In mathematics, an irrationality measure of a real number x {\displaystyle x} is a measure of how "closely" it can be approximated by rationals. If a...

Computable number Diophantine approximation Irrationality measure Proof that e is irrational Proof that π is irrational Square root of 3 Square root of 5 Trigonometric...

irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger or at least equal to the measure of...

that the Tamagawa number τ ( G ) {\displaystyle \tau (G)} , a geometrical measure of a connected linear algebraic group over a global number field, is 1...

irrationality measure (irrationality exponent, approximation exponent, or Liouville–Roth constant) of a real number x {\displaystyle x} is a measure of...

Doron Zeilberger, Zudilin improved the known upper bound of the irrationality measure for π, which as of November 2022 is the current best estimate. "Prof...

p. 41. ISBN 978-0-486-49526-2. MR 0057921. Weisstein, Eric W. "Irrationality Measure". mathworld.wolfram.com. Retrieved 2024-09-14. Murty, M. Ram; Rath...

{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{10^{k!}}}} . Numbers with irrationality measure larger than 2, such as the Champernowne constant C 10 {\displaystyle...

where zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set...

(exponential) periods? How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of specific (suspected) transcendental numbers...

signature, and measure signature) is an indication in music notation that specifies how many note values of a particular type fit into each measure (bar). The...

first to give an explicit irrationality measure for π, in 1953 (of 42). Although some have suggested the irrationality measure of π is likely to be 2, the...

forms L-functions Diophantine equations Diophantine approximation Irrationality measure Simple continued fractions Category List of topics List of recreational...

forms L-functions Diophantine equations Diophantine approximation Irrationality measure Simple continued fractions Category List of topics List of recreational...

Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an irrationality measure of 2. The Prouhet–Thue–Morse constant appears in probability. If...

. Kurt Mahler showed that the constant is transcendental. The irrationality measure of C 10 {\displaystyle C_{10}} is μ ( C 10 ) = 10 {\displaystyle...

is irrational Lindemann–Weierstrass theorem Hilbert's seventh problem Gelfond–Schneider theorem Erdős–Borwein constant Liouville number Irrationality measure...

Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined...

In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers...

context an angle consists of a number and unit of measurement. Angular measure or measure of angle are sometimes used to distinguish between the measurement...

integral can be generalized in a straightforward way to more general spaces, measure spaces, such as those that arise in probability theory. The term Lebesgue...

ergodic for the Lebesgue measure as it has infinitely many finite orbits. On the other hand, if θ {\displaystyle \theta } is irrational then T θ {\displaystyle...

to have infinite measure of irrationality. Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1. Next consider the...

(geōmetría) 'land measurement'; from γῆ (gê) 'earth, land' and μέτρον (métron) 'a measure') is a branch of mathematics concerned with properties of space such as...

and abrupt time signature shifts, coupled with frequent use of "irrational" measure lengths. The production also utilized the use of intricate musical...

Lebesgue measure Lebesgue integration Lebesgue's density theorem Counting measure Complete measure Haar measure Outer measure Borel regular measure Radon...

allowed by Roth's theorem if they were algebraic numbers, having irrationality measure no less than 3. The generating function F ( x ) = ∏ i = 0 ∞ ( 1...

{26}}}{17}}\\[6pt]&=[0;{\overline {2,1,1,1,1,1,1,2}}].\end{aligned}}} Irrationality measure Lagrange number Continued fraction Lagrange spectrum Fernando, Suren...

(but is not equivalent to) the fact that C {\displaystyle C} has irrationality measure 3, which was first observed by Duverney & Shiokawa (2020). To give...

measure is invariant under addition. In the context of topological groups, this measure is an example of a Haar measure. The utility of this measure is...