In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Beatty sequence (or homogeneous Beatty sequence) is the
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s found by taking the
floor
A floor is the bottom surface of a room or vehicle. Floors vary from simple dirt in a cave to many layered surfaces made with modern technology. Floors may be stone, wood, bamboo, metal or any other material that can support the expected load ...
of the positive
multiples of a positive
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
. Beatty sequences are named after
Samuel Beatty, who wrote about them in 1926.
Rayleigh's theorem, named after
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. A ...
, states that the
complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number.
Beatty sequences can also be used to generate
Sturmian words.
Definition
Any irrational number
that is greater than one generates the Beatty sequence
The two irrational numbers
and
naturally satisfy the equation
.
The two Beatty sequences
and
that they generate form a ''pair of complementary Beatty sequences''. Here, "complementary" means that every positive integer belongs to exactly one of these two sequences.
Examples
When
is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
, the complementary Beatty sequence is generated by
. In this case, the sequence
, known as the ''lower Wythoff sequence'', is
and the complementary sequence
, the ''upper Wythoff sequence'', is
These sequences define the optimal strategy for
Wythoff's game
Wythoff's game is a two-player mathematical subtraction game, played with two piles of counters. Players take turns removing counters from one or both piles; when removing counters from both piles, the numbers of counters removed from each pile m ...
, and are used in the definition of the
Wythoff array
In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence ...
.
As another example, for the
square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
,
,
. In this case, the sequences are
For
and
, the sequences are
Any number in the first sequence is absent in the second, and vice versa.
History
Beatty sequences got their name from the problem posed in ''
The American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.
The ''American Mathematical Monthly'' is an e ...
'' by
Samuel Beatty in 1926. It is probably one of the most often cited problems ever posed in the ''Monthly''. However, even earlier, in 1894 such sequences were briefly mentioned by
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. A ...
in the second edition of his book ''The Theory of Sound''.
Rayleigh theorem
Rayleigh's theorem (also known as Beatty's theorem) states that given an irrational number
there exists
so that the Beatty sequences
and
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of positive integers: each positive integer belongs to exactly one of the two sequences.
First proof
Given
let
. We must show that every positive integer lies in one and only one of the two sequences
and
. We shall do so by considering the ordinal positions occupied by all the fractions
and
when they are jointly listed in nondecreasing order for positive integers ''j'' and ''k''.
To see that no two of the numbers can occupy the same position (as a single number), suppose to the contrary that
for some ''j'' and ''k''. Then
=
, a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
, but also,
not a rational number. Therefore, no two of the numbers occupy the same position.
For any
, there are
positive integers
such that
and
positive integers
such that
, so that the position of
in the list is
. The equation
implies
Likewise, the position of
in the list is
.
Conclusion: every positive integer (that is, every position in the list) is of the form
or of the form
, but not both. The converse statement is also true: if ''p'' and ''q'' are two
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
such that every positive integer occurs precisely once in the above list, then ''p'' and ''q'' are irrational and the sum of their reciprocals is 1.
Second proof
Collisions: Suppose that, contrary to the theorem, there are integers ''j'' > 0 and ''k'' and ''m'' such that
This is equivalent to the inequalities
For non-zero ''j'', the irrationality of ''r'' and ''s'' is incompatible with equality, so
which leads to
Adding these together and using the hypothesis, we get
which is impossible (one cannot have an integer between two adjacent integers). Thus the supposition must be false.
Anti-collisions: Suppose that, contrary to the theorem, there are integers ''j'' > 0 and ''k'' and ''m'' such that
Since ''j'' + 1 is non-zero and ''r'' and ''s'' are irrational, we can exclude equality, so
Then we get
Adding corresponding inequalities, we get
which is also impossible. Thus the supposition is false.
Properties
A number
belongs to the Beatty sequence
if and only if
where
denotes the fractional part of
i.e.,
.
Proof:
Furthermore,
.
Proof:
Relation with Sturmian sequences
The
first difference
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...
of the Beatty sequence associated with the irrational number
is a characteristic
Sturmian word over the alphabet
.
Generalizations
If slightly modified, the Rayleigh's theorem can be generalized to positive real numbers (not necessarily irrational) and negative integers as well: if positive real numbers
and
satisfy
, the sequences
and
form a partition of integers. For example, the white and black keys of a piano keyboard are distributed as such sequences for
and
.
The
Lambek–Moser theorem
The Lambek–Moser theorem is a mathematical description of partitions of the natural numbers into two Complement (set theory), complementary sets. For instance, it applies to the partition of numbers into even number, even and odd number, odd, or ...
generalizes the Rayleigh theorem and shows that more general pairs of sequences defined from an integer function and its inverse have the same property of partitioning the integers.
Uspensky's theorem states that, if
are positive real numbers such that
contains all positive integers exactly once, then
That is, there is no equivalent of Rayleigh's theorem for three or more Beatty sequences.
[R. L. Graham]
On a theorem of Uspensky
''Amer. Math. Monthly'' 70 (1963), pp. 407–409.
References
Further reading
*
* Includes many references.
External links
* {{mathworld, title = Beatty Sequence, urlname = BeattySequence
*
Alexander Bogomolny
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and M ...
Beatty Sequences Cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Integer sequences
Theorems in number theory
Diophantine approximation
Combinatorics on words
Articles containing proofs