Irrational Number

A number which cannot be expressed as a Fraction $p/q$ for any Integers $p$ and $q$. Every Transcendental Number is irrational. Numbers of the form $n^{1/m}$ are irrational unless $n$ is the $m$th Power of an Integer.

Numbers of the form $\log_n m$, where $\log$ is the Logarithm, are irrational if $m$ and $n$ are Integers, one of which has a Prime factor which the other lacks. $e^r$ is irrational for rational $r\not=0$. The irrationality of $e$ was proven by Lambert in 1761; for the general case, see Hardy and Wright (1979, p. 46). $\pi^n$ is irrational for Positive integral $n$. The irrationality of $\pi$ was proven by Lambert in 1760; for the general case, see Hardy and Wright (1979, p. 47). Apéry's Constant $\zeta(3)$ (where $\zeta(z)$ 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 $a^b$ is Transcendental (and therefore irrational) if $a$ is Algebraic $\not=0$, 1 and $b$ is irrational and Algebraic. This establishes the irrationality of $e^\pi$ (since $(-1)^{-i} = (e^{i\pi})^{-i}=e^\pi$), $2^{\sqrt{2}}$, and $e\pi$. Nesterenko (1996) proved that $\pi+e^\pi$ is irrational. In fact, he proved that $\pi$, $e^\pi$ and $\Gamma(1/4)$ are algebraically independent, but it was not previously known that $\pi+e^\pi$ was irrational.

Given a Polynomial equation

$$x^m+c_{m-1}x^{m-1}+\ldots+c_0,$$ (1)

where $c_i$ are Integers, the roots $x_i$ are either integral or irrational. If $\cos(2\theta)$ is irrational, then so are $\cos\theta$, $\sin \theta$, and $\tan\theta$.

Irrationality has not yet been established for $2^e$, $\pi^e$, $\pi^{\sqrt{2}}$, or $\gamma$ (where $\gamma$ 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

$$\left\vert{\alpha-{p\over q}}\right\vert<{1\over L_n q^2}$$ (2)

for the best rational approximation possible for an arbitrary irrational number $\alpha$, where the $L_n$ are called Lagrange Numbers and get steadily larger for each ``bad'' set of irrational numbers which is excluded.

The Series

$$\sum_{n=1}^\infty {\sigma_k(n)\over n!},$$ (3)

where $\sigma_k(n)$ is the Divisor Function, is irrational for $k=1$ and 2. The series

$$\sum_{n=1}^\infty {1\over 2^n-1}=\sum_{n=1}^\infty {d(n)\over 2^n},$$ (4)

where $d(n)$ is the number of divisors of $n$, is also irrational, as are

$$\sum_{n=1}^\infty {1\over q^n+r}\qquad{\rm and}\quad\sum_{n=1}^\infty {(-1)^n\over q^n+r}$$ (5)

for $q$ an Integer other than 0 and $\pm 1$, and $r$ a Rational Number other than 0 or $-q^n$ (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

Irrational Numbers

Apéry, R. ``Irrationalité de $\zeta(2)$ et $\zeta(3)$.'' 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 $\zeta(3)$.'' Math. Intel. 1, 196-203, 1979.