ENTRY CONSTANTS Authors: Oliver Knill: March 2000 - March 2004 Literature: Some from Mario Livio "The golden ratio", www.mathworld.com David Wells: "The Penguin Dictionary of Curious and Interesting Numbers". +------------------------------------------------------------ | Archimedes Constant, pi +------------------------------------------------------------ The Archimedes Constant, pi pi=3.14159 is the length of a half circle with radius 1. It is the area of a disc of radius 1. +------------------------------------------------------------ | Bruns constant +------------------------------------------------------------ Bruns constant is the sum of the reciprocals of all twin primes. Brun has proven that this sum converges evenso it is unknown whether there are infinitely many twin primes. +------------------------------------------------------------ | Catalan constant +------------------------------------------------------------ The Catalan constant is defined as the sum (-1)^n/(2n+1)^2=0.91596. +------------------------------------------------------------ | Champernown's number +------------------------------------------------------------ Champernown's number is 0.12345678910111213... whose digits are those of all natural numbers in succession. +------------------------------------------------------------ | Continued fraction constant +------------------------------------------------------------ Continued fraction constant is the number with continued fraction (0,1,2,3,4,5,6,...) it is about 0.697774658. +------------------------------------------------------------ | Euler Mascheroni constant +------------------------------------------------------------ Euler Mascheroni constant is defined as the limit of (1+1/2+1/3+...+1/n)-log(n) as n goes to infinity. +------------------------------------------------------------ | Aperi constant +------------------------------------------------------------ Aperi constant It is an irrational number zeta(3)=1.20206, the value of the zeta function at 3. % Feigenbaum constant When iterating maps f(x)=ax(1-x) on the unit interval the stable periodic orbits bifurcate when varing a. If a_n are the bifurcation values, then delta=lim (a_n-a_n-1)/(a_n+1-a_n) is a Feigenbaum constant. +------------------------------------------------------------ | golden ratio +------------------------------------------------------------ The golden ratio is tau=(1+sqrt5)/2 = 0.618... If 1,1,2,3,5,8,13,21... are the Fibonnachi numbers (the next number is always the sum of the two previous once), then the ratio of neighboring entries approaches the golden mean. 13/21=0.61904 is already quite close to the golden mean. The Golden ratio has the continued fraction expansion 1,1,1,1... which means that the number can be written as tau = 1+1/(1+1/(1+...)). The golden mean is an example of a Diophantine number, a number which can not be approximated well by rational numbers. Especially, it is irrational. The golden ratio is also called "golden mean" or "divine constant". +------------------------------------------------------------ | golden mean +------------------------------------------------------------ The golden mean see golden ratio. +------------------------------------------------------------ | Khinchin constant +------------------------------------------------------------ The Khinchin constant is defined as the limit (a_1 a_2 ... a_n)^1/n where a_1,a_2,... is the continued fraction of a random number in the sense that the limit is known to exist for almost all real numbers. It is not known for example, if pi is a typical number in the sense that it produces the Khinchin constant. +------------------------------------------------------------ | natural logarithmic base +------------------------------------------------------------ The natural logarithmic base e=2.7182818... can be defined as exp(1)=1+1/1!+1/2!+1/3!+... or lim_n to infinity (1+1/n)^n. +------------------------------------------------------------ | number of the beast +------------------------------------------------------------ The number of the beast is the integer 666. The "beast" is associated with the "antichrist". The origin of the association is the bible: the book of revelations (13:18) reads: "this calls for wisdom: let anyone with understanding calculate the number of the beast, for it is the number of a man. Its number is six hundred and sixty six." +------------------------------------------------------------ | Pythagoras constant +------------------------------------------------------------ The Pythagoras constant is the square root of 2 x=sqrt2=1.41421 .... It is the length of the diagonal of the unit square. It is irrational because x=p/q would imply 2 q^2 = x^2 q^2 = p^2, which is impossible because the prime factorization on the left contains an odd number of 2's, while it contains an even number of 2's on the right. +------------------------------------------------------------ | Smith number +------------------------------------------------------------ Smith number Smith numbers are integers n such that the sum of its digits in the decimal exapansion of n is qual to the sum of the digits of its prime factorization, excluding 1. Smith numbers were defined by A. Wilansky. He called it Smith numbers after his brother in law H. Swmith, whose telephone number 4937775 = 3*5*5*65'837 is a Smith number. Here are the first Smith numbers: 4,22,27,58,85,94,121,166,202,265.... +------------------------------------------------------------ | Wallis constant +------------------------------------------------------------ The Wallis constant is the real solution to the polynomial x^3-2x+5 which is 2.0945514815.... This equation was solved by the English mathematicaian John Wallis 1616-1703 to illustrate Newton's method for the numerical solution of equations. It has since served as a test for many subsequent methods of approximation. +------------------------------------------------------------ | Zero +------------------------------------------------------------ Zero The integer zero 0 is the neutral element in the additive group of integers n+0=n. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 16 entries in this file. COUNT: 16