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Quickly-growing integer sequence
In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers
Irrationality_sequence
Doubly exponential integer sequence
This sequence provides an example showing that double-exponential growth is not enough to cause an integer sequence to be an irrationality sequence. To
Sylvester's_sequence
Function that quantifies how near a number is to being rational
of their irrationality exponent is not known. Below is a table of known upper and lower bounds. The irrationality base or Sondow irrationality measure
Irrationality_measure
Two raised to an integer power
{1}{16x_{2}}}+\cdots } converges to an irrational number. Despite the rapid growth of this sequence, it is the slowest-growing irrationality sequence known. Since it is common
Power_of_two
{\displaystyle \pi ^{2}} is irrational. As in many proofs of irrationality, it is a proof by contradiction. Consider the sequences of real functions A n {\displaystyle
Proof_that_pi_is_irrational
Numbers obtained by adding the two previous ones
inverses de certaines suites récurrentes" [Irrationality of the sum of the reciprocals of certain recurrence sequences], Comptes Rendus de l'Académie des Sciences
Fibonacci_sequence
Mathematical sequence
mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl: The sequence of all multiples of an irrational α, 0, α, 2α
Weyl_sequence
Integers formed by rounding down the integer multiples of a positive irrational number
Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of an irrational number
Beatty_sequence
Number that is not a ratio of integers
as irrational magnitudes. He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:
Irrational_number
Finite or infinite ordered list of elements
In mathematics, a sequence is a collection of objects possibly with repetition, that come in a specified order. Like a set, it contains members (also called
Sequence
Sequence of points that get progressively closer to each other
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given
Cauchy_sequence
Integer sequence
look-and-say sequence is the sequence of integers beginning as follows: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ... (sequence A005150
Look-and-say_sequence
Mathematical concept
number is irrational. The square root of 2 was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square
Quadratic_irrational_number
Kind of infinitely long sequence of characters
as cutting sequences for lines of irrational slope or codings for irrational rotations. They are traditionally taken to be infinite sequences on the alphabet
Sturmian_word
Unique positive real number which when multiplied by itself gives 2
be irrational. The fraction 99/70 (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence A002193
Square_root_of_2
Irrational numbers which appear to be rational
appearing to be rational for periods, and then disintegrating into irrationality. This is illustrated by the first 500 digits of √f (49): 1111111111111111111111111
Schizophrenic_number
14th episode of the 37th season of The Simpsons
"Irrational Treasure" is the fourteenth episode of the thirty-seventh season of the American animated television series The Simpsons, and the 804th episode
Irrational_Treasure
Type of number sequence
The sequence log(n) is not uniformly distributed modulo 1. This fact is related to Benford's law. The sequence of all multiples of an irrational α by
Equidistributed_sequence
Type of number related to Diophantine approximation
Lejeune Dirichlet's criterion on irrationality to the statement that a real number α {\displaystyle \alpha } is irrational if and only if there are infinitely
Lagrange_number
Number, approximately 1.3247
equation x3 = x + 1. Its decimal expansion begins with 1.324717957244746... (sequence A060006 in the OEIS). The adjective plastic does not refer to the artificial
Plastic_ratio
Rotation of a circle by an angle of π times an irrational number
In the mathematical theory of dynamical systems, an irrational rotation is a map T θ : [ 0 , 1 ] → [ 0 , 1 ] , T θ ( x ) ≜ x + θ mod 1 , {\displaystyle
Irrational_rotation
Number, approximately 1.46557
equation x3 = x2 + 1. Its decimal expansion begins with 1.465571231876768... (sequence A092526 in the OEIS). The name supergolden ratio is by analogy with the
Supergolden_ratio
Number representing a continuous quantity
in c. 600 BC include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians
Real_number
Increasing sequence of reduced fractions
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which
Farey_sequence
Italian mathematician (c. 1170 – c. 1240/50)
of Liber Abaci (Book of Calculation) and also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci. Fibonacci
Fibonacci
types of limit of a sequence. Euler proved that the number e is represented as the infinite simple continued fraction (sequence A003417 in the OEIS):
List_of_representations_of_e
Type of mathematical sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N {\displaystyle N} , its subsequence x 1 , … , x N {\displaystyle
Low-discrepancy_sequence
Online database of integer sequences
numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 (A006842) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 (A006843). Important irrational numbers such
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Sum of the reciprocal of the Mersenne numbers
showed that the constant E is an irrational number. Later, Borwein provided an alternative proof. Despite its irrationality, the binary representation of
Erdős–Borwein_constant
Class of sequences of natural numbers
In number theory, a Sidon sequence is a sequence A = { a 0 , a 1 , a 2 , … } {\displaystyle A=\{a_{0},a_{1},a_{2},\dots \}} of natural numbers in which
Sidon_sequence
Class of irrational numbers
that is to say, it is a dense Gδ set. The Liouville–Roth irrationality measure (irrationality exponent, approximation exponent, or Liouville–Roth constant)
Liouville_number
2.71828...; base of natural logarithms
number e is one of only a few transcendental numbers for which the exact irrationality exponent is known (given by μ ( e ) = 2 {\displaystyle \mu (e)=2} )
E_(mathematical_constant)
Integer multiples of any irrational mod 1 are uniformly distributed on the circle
the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the circle R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , when a is an irrational number
Equidistribution_theorem
Sum of the inverses of the positive integers cubed is irrational
the well-known irrationality criterion from Peter Gustav Lejeune Dirichlet, which states that a number ξ {\displaystyle \xi } is irrational if there are
Apéry's_theorem
New Series, 12: 63–66, MR 0029405 Borwein, Peter B. (1992), "On the irrationality of certain series", Mathematical Proceedings of the Cambridge Philosophical
List_of_numbers
Quotient of two integers
(example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...). This statement
Rational_number
Number, approximately 2.20557
x3 = 2x2 + 1. Its decimal expansion begins with 2.2055694304005903... (sequence A356035 in the OEIS). The name supersilver ratio is by analogy with the
Supersilver_ratio
Natural number
N. J. A. (ed.). "Sequence A001358". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A006753 (Smith numbers)"
22_(number)
Records individual grid lines crossed ("cut") as a curve crosses a square grid
Sturmian words are a special case of cutting sequences where the curves are straight lines of irrational slope. Notes Monteil, T. (2011). "The complexity
Cutting_sequence
Transcendental number(s) with all positive integers in order
base 9. Kurt Mahler showed that the constant is transcendental. The irrationality measure of C 10 {\displaystyle C_{10}} is μ ( C 10 ) = 10 {\displaystyle
Champernowne_constant
Function that is discontinuous at rationals and continuous at irrationals
= p q ( x is rational), with p ∈ Z and q ∈ N coprime 0 if x is irrational. {\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac
Thomae's_function
Cheat code, originally for Konami games
following sequence of buttons on the game controller to enable a cheat or other effects: ↑↑↓↓←→←→BA; sometimes Start and/or Select is added to the sequence. The
Konami_Code
Number represented as a0+1/(a1+1/...)
the continued fraction of an irrational number is infinite. 1761 Johann Lambert – gave the first proof of the irrationality of π using a continued fraction
Simple_continued_fraction
Number, approximately 1.618
the Penrose tiling. The golden ratio is an irrational number. Below are two short proofs of irrationality: This is a proof by infinite descent. Recall
Golden_ratio
Infinite sequence in mathematics
Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols {1,2} that is the sequence of run lengths
Kolakoski_sequence
Fixed number that has received a name
{\displaystyle G} having been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven." There
Mathematical_constant
Used to count, measure, and label
likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent
Number
In mathematics, a non-algebraic number
{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{10^{k!}}}} . Numbers with irrationality measure larger than 2, such as the Champernowne constant C 10 {\displaystyle
Transcendental_number
sum-free sequence is less than 2.8570 . The sum of the reciprocals of the heptagonal numbers converges to a known value that is not only irrational but also
List_of_sums_of_reciprocals
Number used to approximate the square root of 2
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational
Pell_number
Sum of the inverses of the positive cubes
38161511449990764986292… (sequence A002117 in the OEIS). It is named after Roger Apéry, who proved that it is an irrational number. Apéry's constant arises
Apéry's_constant
Unique positive real number which when multiplied by itself gives 3
root of 3 is an irrational number. It is also known as Theodorus's constant, after Theodorus of Cyrene, who proved its irrationality. In 2013, its numerical
Square_root_of_3
Mathematical constant
1. Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an irrationality measure of 2. The Prouhet–Thue–Morse constant appears in probability
Prouhet–Thue–Morse_constant
numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational. A standard
Hermite's_problem
