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A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
This ratio of densities, and other ratios (using four fundamental constants: speed of light in vacuum c, Newtonian constant of gravity G, reduced Planck constant ℏ, and Hubble constant H) computes to an exact number, 32.8·10 120. This provides evidence of the Dirac large numbers hypothesis by connecting the macro-world and the micro-world.
A mathematical coincidence often involves an integer, and the surprising feature is the fact that a real number arising in some context is considered by 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.
Coincidences involving the letter E, for example, are relatively likely. So when any two English texts are compared, the coincidence count will be higher than when an English text and a foreign-language text are used. This effect can be subtle. For example, similar languages will have a higher coincidence count than dissimilar languages.
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of ¯ with values in the field of -adic numbers, where is a prime coprime to . If X {\displaystyle X} is smooth and equidimensional , this formula can be rewritten in terms of the arithmetic Frobenius Φ q {\displaystyle \Phi _{q}} , which acts as ...
The circumference of a circle with diameter 1 is π.. A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
The number e π − π is also very close to an integer, its decimal expansion being given by: . e π − π = 19.999 099 979 189 475 767 26... (sequence A018938 in the OEIS). The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to Jacobi theta functions as follows: = =