Irrational Number
An irrational number is a number that cannot be expressed as a fraction 
for any integers 
and 
. Irrational numbers
have decimal expansions that neither terminate
nor become periodic. Every transcendental number
is irrational.
There is no standard notation for the set of irrational numbers, but the notations 
, 
, or 
, where the bar,
minus sign, or backslash indicates the set complement of the rational
numbers 
over the reals 
, could all be
used.
The most famous irrational number is 
, sometimes
called Pythagoras's constant. Legend has
it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate
the irrationality of 
while at sea and, upon notifying
his comrades of his great discovery, was immediately thrown overboard by the fanatic
Pythagoreans. Other examples include 
, 
, 
, etc. The Erdős-Borwein
constant
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
(OEIS A065442; Erdős 1948, Guy 1994), where 
is the numbers of divisors of 
, and a set of generalizations (Borwein 1992) are
also known to be irrational (Bailey and Crandall 2002).
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 
. 
is irrational
for every rational number 
(Niven 1956,
Stevens 1999), and 
(for 
measured in
degrees) is irrational for every rational 
with the exception of 
(Niven 1956). 
is irrational for every rational
(Stevens 1999).
The irrationality of e was proven by Euler in 1737; for the general case, see Hardy and Wright (1979, p. 46). 
is irrational
for positive integral 
. The irrationality
of pi itself 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 (1979; van der Poorten 1979). In addition, T. Rivoal (2000)
recently proved that there are infinitely many integers 
such that 
is irrational. Subsequently, he also showed
that at least one of 
, 
, ..., 
is irrational (Rivoal 2001).
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 Gelfond's constant 
(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
 |
(4)
|
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
 |
(5)
|
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
 |
(6)
|
where 
is the divisor
function, is irrational for 
and 2.
Apéry, R. "Irrationalité de 
et 
." Astérisque 61,
11-13, 1979.
Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.
Preprint dated Feb. 22, 2003 available at https://www.nersc.gov/~dhbailey/dhbpapers/bcnormal.pdf.
Borwein, P. "On the Irrationality of Certain Series." Math. Proc. Cambridge Philos. Soc. 112, 141-146, 1992.
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.
Erdős, P. "On Arithmetical Properties of Lambert Series." J. Indian Math. Soc. 12, 63-66, 1948.
Gourdon, X. and Sebah, P. "Irrationality Proofs." https://numbers.computation.free.fr/Constants/Miscellaneous/irrationality.html.
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.
Huylebrouck, D. "Similarities in Irrationality Proofs for 
, 
, 
, and 
." Amer. Math. Monthly 108,
222-231, 2001.
Manning, H. P. Irrational Numbers and Their Representation by Sequences and Series. New York: Wiley, 1906.
Nagell, T. "Irrational Numbers" and "Irrationality of the numbers 
and 
." §12-13
in Introduction
to Number Theory. New York: Wiley, pp. 38-40, 1951.
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.
Rivoal, T. "La fonction Zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs." Comptes Rendus Acad. Sci. Paris 331, 267-270, 2000.
Rivoal, T. "Irrationalité d'au moins un des neuf nombres 
, 
, ..., 
." 25 Apr 2001. https://arxiv.org/abs/math.NT/0104221.
Sloane, N. J. A. Sequence A065442 in "The On-Line Encyclopedia of Integer Sequences."
Stevens, J. "Zur Irrationalität von 
." Mitt.
Math. Ges. Hamburg 18, 151-158, 1999.
van der Poorten, A. "A Proof that Euler Missed... Apéry's Proof of the Irrationality of 
." Math. Intel. 1,
196-203, 1979.
Weisstein, E. W. "Books about Irrational Numbers." https://www.ericweisstein.com/encyclopedias/books/IrrationalNumbers.html.
Weisstein, Eric W. "Irrational Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IrrationalNumber.html
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