Number with all digits equally frequent
log r / log s irrational, there are uncountably many numbers normal in each base but not the other. A disjunctive sequence is a sequence in which every
Normal_number
Mathematical theory related to general topology
the rational sequence topology is an example of a topology given to the set R of real numbers. For each irrational number x take a sequence of rational
Rational_sequence_topology
Natural number
grapheme. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses
1
Mathematical constant described by Marvin Ray Burns
constant is a mathematical constant, with decimal expansion 0.187859… (sequence A037077 in the OEIS). The constant is named after its discoverer, Marvin
MRB_constant
Sequence of n-tuples of integers
A Ducci sequence is a sequence of n-tuples of integers, sometimes known as "the Diffy game", since it is based on differences (subtractions). Given an
Ducci_sequence
Indicator function of rational numbers
function of the set of the first n terms of this sequence of rational numbers. The increasing sequence of functions fn (which are nonnegative, Riemann-integrable
Dirichlet_function
Number, approximately 3.14
rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger or at least
Pi
Function with unusual fractal properties
{1}{64}}+\cdots =\pi .} There is a different way of interpreting the same sequence, however, using continued fractions. Interpreting the fractional part "0
Minkowski's question-mark function
Minkowski's_question-mark_function
Number, sum of distinct powers of 4
In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct
Moser–de_Bruijn_sequence
Natural number
third of the form (32, q) has an aliquot sum of 57, within an aliquot sequence of two composite numbers (99,57,23,1,0), to the Prime in the 23-aliquot
99_(number)
Natural number
Pierpont prime as it is 3 × 25 + 1. the eleventh member of the Mian–Chowla sequence. a self number in base 10, since there is no integer that added to its
97_(number)
Concept in number theory
shows that any irrational number has irrationality exponent at least 2. The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent
Dirichlet's approximation theorem
Dirichlet's_approximation_theorem
Bounds of a sequence
inferior and limes superior) of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a
Limit inferior and limit superior
Limit_inferior_and_limit_superior
Decimal representation of a number whose digits are periodic
periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there are only
Repeating_decimal
Irrational number based on primes
approximately 0.235711131719232931374143... (sequence A033308 in the OEIS). The constant is irrational; this can be proven with Dirichlet's theorem on
Copeland–Erdős_constant
Natural number
(ed.). "Sequence A094133 (Leyland numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A094133
17_(number)
Integer side lengths of a right triangle
N. J. A. (ed.), "Sequence A001652", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation; Sloane, N. J. A. (ed.), "Sequence A001653", The On-Line
Pythagorean_triple
Positional numeral system
1, and avoiding the digit sequence "11" – this is called a standard form. A base-φ numeral that includes the digit sequence "11" can always be rewritten
Golden_ratio_base
French mathematician (1916–1994)
until his retirement. In 1979 he published an unexpected proof of the irrationality of ζ(3), which is the sum of the inverses of the cubes of the positive
Roger_Apéry
2026 Indian film by Vivek Daschaudary
production, Archana Puran Singh suffered a fall while shooting an early morning sequence in Virar, resulting in a fractured right wrist. She underwent surgery and
Toaster_(film)
Positive real number which when multiplied by itself gives 5
its decimal expansion are: 2.236067977499789696409173668731276235440... (sequence A002163 in the OEIS). A length of 5 {\displaystyle {\sqrt {5}}} can
Square_root_of_5
Mathematical expression
Elsevier: 904–911. doi:10.1016/j.jnt.2009.12.003. Angell, David (2021). Irrationality and Transcendence in Number Theory. Chapman and Hall/CRC. ISBN 9780367628376
Continued_fraction
American animated sitcom
Claypool, and the show's four central characters during the opening title sequence. Kenny's muffled lines are altered after every few seasons. His lines are
South_Park
is irrational Lindemann–Weierstrass theorem Hilbert's seventh problem Gelfond–Schneider theorem Erdős–Borwein constant Liouville number Irrationality measure
List_of_number_theory_topics
Series related to Ramanujan's pi formulas
4. It involved Apéry numbers which were first used to establish the irrationality of ζ ( 3 ) {\displaystyle \zeta (3)} . First, define, j 6 A ( τ ) =
Ramanujan–Sato_series
mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite 2-automatic sequence named after Marcel Golay, Harold
Rudin–Shapiro_sequence
Concept in set theory
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology, called the product topology. This space
Baire_space_(set_theory)
Difference between logarithm and harmonic series
and e makes the irrationality of γ a major open question in mathematics. Unsolved problem in mathematics Is Euler's constant irrational? If so, is it transcendental
