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Irrational numbers

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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:
 2 ℵ0
) irrational numbers are transcendental while only countably many,
 ℵ0
, 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
 ξ
, there are infinitely many rational approximations
 m /n
with
${\displaystyle \left|\xi -{\frac {m}{n}}\right|<{\frac {1}{n^{2}{\sqrt {5}}}}}$
and this theorem is sharp in the sense that
 2√  5
cannot be replaced with a larger number, nor can the exponent
 2
be replaced with a larger number (even allowing an arbitrarily small positive number in place of
 2√  5
). However, by omitting certain classes of algebraic numbers (such as the golden ratio
 φ
), the constant can be improved to
 2√  9 − 4 / [A002559(n)] 2
. For example, for any irrational number
 ξ
not of the form
${\displaystyle {\frac {a\varphi +b}{c\varphi +d}},\quad ad-bc=\pm 1,}$
there are infinitely many rational approximations
 m /n
with
${\displaystyle \left|\xi -{\frac {m}{n}}\right|<{\frac {1}{n^{2}{\sqrt {8}}}}.}$ For this reason
 φ
is sometimes considered "the most irrational number": the partial denominators of its simple continued fraction being
 1
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
 2√  2
is irrational, while it wasn't until the 18th century that it was proved that
 e
and
 π
are irrational (and transcendental), the 20th century for Apéry's constant
 ζ (3)
, and the rationality of the Euler-Mascheroni constant
 γ
is an open problem.

Notes

1. A. Hurwitz, Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche, Mathematische Annalen 39:2 (June 1891), pp. 279-284.
2. 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.]
3. K. Th. Vahlen, Ueber Näherungswerte und Kettenbrüche, J. Reine Angew. Math. 115 (1895), pp. 221-233.
4. Weisstein, Eric W., Hurwitz's Irrational Number Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/HurwitzsIrrationalNumberTheorem.html]