“Irrational Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

Irrational Number

释义

Irrational Number

A number which cannot be expressed as a Fraction for any Integers and . EveryTranscendental Number is irrational. Numbers of the form are irrational unless is the th Powerof an Integer.

Numbers of the form , where is the Logarithm, are irrational if and are Integers, one of which has a Prime factor which the other lacks. is irrational forrational . The irrationality of was proven by Lambert in 1761; for the general case, see Hardyand Wright (1979, p. 46). is irrational for Positive integral . The irrationality of was proven byLambert in 1760; for the general case, see Hardy and Wright (1979, p. 47). Apéry's Constant (where is the Riemann Zeta Function) was proved irrational by Apéry (Apéry 1979,van der Poorten 1979).

From Gelfond's Theorem, a number of the form is Transcendental (and thereforeirrational) if is Algebraic , 1 and is irrational and Algebraic. This establishes the irrationality of (since ), , and .Nesterenko (1996) proved that is irrational. In fact, he proved that , and arealgebraically independent, but it was not previously known that was irrational.

Given a Polynomial equation

(1)

where

are Integers, the roots

are either integral or irrational. If

is irrational,then so are

, and

Irrationality has not yet been established for , , , or (where is theEuler-Mascheroni Constant).

Quadratic Surds are irrational numbers which have periodic Continued Fractions.

Hurwitz's Irrational Number Theorem gives bounds of the form

(2)

for the best rational approximation possible for an arbitrary irrational number

, where the

are calledLagrange Numbers and get steadily larger for each ``bad''set of irrational numbers which is excluded.

The Series

(3)

where

is the Divisor Function, is irrational for

and 2. The series

(4)

where

is the number of divisors of

, is also irrational, as are

(5)

for

an Integer other than 0 and

, and

a Rational Number other than 0 or

(Guy 1994).

See also Algebraic Integer, Algebraic Number, Almost Integer, Dirichlet Function,Ferguson-Forcade Algorithm, Gelfond's Theorem, Hurwitz's Irrational Number Theorem, Near NobleNumber, Noble Number, Pythagoras's Theorem, Quadratic Irrational Number, Rational Number,Segre's Theorem, Transcendental Number

References

Irrational Numbers

Apéry, R. ``Irrationalité de et .'' Astérisque 61, 11-13, 1979.

Courant, R. and Robbins, H. ``Incommensurable Segments, Irrational Numbers, and the Concept of Limit.'' §2.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 58-61, 1996.

Guy, R. K. ``Some Irrational Series.'' §B14 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 69, 1994.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Manning, H. P. Irrational Numbers and Their Representation by Sequences and Series. New York: Wiley, 1906.

Nesterenko, Yu. ``Modular Functions and Transcendence Problems.'' C. R. Acad. Sci. Paris Sér. I Math. 322, 909-914, 1996.

Nesterenko, Yu. V. ``Modular Functions and Transcendence Questions.'' Mat. Sb. 187, 65-96, 1996.

Niven, I. M. Irrational Numbers. New York: Wiley, 1956.

Niven, I. M. Numbers: Rational and Irrational. New York: Random House, 1961.

Pappas, T. ``Irrational Numbers & the Pythagoras Theorem.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 98-99, 1989.

van der Poorten, A. ``A Proof that Euler Missed... Apéry's Proof of the Irrationality of .'' Math. Intel. 1, 196-203, 1979.

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