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Irrational numbers are numbers which can't be expressed as the ratio of two integers (not rational numbers); in other words they are not the root of any linear polynomial, i.e. not algebraic numbers of degree one. Irrational numbers are either
Transcendental numbers are obviously irrational; most (uncountably many:
) irrational numbers are transcendental while only countably many,
, are
algebraic.
Rational approximations of irrational numbers
Rational numbers can be used to approximate irrational numbers. The best rational approximations of a number are obtained from the convergents from simple continued fractions.
A theorem of Hurwitz
[1], improving on earlier work by Dirichlet
[2] and Vahlen
[3][4], states that for any irrational number
, there are infinitely many rational approximations
with
and this theorem is sharp in the sense that
cannot be replaced with a larger number, nor can the exponent
be replaced with a larger number (even allowing an arbitrarily small positive number in place of
). However, by omitting certain classes of algebraic numbers (such as the
golden ratio ), the constant can be improved to
. For example, for any irrational number
not of the form
there are infinitely many rational approximations
with
- For this reason is sometimes considered "the most irrational number": the partial denominators of its simple continued fraction being makes it the worst case for approximation by convergents.
Irrationality of a number
The irrationality of a given number is not always known for certain. Since the time of
Pythagoras, it has been known that
is irrational, while it wasn't until the 18
th century that it was proved that
and
are irrational (and transcendental), the 20
th century for
Apéry's constant , and the rationality of the
Euler-Mascheroni constant is an
open problem.
Notes
- ↑ A. Hurwitz, Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche, Mathematische Annalen 39:2 (June 1891), pp. 279-284.
- ↑ P. G. L. Dirichlet, Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen, SBer. Kgl. Preuß. Akad. Wiss. Berlin (1842), pp. 93-95. Reprinted in P. G. L. Dirichlet, Werke, vol. 1, Springer, Berlin (1889), pp. 633-638.]
- ↑ K. Th. Vahlen, Ueber Näherungswerte und Kettenbrüche, J. Reine Angew. Math. 115 (1895), pp. 221-233.
- ↑ Weisstein, Eric W., Hurwitz's Irrational Number Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/HurwitzsIrrationalNumberTheorem.html]