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Topics referred to by the same term
Rational approximation may refer to: Diophantine approximation, the approximation of real numbers by rational numbers Padé approximation, the approximation
Rational_approximation
Rational-number approximation of a real number
number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of
Diophantine_approximation
Number represented as a0+1/(a1+1/...)
process continues indefinitely. This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as
Simple_continued_fraction
Unique positive real number which when multiplied by itself gives 2
The fraction 99/70 (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence A002193 in the On-Line
Square_root_of_2
Formula to estimate the sine function
In mathematics, Bhāskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the
Bhāskara I's sine approximation formula
Bhāskara_I's_sine_approximation_formula
Irrational algebraic number
equal to 3.16. Historically, the square root of 10 has been used as an approximation for the mathematical constant π, with some mathematicians erroneously
Square_root_of_10
Simple rational approximation (SRA) is a subset of interpolating methods using rational functions. Especially, SRA interpolates a given function with a
Simple_rational_approximation
Positive real number which when multiplied by itself gives 5
sequence of best rational approximations to 5 {\displaystyle {\sqrt {5}}} , each more accurate than any other rational approximation with the same or
Square_root_of_5
Mathematical expression
referred to as a simple (or regular) continued fraction. Any positive rational number can be expressed as a finite simple continued fraction, and any
Continued_fraction
'Best' approximation of a function by a rational function of given order
mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique
Padé_approximant
Mathematical concept
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation
Approximation_error
Pi approximations by astronomer Zu Chongzhi
less than 1/3748629. The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52163/16604, though it
Milü
convenient rational approximation for the square root of 8 is 17/6 (≈ 2.8333), accurate to within approximately 0.17%. The rational approximation 82/29
Square_root_of_8
Something roughly the same as something else
analysis. Diophantine approximation deals with approximations of real numbers by rational numbers. Approximation usually occurs when an exact form or an exact
Approximation
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
theorem can be used to prove Dirichlet's theorem on simultaneous rational approximation. Another application of Minkowski's theorem is the result that every
Minkowski's_theorem
Number used to approximate the square root of 2
comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12
Pell_number
Varying methods used to calculate pi
other approximations of π: π ≈ 22⁄7 and π ≈ 355⁄113, which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation
Approximations_of_pi
Algorithms for calculating square roots
compute the square root digit by digit, or using Taylor series. Rational approximations of square roots may be calculated using continued fraction expansions
Square_root_algorithms
Concept in geometry
table has 355⁄113 as one of its best rational approximations; i.e., there is no better approximation among rational numbers with denominator up to 113.
Area_of_a_circle
Data type approximating a real number
Most often, a computer will use a rational approximation to a real number. The most general data type for a rational number (a number that can be expressed
Real_data_type
Unique positive real number which when multiplied by itself gives 3
it can be approximated arbitrarily closely by such rational numbers. Particularly good approximations are the integer solutions of Pell's equations, x 2
Square_root_of_3
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
sine function. The pulse function may also be expressed as a limit of a rational function: Π ( t ) = lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 . {\displaystyle
Rectangular_function
involving a square root operator. They are obtained by applying rational approximations to the square root operator. After a one-way model is obtained
Beam_propagation_method
Positive real number which when multiplied by itself gives 7
root of seven have been published. The extraction of decimal-fraction approximations to square roots by various methods has used the square root of 7 as
Square_root_of_7
Function that quantifies how near a number is to being rational
{\displaystyle q>0} that satisfy the inequality. For example, whenever a rational approximation p q ≈ x {\displaystyle {\frac {p}{q}}\approx x} with p , q ∈ N {\displaystyle
Irrationality_measure
Number, approximately 3.14
widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer
Pi
Concept in number theory
fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence
Dirichlet's approximation theorem
Dirichlet's_approximation_theorem
Indian mathematician and astronomer (600–680)
with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work. This
Bhāskara_I
Fraction with denominator a power of two
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example
Dyadic_rational
Chinese mathematician-astronomer (429–500)
known as "Zu's ratio". Zu's ratio is a best rational approximation to π, and is the closest rational approximation to π from all fractions with denominator
Zu_Chongzhi
Mathematical constants related to chaotic behavior
669\,201\,609\,102\,990\,671\,853\,203\,820\,466\ldots } A simple rational approximation is 621/133, which is correct to 5 significant values (when rounding)
Feigenbaum_constants
Type of equation involving matrix-valued functions
with rational approximation by set-valued AAA. The MATLAB toolbox RKToolbox (Rational Krylov Toolbox) contains implementations of the rational Krylov
Nonlinear_eigenproblem
Coincidence in mathematics
some standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number with a small denominator
Mathematical_coincidence
Problem of constructing equal-area shapes
increasingly accurate rational approximations for π {\displaystyle \pi } . Jacob de Gelder published in 1849 a construction based on the approximation π ≈ 355 113
Squaring_the_circle
Class of irrational numbers
be approximated by rational numbers, and he defined Liouville numbers specifically so that they would have rational approximations better than the ones
Liouville_number
Method for estimating new data outside known data points
(2003). Claude Brezinski and Michela Redivo-Zaglia : "Extrapolation and Rational Approximation", Springer Nature, Switzerland, ISBN 9783030584177, (2020).
