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

• irrational algebraic numbers (a root of a polynomial of minimal degree two or higher, i.e. algebraic numbers of degree two or higher); or
• transcendental numbers (a root of an infinite power series).

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, improving on earlier work by Dirichlet and Vahlen, states that for any irrational number  

ξ

, there are infinitely many rational approximations  

m /n

with

$|\xi -\frac{m}{n}|<\frac{1}{n^2\sqrt{5}}$

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

$\frac{a\phi +b}{c\phi +d},ad-bc=\pm 1,$

there are infinitely many rational approximations  

m /n

with

$|\xi -\frac{m}{n}|<\frac{1}{n^2\sqrt{8}}.$ 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 18^th century that it was proved that  

e

and  

π

are irrational (and transcendental), the 20^th century for Apéry's constant  

ζ (3)

, and the rationality of the Euler-Mascheroni constant  

γ

is an open problem.