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...

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

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...

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

number, in particular: Liouville's constant. Numbers with large irrationality measure, such as the Champernowne constant C 10 {\displaystyle C_{10}} (by...

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...

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...

in 2001. With Doron Zeilberger, Zudilin improved upper bound of irrationality measure for π, which as of November 2022 is the current best estimate. "Wadim...

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

shows that any irrational number has irrationality measure at least 2. The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent...

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

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...

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

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

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...

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

C2-diffeomorphism of the circle with an irrational rotation number θ is topologically conjugate to Tθ. An irrational rotation is a measure-preserving ergodic transformation...

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...

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...

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

and in fact their difference is bounded by a universal constant. Irrationality measure Markov constant Brjuno, Alexander D. (1971), "Analytic form of differential...

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...

likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent...

(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...

The magnitude of an angle is called an angular measure or simply "angle". Angle of rotation is a measure conventionally defined as the ratio of a circular...

(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...

New Series, 12: 63–66, MR 0029405 Borwein, Peter B. (1992), "On the irrationality of certain series", Mathematical Proceedings of the Cambridge Philosophical...