Extrapolation
Natural number
integer. 161 is a palindromic number. 161/72 is a commonly used rational approximation of the square root of 5 and is the closest fraction with denominator
161_(number)
Method for estimating new data within known data points
Newton–Cotes formulas Radial basis function interpolation Simple rational approximation Smoothing Sheppard, William Fleetwood (1911). "Interpolation" .
Interpolation
Approximation to a sine curve
rule relies on the approximation of tan 60° or √3 (~1.732) with 5/3 (~1.667) yielding 3.77% error. The next best rational approximation, 7/4 (1.75) yields
Rule_of_twelfths
Algorithm in computational number theory
algorithms for factorizing polynomials with rational coefficients, for finding simultaneous rational approximations to real numbers, and for solving the integer
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Algorithm for computing greatest common divisors
theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. Finally, it can be used as a basic tool for proving
Euclidean_algorithm
Approximating an arbitrary function with a well-behaved one
In general, a function approximation problem asks us to select a function that closely matches ("approximates") a function in a task-specific way.[better source needed]
Function_approximation
Theory of getting acceptably close inexact mathematical calculations
typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual
Approximation_theory
Mathematical constant
Constant". MathWorld. Bugeaud, Yann; Queffélec, Martine (2013). "On Rational Approximation of the Binary Thue-Morse-Mahler Number". Journal of Integer Sequences
Prouhet–Thue–Morse_constant
Mathematical approximation of a function
– best approximation by a rational function Puiseux series – power series with rational exponents Approximation theory Function approximation Banner 2007
Taylor_series
Pair of polynomial sequences
properties and applications of Chebyshev polynomial and rational approximation". Rational Approximation and Interpolation. Lecture Notes in Mathematics. Vol
Chebyshev_polynomials
Algebraic numbers are not near many rationals
diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational approximations that are
Roth's_theorem
Question in geometric probability
to replicate the already well-known approximation of 355/113 (in fact, there is no better rational approximation with fewer than five digits in the numerator
Buffon's_needle_problem
Calculation of signal delay times in integrated circuits
multiple moments in the time domain or finding a good rational approximation (a Padé approximation) in the frequency domain. (These are very closely related
Delay_calculation
Mathematical formula involving a given set of operations
numbers (not to be confused with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. The Liouvillian numbers, denoted
Closed-form_expression
Natural number
Primes found among the denominators of the continued fraction rational approximations to sqrt(2)". The On-Line Encyclopedia of Integer Sequences. OEIS
29_(number)
Japanese mathematician
π 2 {\displaystyle \pi ^{2}} . In 1766, he found the following rational approximation of π {\displaystyle \pi } , correct to 29 digits: π ≈ 428224593349304
Arima_Yoriyuki
German mathematician (1849–1917)
biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the
Ferdinand_Georg_Frobenius
Development of mathematics in South Asia
Laghu-bhaskariya. He produced: Solutions of indeterminate equations. A rational approximation of the sine function. A formula for calculating the sine of an acute
Indian_mathematics
Approximation for factorials
mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate
Stirling's_approximation
Natural number
x 5 = 75). It is the numerator of, 355/113, the best simplified rational approximation of pi having a denominator of four digits or fewer, known as Milü
300_(number)
Number in {..., –2, –1, 0, 1, 2, ...}
integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers
Integer
Algebraic irrational number
[1: 16, 1, 4, 2, 7, 1, 1, 2, 2, 7, 4, 1, 2, 1, ...], so a simple rational approximation is 18/17. A musical interval is a ratio of frequencies and the
Twelfth_root_of_two
question asked in weak approximation is whether the embedding of G(k) in G(AS) has dense image. If the group G is connected and k-rational, then it satisfies
