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List of ISO 3166 country codes. The International Organization for Standardization ( ISO) created and maintains the ISO 3166 standard – Codes for the representation of names of countries and their subdivisions. [1] . The ISO 3166 standard contains three parts:
The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although ...
Bibliography. External links. List of trigonometric identities. In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.
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The number e is the limit It is the sum of the infinite series It is the unique positive number a such that the graph of the function y = ax has a slope of 1 at x = 0. One has The logarithm of base b can be defined as the inverse function of the function x ↦ b x . {\displaystyle x\mapsto b^{x}.} Since b = b 1 , {\displaystyle b=b^{1},} one has log ...
The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base e {\displaystyle e} . It is assumed that the table was written by William Oughtred. In 1661, Christiaan Huygens studied how to ...
Compound interest
Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest: If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00 × 1.254 = $2.44140625, and compounding monthly yields $1.00 × (1 + 1/12)12 = $2.613035.... If there are n compounding intervals, the interest for each interval will...
Bernoulli trials
The number e itself also has applications in probability theory, in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. As n increases, the probability that gambler will lose all n bets approaches 1/e. For n = 20, this is already approximately 1/2.789509.... This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in n chance of wi...
Exponential growth and decay
Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). If the constant of proportiona...
Calculus
The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. A general exponential function y = ax has a derivative, given by a limit: 1. d d x a x = lim h → 0 a x + h − a x h = lim h → 0 a x a h − a x h = a x ⋅ ( lim h → 0 a h − 1 h ) . {\displaystyle {\begin{aligned}{\frac {d}{dx}}a^{x}&=\lim _{h\to 0}{\frac {a^{x+h}-a^{x}}{h}}=\lim _{h\to 0}{\frac {a^{x}a^{h}-a^{x}}{h}}\\&=a^{x}\...
Inequalities
The number eis the unique real number such that Also, we have the inequality
Exponential-like functions
Steiner's problem asks to find the global maximumfor the function This maximum occurs precisely at x = e. (One can check that the derivative of ln f(x) is zero only for this value of x.) Similarly, x = 1/e is where the global minimumoccurs for the function The infinite tetration 1. x x x ⋅ ⋅ ⋅ {\displaystyle x^{x^{x^{\cdot ^{\cdot ^{\cdot }}}}}} or ∞ x {\displaystyle {^{\infty }}x} converges if and only if x ∈ [(1/e)e, e1/e] ≈ [0.06599, 1.4447] , shown by a theorem of Leonhard Euler.
The number e can be represented in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. In addition to the limit and the series given above, there is also the continued fraction 1. e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , . . . , 1 , 2 n , 1 , . . . ] , {\displaystyle e=[2;1,2,1,1,4,1,1...
One way to compute the digits of eis with the series A faster method involves two recursive functions p ( a , b ) {\displaystyle p(a,b)} and q ( a , b ) {\displaystyle q(a,b)} . The functions are defined as The expression
During the emergence of internet culture, individuals and organizations sometimes paid homage to the number e. In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach e. The versions are 2, 2.7, 2.71, 2.718, and so forth. In another instance, the IPO filing for Google in 2004, rather than a ...
Maor, Eli; e: The Story of a Number, ISBN 0-691-05854-7Commentary on Endnote 10 of the book Prime Obsessionfor another stochastic representationMcCartin, Brian J. (2006). "e: The Master of All" (PDF). The Mathematical Intelligencer. 28 (2): 10–21. doi:10.1007/bf02987150. S2CID 123033482.The number e to 1 million places and NASA.gov2 and 5 million placese Approximations– Wolfram MathWorld"The story of e", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
Definition. The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling (see preceding image) The Fibonacci numbers may be defined by the recurrence relation [6] and for n > 1 .
Many star names are, in origin, descriptive of the part in the constellation they are found in; thus Phecda, a corruption of Arabic فخذ الدب ( fakhdh ad-dubb, 'thigh of the bear'). Only a handful of the brightest stars have individual proper names not depending on their asterism; so Sirius ('the scorcher'), Antares ('rival of Ares ', i.e ...
