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This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources. This is a list of articles about prime numbers. 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 ...
Fibonacci sequence. A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21. In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn .
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...
Vance Drummond (1927–1967) was a New Zealand–born Australian pilot who fought in the Korean and Vietnam Wars. Posted to No. 77 Squadron in Korea, he flew Gloster Meteor jet fighters and earned the US Air Medal for his combat skills. He was shot down in 1951 and imprisoned for almost two years. He was awarded the Air Force Cross in 1965 ...
- Definitions
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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)Current ISO 3166 country codes. The sortable table below contains the three sets of ISO 3166-1 country codes for each of its 249 countries, links to the ISO 3166-2 country subdivision codes, and the Internet country code top-level domains (ccTLD) which are based on the ISO 3166-1 alpha-2 standard with the few exceptions noted.
A high school student explains her engineering project to a judge in Sacramento, California, United States (2015). Science, technology, engineering, and mathematics ( STEM) is an umbrella term used to group together the distinct but related technical disciplines of science, technology, engineering, and mathematics.