Approximation in algebraic groups
Approximation_in_algebraic_groups
Mathematical power series of arctangent
Siddiqui, Rajinder Kumar Jagpal and Brendan M. Quine (2024), "A rational approximation of the two-term Machin-like formula for π", AppliedMath, 4 (3):
Arctangent_series
Used to count, measure, and label
number is necessarily a rational number, of which there are only countably many. All measurements are, by their nature, approximations, and always have a margin
Number
Mathematical theorem in the study of analysis
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly
Stone–Weierstrass_theorem
Philosphical view that existence proofs must be constructive
closer together. We can use ƒ and g together to compute as close a rational approximation as we like to the real number they represent. Under this definition
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
Statistical function that defines the quantiles of a probability distribution
Thorough composite rational and polynomial approximations have been given by Wichura and Acklam. Non-composite rational approximations have been developed
Quantile_function
Ordered binary tree of rational numbers
provides a sequence of approximations to q with smaller denominators than q. Because the tree contains each positive rational number exactly once, a breadth
Stern–Brocot_tree
2.71828...; base of natural logarithms
function use range reduction, lookup tables, and polynomial or rational approximations (such as Padé approximants or Taylor expansions) to achieve accurate
E_(mathematical_constant)
American mathematician (1931–2019)
algebras and then applies it in the last two chapters to the theory of rational approximation in the plane, and to finding analytic structure in the spectrum
Andrew_Browder
Lists of values of mathematical functions
combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, Padé approximation, and typically for higher
Trigonometric_table
Functional equation
r_{n}:\mathbb {R} \to \mathbb {Q} ,n\in \mathbb {N} } denote a series of rational approximation functions such that for all z ∈ R {\displaystyle z\in \mathbb {R}
Cauchy's_functional_equation
Theorem in complex analysis
contour. This gives a uniform approximation by a rational function with poles on Γ. To modify this to an approximation with poles at specified points
Runge's_theorem
Finitely many for a smooth algebraic curve of genus > 0 defined over a number field
Thue's method in diophantine approximation also is ineffective in describing possible very good rational approximations to almost all algebraic numbers
Siegel's theorem on integral points
Siegel's_theorem_on_integral_points
British mathematician (1925–2015)
sequences. The subject of Diophantine approximation seeks accurate approximations of irrational numbers by rational numbers. The question of how accurately
Klaus_Roth
Procedure to solve equations of second degree
roots are real, there is an alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method
Solving quadratic equations with continued fractions
Solving_quadratic_equations_with_continued_fractions
Russian mathematician
University under Victor Havin with thesis Hardy Classes Hp for p∈(0,1) (Rational Approximation, Backward Shift Operator, Cauchy-Stieltjes Type Integral (title
Alexei_Borisovich_Aleksandrov
which is the best rational approximation. Zu ultimately determined the value for π to be between 3.1415926 and 3.1415927. Zu's approximation was the most accurate
List_of_Chinese_discoveries
Treatise by Archimedes
setting of the regular dodecagon. In this case, the task is to give rational approximations to the tangent of π/12. Knorr, Wilbur R. (1986-12-01). "Archimedes'
Measurement_of_a_Circle
Ratio of polynomial functions
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator
Rational_function
Computer-aided geometric design
on the original curve. Then, the initial polynomial approximation curve or rational approximation curve of the offset curve is generated from these sampled
Progressive-iterative approximation method
Progressive-iterative_approximation_method
Hindu astronomy, mathematics, science school in India
}{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots } Their rational approximation of the error for the finite sum of their series are of particular
Kerala school of astronomy and mathematics
