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Irrational Numbers

Ivan Niven
Volume: 11
Copyright Date: 1985
Edition: 1
Pages: 177
https://www.jstor.org/stable/10.4169/j.ctt5hh8zn

Table of Contents

1. Front Matter
(pp. i-vi)
2. PREFACE
(pp. vii-x)
Ivan Niven
3. Table of Contents
(pp. xi-xii)
4. CHAPTER 1 RATIONALS AND IRRATIONALS
(pp. 1-14)

Our general intention in this book is to characterize, classify, and exhibit irrational numbers in various ways, not only in the framework of real numbers but also in the larger setting of complex numbers. We do not examine the axiomatic foundations of our subject, preferring simply to take for granted the following basic classification. A rational number is one that can be put in the formh/k,wherehandkare integers withk≠ 0. Real numbers like$\sqrt 2$which are not rational are said to be irrational.†

The first three sections of this chapter are devoted to...

5. CHAPTER 2 SIMPLE IRRATIONALITIES
(pp. 15-27)

In this chapter we establish the irrationality of some well-known numbers of elementary mathematics. The general criteria for irrationality given in Theorems 1.7 and 1.9 will not suffice for these numbers, and so in the next section we introduce a technique of Diophantine approximation. However, before getting into that, we set forth one more elementary criterion for irrationality, the Gaussian

-generalization of the usual proof that$\sqrt 2$is irrational.

Theorem 2.1. If the real numberxsatisfies an equation

${x^n} + {c_1}{x^{n - 1}} + {\text{. }}{\text{. }}{\text{.}} + {c_n} = 0$

with integral coefficients, thenxis either an integer or an irrational number.

Proof. Suppose that the numberxis...

6. CHAPTER 3 CERTAIN ALGEBRAIC NUMBERS
(pp. 28-41)

The trigonometric functions for rational values of the argument or angle were discussed in the preceding chapter. Now we examine the values of the trigonometric functions for those angles that are rational multiples of π: i.e., for angles that are rational when measured in degrees. Apart from trivial exceptions these values are irrational numbers, but they are also algebraic numbers whose degrees we shall calculate. We now state the prerequisite material needed up to § 4 of this chapter; one or two additional results are required in § 5 and are given there.

Analgebraic numberis one that satisfies...

7. CHAPTER 4 THE APPROXIMATION OF IRRATIONALS BY RATIONALS
(pp. 42-50)

Given an irrational number α, it is clear that there are rational numbersh/kclose to α, so that |a —h/k| is small. How small? Since by Theorem 1.5 the rationale are dense, we can chooseh/kso that for any arbitrary positive ε we have |α —h/k| < ε. If we presumekpositive, this can be written as |kα —h| < εk. This inequality suggests, but does not solve, the problem of trying to select the integerkso thatkα is arbitrarily close to an integer. Stated completely the problem is this: Given an irrational number α,...

8. CHAPTER 5 CONTINUED FRACTIONS
(pp. 51-67)

Consider a pair of integers u₀ and u₁, with${u_1} \ne 0$and (u₀, u₁) = 1. The division algorithm shows that, if u₀ is divided by u₁, there is a unique quotient [u₀/u₁] andaunique remainder, say u₂, with$0\underline \leqslant {u_2} < {u_1}$. If${u_2} \ne 0$, the process continues with u₁ divided by u₂, and in this way we get the Euclidean algorithm

${u_0} = {u_1}[{u_0}/{u_1}] + {u_2}$

${u_0} = {u_2}[{u_1}/{u_2}] + {u_3}$

. . . . . . . . . . .

(5.1)${u_{i - 1}} = {u_i}[{u_{i - 1}}/{u_i}] + {u_{i + 1}}$

. . . . . . . . . . .

${u_{m - 1}} = {u_m}[{u_{m - 1}}/{u_m}] + {u_{m + 1}}$

${u_m} = {u_{m + 1}}[{u_m}/{u_{m + 1}}]$

The remainders satisfy the inequality$0 < {u_{i + 1}} < {u_i}$for$1\underline \leqslant i\underline \leqslant m$. The lasr non-zero remainder,${u_{m + 1}}$,...