Euler's_constant
Function that counts distinct factors of a string
is the complexity function p(x,b,n) of the sequence of digits of x written in base b. If x is an irrational number then p(x,b,n) ≥ n+1; if x is rational
Complexity_function
Sum of an infinite series, about 0.6434
Cahen–Mellin integral), who was the first to introduce it and prove its irrationality. The majority of naturally occurring mathematical constants have no
Cahen's_constant
insists that is the number. Malia and Lydia later figure out the number sequence spells out AIDEN on a telephone pad, cracking the second third . In "Orphaned"
List of Teen Wolf (2011 TV series) secondary characters
List_of_Teen_Wolf_(2011_TV_series)_secondary_characters
Relationship between two numbers of the same kind
integers and hence are rational, the limit of the sequence of these rational ratios is the irrational golden ratio. Similarly, the silver ratio of a and
Ratio
Base sixty numeral system
in which days or years are named by positions in a sequence of ten stems and in another sequence of 12 branches. The same stem and branch repeat every
Sexagesimal
Order-preserving mathematical function
may even be dense in an interval (a, b). For example, for any summable sequence ( a i ) (a_{i}) of positive numbers and any enumeration ( q i ) {\displaystyle
Monotonic_function
Integer Sequences. 16 (13.2.3). Cuyt et al. 2008, p. 185. Cuyt et al. 2008, p. 186. Wolf, Marek (22 February 2010). "Remark on the irrationality of the
List of mathematical constants
List_of_mathematical_constants
Metric geometry
has this property. However, considered as a sequence of real numbers, it does converge to the irrational number 2 {\displaystyle {\sqrt {2}}} . The open
Complete_metric_space
Mathematical function whose set of values is bounded
is a bounded sequence, where X {\displaystyle X} is taken to be the set N {\displaystyle \mathbb {N} } of natural numbers. Thus a sequence f = ( a 0 ,
Bounded_function
1998 thriller film by Darren Aronofsky
and contrasting two seemingly irreconcilable entities: the imperfect irrationality of humanity and the rigor and regularity of mathematics, specifically
Pi_(film)
Sequence in which every finite string appears as a subsequence
A disjunctive sequence is an infinite sequence of characters drawn from a finite alphabet, in which every finite string appears as a substring. For instance
Disjunctive_sequence
Mathematical constant
{\displaystyle \psi =3.359885666243177553172011302918927179688905133732\dots } (sequence A079586 in the OEIS). With k terms, the series gives O(k) digits of accuracy
Reciprocal_Fibonacci_constant
Embracing of scientifically unsound ideas by Nobel laureates
general intelligence, such characteristics are not incompatible with irrationality. Nobel disease also serves to demonstrate that, for some prize winners
Nobel_disease
Alternative decimal expansion of 1
value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, and so on. It can be proved that this number is 1; that
0.999...
2024 American fantasy comedic television series
is prone to many paranoid theories and irrational fears. He was previously named Steve in the opening sequence in Ted. Giorgia Whigham as Blaire Bennett
Ted_(TV_series)
Root-finding algorithm
methods, after Newton's method. Like the latter, it iteratively produces a sequence of approximations to the root; their rate of convergence to the root is
Halley's_method
Rational-number approximation of a real number
p. 18 See Michel Waldschmidt: Introduction to Diophantine methods irrationality and transcendence Archived 2012-02-09 at the Wayback Machine, pp 24–26
Diophantine_approximation
Result of repeatedly applying a mathematical function
(an)m = amn. The sequence of functions f n is called a Picard sequence, named after Charles Émile Picard. For a given x in X, the sequence of values fn(x)
Iterated_function
IRRATIONALITY SEQUENCE
IRRATIONALITY SEQUENCE
Surname or Lastname
English
English : from a medieval male personal name (from Latin Hilarius, a derivative of hilaris ‘cheerful’, ‘glad’, from Greek hilaros ‘propitious’, ‘joyful’). The Latin name was chosen by many early Christians to express their joy and hope of salvation, and was borne by several saints, including a 4th-century bishop of Poitiers noted for his vigorous resistance to the Arian heresy, and a 5th-century bishop of Arles. Largely due to veneration of the first of these, the name became popular in France in the forms Hilari and Hilaire, and was brought to England by the Norman conquerors.English : from the much rarer female personal name Eulalie (from Latin Eulalia, from Greek eulalos ‘eloquent’, literally well-speaking, chosen by early Christians as a reference to the gift of tongues), likewise introduced into England by the Normans. A St. Eulalia was crucified at Barcelona in the reign of the Emperor Diocletian and became the patron of that city. In England the name underwent dissimilation of the sequence -l-l- to -l-r- and the unfamiliar initial vowel was also mutilated, so that eventually the name was considered as no more than a feminine form of Hilary (of which the initial aspirate was in any case variable).
