|
Representations Of E
The mathematical constant can be represented in a variety of ways as a real number. Since is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, may also be represented as an infinite series, infinite product, or other types of limit of a sequence. As a continued fraction Euler proved that the number is represented as the infinite simple continued fraction : :e = ; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, \ldots, 1, 2n, 1, \ldots Its convergence can be tripled by allowing just one fractional number: : e = ; 1/2, 12, 5, 28, 9, 44, 13, 60, 17, \ldots, 4(4n-1), 4n+1, \ldots Here are some infinite generalized continued fraction expansions of . The second is generated from the first by a simple equivalence transformation. : e= 2+\cfrac = 2+\cfrac :e = 2+\cfrac = 1+\cfrac This last, equivalent to ; 0.5, 12, 5, 28, 9, ... is a special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Constant
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. Constants arise in many areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory, statistics, and calculus. What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, with some mathematical constants being notable more for historical reasons than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places. All named mathematical constants are definable numbers, and usually are also computable numbers (Chaitin's constant being a significant exception). Basic mathematical constants These are constants which one is likely to encounter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