Kerala_school_of_astronomy_and_mathematics
Notable events in the history of geometry
intervals from 0 to 90 degrees) 7th century – Bhaskara I gives a rational approximation of the sine function 8th century – Virasena gives explicit rules
Timeline_of_geometry
Study of numbers that are not solutions of polynomials with rational coefficients
(1921). "Approximation algebraischer Zahlen". Mathematische Zeitschrift. 10 (3–4): 172–213. doi:10.1007/BF01211608. Roth, K. F. (1955). "Rational approximations
Transcendental_number_theory
Mathematical technique for improving convergence
Brezinski Claude and Redivo-Zaglia Michela : "Extrapolation and Rational Approximation", Springer, ISBN 978-3-030-58417-7 (2020). Convergence acceleration
Series_acceleration
Koukoulopoulos and James Maynard Duffin–Schaeffer conjecture number theory Rational approximation of irrational numbers 2019 Hao Huang Sensitivity conjecture computational
List_of_conjectures
al. 2014, p. 190. Bugeaud, Yann; Queffélec, Martine (2013). "On Rational Approximation of the Binary Thue-Morse-Mahler Number". Journal of Integer Sequences
List of mathematical constants
List_of_mathematical_constants
Model of humans as rational, self-interested agents
economic man, is the portrayal of humans as agents who are consistently rational and narrowly self-interested, and who pursue their subjectively defined
Homo_economicus
related fields, relaxation is a modeling strategy. A relaxation is an approximation of a difficult problem by a nearby problem that is easier to solve.
Relaxation_(approximation)
Property of an irrational number
Dirichlet's approximation theorem can be improved for α {\displaystyle \alpha } . Certain numbers can be approximated well by certain rationals; specifically
Markov_constant
On graph drawing with integer edge lengths
MR 2419522, S2CID 1856482. Benediktovich, Vladimir I. (2013), "On rational approximation of a geometric graph", Discrete Mathematics, 313 (20): 2061–2064
Harborth's_conjecture
Theorem in complex analysis
this further rational approximation problem was also suggested by Mergelyan in 1952. Further deep results on rational approximation are due to, in particular
Mergelyan's_theorem
Mathematical functions having established names and notations
representation may converge slowly or not at all. In algorithmic languages, rational approximations are typically used, although they may behave badly in the case
Special_functions
Belgian mathematician and computer scientist
and their Padé Approximations (Operator Theory: Advances and Applications 27, Birkhäuser, 1987) Linear Algebra, Rational Approximation and Orthogonal
Adhemar_Bultheel
Gibbs phenomenon Simple rational approximation Polynomial and rational function modeling — comparison of polynomial and rational interpolation Wavelet Continuous
List of numerical analysis topics
List_of_numerical_analysis_topics
real numbers reduces to the previous case by taking the limit of rational approximations. The case where α 0 < ⋯ < α n {\displaystyle \alpha _{0}<\dots
Totally_positive_matrix
(1952) degrees from Cambridge. Chisholm developed a method for rational approximations of two variable functions generalising Padé approximant. Roy Chisholm
John_Stephen_Roy_Chisholm
Numbers obtained by adding the two previous ones
from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F( j):F( j + 1), the nearest
Fibonacci_sequence
Arithmetical operation
141,\ldots \}.} A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular,
Multiplication
Number in base-10 numeral system
digits after the decimal separator, for example, that "3.14 is the approximation of π to two decimals" or "two decimal places." The numbers that may
Decimal
Theorem about Diophantine approximations
theorem is a theorem about diophantine approximation, introduced by Leopold Kronecker (1884). Kronecker's approximation theorem had been firstly proved by
Kronecker's_theorem
botanist Philibert Commerçon is on board. Arima Yoriyuki finds a rational approximation of π {\displaystyle \pi } , correct to 29 digits. Euler gives up
1766_in_science
Mathematical constant
1,1,1,3,10,1,1,1,2,1,1,1,1,3,2,3,1,...\right]} , which yields rational approximations, the first few of which are 0, 1, 2/3, 7/10, 9/13 and 61/88. This
Natural_logarithm_of_2
RATIONAL APPROXIMATION
RATIONAL APPROXIMATION
Boy/Male
English
National protector.
