## Irrational Number

A number which cannot be expressed as a Fraction for any Integers and . Every Transcendental Number is irrational. Numbers of the form are irrational unless is the th Power of 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 for rational . The irrationality of was proven by Lambert in 1761; for the general case, see Hardy and Wright (1979, p. 46). is irrational for Positive integral . The irrationality of was proven by Lambert 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 therefore irrational) 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 are algebraically 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 the Euler-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 called Lagrange 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 Noble Number, Noble Number, Pythagoras's Theorem, Quadratic Irrational Number, Rational Number, Segre's Theorem, Transcendental Number

References

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.