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Sequence
Boy/Male
Indian, Sikh
Music; In-sequence
Surname or Lastname
English
English : habitational name from the village of Brattle, near Ashford in Kent.Thomas Brattle (c.1624–83) was reckoned, at the time of his death, to be the wealthiest man in New England. His son, also called Thomas Brattle (1658–1713), treasurer of Harvard College from 1693 to 1713, was a man noted for his rationality and humanism, which included opposition to the Salem withccraft trials of 1692.
Boy/Male
Australian, Dutch, Finnish, German
God of Irrationality
Girl/Female
Greek
Goddess of irrationality.
Boy/Male
Indian, Sanskrit
Order; Sequence
Girl/Female
Tamil
Anuloma | அநà¯à®²à¯‹à®®à®¾
Sequence
IRRATIONALITY SEQUENCE
IRRATIONALITY SEQUENCE
Girl/Female
Indian, Telugu
Earth
Girl/Female
Indian
Perception, Intelligence, Life, Vigour
Boy/Male
Tamil
Boy/Male
Tamil
Viswanathan | விஸà¯à®µà®¨à®¾à®¤à®¾à®¨Â
God of universe, Worlds owner or rich
Girl/Female
Arabic, Australian
Gift
Boy/Male
Danish, German, Scandinavian, Swedish
Brother
Surname or Lastname
English
English : patronymic from Hitchen 1.
Boy/Male
Greek
Victorious.
Girl/Female
Tamil
Girl/Female
Hebrew Biblical
God's glory.
IRRATIONALITY SEQUENCE
IRRATIONALITY SEQUENCE
IRRATIONALITY SEQUENCE
IRRATIONALITY SEQUENCE
IRRATIONALITY SEQUENCE
n.
The manner in which a thing is conceived or constituted by an act of human thinking; the result of such an act; as, animality and rationality are formalities.
n.
A form of melody in which a phrase or passage is successively repeated, each time a step or half step higher; a melodic sequence.
n.
A number of things or events standing or succeeding in order, and connected by a like relation; sequence; order; course; a succession of things; as, a continuous series of calamitous events.
n.
The quality or state of being irrational.
n.
The quality or state of succession in a series; sequence.
n.
A hymn introduced in the Mass on certain festival days, and recited or sung immediately before the gospel, and after the gradual or introit, whence the name.
n.
Three or more cards of the same suit in immediately consecutive order of value; as, ace, king, and queen; or knave, ten, nine, and eight.
adv.
In an irrational manner.
n.
All five cards, of a hand, in consecutive order as to value, but not necessarily of the same suit; when of one suit, it is called a sequence flush.
v. t.
To mimic, as an ape imitates human actions; to imitate or follow servilely or irrationally.
n.
That which follows as a result; a sequence.
v. i.
To wander in mind or intellect; to be delirious; to talk or act irrationally; to be wild, furious, or raging, as a madman.
n.
Simple succession, or the coming after in time, without asserting or implying causative energy; as, the reactions of chemical agents may be conceived as merely invariable sequences.
n.
A melodic phrase or passage successively repeated one tone higher; a rosalia.
n.
The quality or state of being rational; rationality.
a.
Talking irrationally and wildly; as, a raving lunatic.
pl.
of Rationality
n.
Irrationality.
n.
The quality or state of being rational; agreement with reason; possession of reason; due exercise of reason; reasonableness.
n.
A sequence of three playing cards of the same suit. Tierce of ace, king, queen, is called tierce-major.