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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
, improving on earlier work by Dirichlet
, states that for any irrational number
, there are infinitely many rational approximations
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
- 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 18th
century that it was proved that
are irrational (and transcendental), the 20th
century for Apéry's constant
, and the rationality of the Euler-Mascheroni constant
is an open problem
- ↑ 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]