9. CHAPTER 6 FURTHER DIOPHANTINE APPROXIMATIONS
(pp. 68-82)

This section is devoted to an improvement of Theorem 4.1, and this improvement is shown to be best possible in the next section. The last part of this chapter deals with the distribution of the fractional parts of the positive integral multiples of an irrational number.

THEOREM 6.1.For any irrational ε there exist infinitely many rational numbers h/k such that

$\left| {\xi - \frac{h} {k}} \right| < \frac{1} {{\sqrt {5{k^2}} }}$.

Proof. We make use of the continued fraction expansion$[{\alpha _0},{\alpha _1},{\alpha _2},...]$ofε, and the convergents${h_n}/{k_n}$there-to. The proof consists in establishing that, of any three consecutive convergents, at least one satisfies the inequality of the theorem....

10. CHAPTER 7 ALGEBRAIC AND TRANSCENDENTAL NUMBERS
(pp. 83-93)

Algebraic and transcendental numbers were defined at the beginning of Chapter 3, and a few basic ideas were outlined. It is customary to separate thecomplexnumbers into the types algebraic and transcendental, whereas it is therealnumbers that are classified as rational and irrational. These usages are readily extended, of course. We would say thata+biis a complex rational if bothaandbare rational. Likewise a real algebraic number is simply one that is simultaneously real and algebraic. If a complex number is algebraic, must its real and complex parts be algebraic? The...

11. CHAPTER 8 NORMAL NUMBERS
(pp. 94-116)

To put the matter roughly at first, a normal number is one in whose decimal expansion all digits occur with equal frequency, and in fact all blocks of digits of the same length occur with equal frequency. For example, if the base is 10, then such a specified digit as 7 occurs with frequency${10^{ - 1}}$, and any triplet of digits such as 357 occurs with frequency${10^{ - 3}}$. As we shall see, almost all numbers have this property, which somewhat justifies the use of the adjective “normal.”

We now formulate the definition exactly. Let$x = {x_1}{x_2}{x_3}...$be an infinite decimal to base...

12. CHAPTER 9 THE GENERALIZED LINDEMANN THEOREM
(pp. 117-133)

Transcendental numbers were first exhibited by Liouville, using a technique which we set forth in Theorem 7.9. Later, Hermite in 1873 proved thateis transcendental (Theorem 2.12), and Lindemann extended the method to π in 1882. The transcendence ofeand π are special cases of a more general theorem of Lindemann which is the subject of this chapter. We state two equivalent forms.

THEOREM 9.1.Given any distinct algebraic numbers${\alpha _1}$, the values${\alpha _1},{\alpha _2},...,{\alpha _m}$the values${e^{\alpha 1}},{e^{\alpha 2}},...,{e^{\alpha m}}$are linearly independent over the field of algebraic numbers.

Alternative statement.Given any distinct algebraic numbers${\alpha _1},{\alpha _2},...,{\alpha _m}$, the equation

(9.1)$\sum\limits_{j = 1}^m {{a_j}{e^{\alpha j}}} = 0$...

13. CHAPTER 10 THE GELFOND-SCHNEIDER THEOREM
(pp. 134-150)

In 1900 David Hubert announced a list of twenty-three outstanding unsolved problems. The seventh problem was settled by the publication of the following result in 1934 by A. O. Gelfond, which was followed by an independent proof by Th. Schneider in 1935.

THEOREM 10.1.If α and β are algebraic numbers with$\alpha \ne 0$,$\alpha \ne 1$, and if β is not a real rational number, then any value of${\alpha ^\beta }$is transcendental.

Remarks.The hypothesis that “βis not a real rational number” is usually stated in the form “βis irrational.” Our wording is an attempt to avoid the suggestion that...

14. LIST OF NOTATION
(pp. 151-152)
15. GLOSSARY
(pp. 153-156)
16. REFERENCE BOOKS
(pp. 157-158)
17. INDEX OF TOPICS
(pp. 159-162)
18. INDEX OF NAMES
(pp. 163-164)