Boy/Male
Gujarati, Hindu, Indian
Lord of Pleasure
Girl/Female
Hindu, Indian
Rational
Girl/Female
Indian
Optional
Boy/Male
Indian
Talker, Speaker, Rational
Boy/Male
Muslim/Islamic
Categorical (decision) talker, speaker, rational
Girl/Female
Christian, German, Greek, Hebrew
Noble; Kind; Rational; Great Happiness
Boy/Male
Tamil
Rational
Boy/Male
Hindu
Rational
Boy/Male
American, Anglo, British, English, Teutonic
National Protector; Wealthy Defender
Girl/Female
Hindu, Indian
Rational
Boy/Male
Arabic, Muslim
National Leader
Boy/Male
Tamil
Rational
Boy/Male
Hindu, Indian
National Player
Boy/Male
Hindu
Rational
Boy/Male
Hindu, Indian, Tamil
Revolving; Pearl
Boy/Male
Indian, Tamil
National Boy; Lord Krishna
Boy/Male
Muslim
Talker, Speaker, Rational
Girl/Female
German, Greek
Noble; Kind; Rational
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Animated; Rational
RATIONAL APPROXIMATION
RATIONAL APPROXIMATION
Girl/Female
German, Hawaiian, Hebrew, Spanish
Noble One; Refuge of God; Just; Of the Nobility
Boy/Male
Muslim
Mars. Planet.
Male
German
Low German form of Latin Stephanus, STEFFEN means "crown."
Girl/Female
Arthurian Legend American
Mother of Lancelot.
Girl/Female
Tamil
Padmasree | பதà¯à®®à®·à¯à®°à¯€
Divine lotus
Girl/Female
Gujarati, Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
Dear-eyed
Girl/Female
Latin Spanish
Modest.
Girl/Female
Indian
Helpful; Goddess Parvati
Boy/Male
Hindu
Lord Krishna
Male
Native American
Native American Hopi name KWAHU means "eagle."
RATIONAL APPROXIMATION
RATIONAL APPROXIMATION
RATIONAL APPROXIMATION
RATIONAL APPROXIMATION
RATIONAL APPROXIMATION
adv.
In a rational manner.
v. t.
To supply with rations, as a regiment.
a.
Attached to one's own country or nation.
a.
Involving an option; depending on the exercise of an option; left to one's discretion or choice; not compulsory; as, optional studies; it is optional with you to go or stay.
v. t.
To form a rational conception of.
a.
Notional.
a.
Of or pertaining to fractions or a fraction; constituting a fraction; as, fractional numbers.
a.
Relating to the reason; not physical; mental.
n.
A rational being.
a.
Fractional.
a.
Given to foolish or visionary expectations; whimsical; fanciful; as, a notional man.
a.
Relatively small; inconsiderable; insignificant; as, a fractional part of the population.
a.
An explanation or exposition of the principles of some opinion, action, hypothesis, phenomenon, or the like; also, the principles themselves.
a.
Agreeable to reason; not absurd, preposterous, extravagant, foolish, fanciful, or the like; wise; judicious; as, rational conduct; a rational man.
a.
Not rational; void of reason or understanding; as, brutes are irrational animals.
a.
Having reason, or the faculty of reasoning; endowed with reason or understanding; reasoning.
a.
Expressing the type, structure, relations, and reactions of a compound; graphic; -- said of formulae. See under Formula.
a.
Involving surds; not capable of being expressed in rational numbers; radical; irrational; as, a surd expression or quantity; a surd number.
n.
The state of being national; national attachment; nationality.
a.
Of or pertaining to a nation; common to a whole people or race; public; general; as, a national government, language, dress, custom, calamity, etc.