationals
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- 1 The fundamental theorem of arithmetic
- 2 Modular addition and Euler's ɸ function
- 3 Modular multiplication
- 4 Quadratic residues
- 5 The equation xn + yn = zn, for n = 2, 3, 4
- 6 Sums of squares
- 7 Partitions
- 8 Quadratic forms
- 9 Geometry of numbers
- 10 Continued fractions
- 11 Approximation of irrationals by rationals
- Bibliography
- Index
Summary
Naive approach
1 What is the integer nearest to √2? What is the integer nearest to √3?
2 If a is a real number, what can be said about the value of α-[α]? Is there necessarily an integer n such that? If « is irrational, how can this inequality be improved?
3 If the integer points, n, and the points mid-way between them, are marked on a number line, how far away from the nearest of such points can any point on the number line be? Find an integer m such that, and an integer k such that.
For any real number a, must there be an integer m such that
4 If for all integers n, the points and n are marked on a number line, how far away from the nearest of such points can any point on the number line be?
Find an integer m such that, and an integer k such that
For any irrational number a, must there be an integer m such that
5 For any irrational number a must there be an integer m such that What length of the closed interval [1, 2] is within of one of the points 1, or 2?
Find a real number a such that
for all integers m.
6 If α is an irrational number and q is a given positive integer, does there always exist an integer p such that and an integer p such that?
7 Use a pocket calculator to determine the eleven numbers to two places of decimals for n = 1, 2, …, 11.
Give reasons why no two of these eleven numbers are equal, and why none of them have a value for any integer k.
Deduce that each of the eleven numbers belongs to exactly one of the open intervals .
Select two of these numbers which lie in the same open interval and determine integers p and q such that.
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- A Pathway Into Number Theory , pp. 242 - 256Publisher: Cambridge University PressPrint publication year: 1996