Meta tags:

Headings (most frequently used words):

edit, in, calculus, representations, mathematical, constant, contents, history, applications, properties, computer, culture, notes, further, reading, external, links, navigation, menu, compound, interest, bernoulli, trials, standard, normal, distribution, derangements, optimal, planning, problems, asymptotics, alternative, characterizations, inequalities, exponential, like, functions, number, theory, complex, numbers, differential, equations, stochastic, known, digits, search,

Text of the page (most frequently used words):
the (335), and (73), displaystyle (65), frac (59), for (55), number (49), that (43), with (35), euler (28), this (27), edit (27), #constant (26), #function (24), log (23), lim (22), which (21), interest (21), exponential (21), from (20), series (19), right (19), base (18), value (18), bernoulli (16), left (16), logarithm (15), page (14), mathematics (14), also (14), probability (14), cfrac (14), mathematical (13), digits (13), calculus (13), problem (13), are (13), was (12), articles (12), natural (12), one (12), limit (12), sin (12), cos (12), has (11), isbn (11), sequence (11), wikipedia (10), all (10), numbers (10), retrieved (10), see (10), compounding (10), sum (10), unique (10), leonhard (9), first (9), its (9), functions (9), will (9), equal (9), year (9), known (9), such (9), infinite (9), representations (9), infty (9), can (9), positive (9), other (8), more (8), not (8), real (8), digit (8), 100 (8), standard (8), jacob (8), cdot (8), pdf (7), history (7), logarithms (7), est (7), 978 (7), where (7), letter (7), end (7), there (7), distribution (7), any (7), maximum (7), occurs (7), about (6), related (6), ratio (6), prime (6), irrational (6), places (6), another (6), computer (6), shanks (6), john (6), problems (6), theorem (6), complex (6), expression (6), out (6), approximately (6), normal (6), derivative (6), characterizations (6), each (6), hats (6), after (6), compound (6), under (5), here (5), article (5), account (5), transcendental (5), root (5), 2007 (5), google (5), william (5), furthermore (5), example (5), theory (5), unit (5), some (5), every (5), then (5), than (5), equation (5), sometimes (5), time (5), decimal (5), cdots (5), continued (5), fraction (5), times (5), like (5), given (5), two (5), proof (5), global (5), area (5), sqrt (5), into (5), may (4), using (4), commons (4), links (4), special (4), main (4), unsourced (4), statements (4), 2020 (4), constants (4), stochastic (4), 000 (4), have (4), computation (4), common (4), between (4), 1991 (4), vol (4), proved (4), approach (4), int (4), dyds (4), rdx (4), written (4), part (4), most (4), starts (4), needed (4), above (4), ways (4), used (4), differential (4), formula (4), case (4), identity (4), only (4), inequality (4), geq (4), properties (4), following (4), thus (4), equals (4), aligned (4), asymptotics (4), optimal (4), guests (4), trials (4), applications (4), yields (4), while (4), credited (4), napier (4), terms (3), non (3), wikimedia (3), use (3), 2022 (3), simple (3), english (3), wikidata (3), random (3), navigation (3), search (3), identifiers (3), october (3), org (3), square (3), representation (3), marcus (3), 2004 (3), com (3), found (3), consecutive (3), knuth (3), 205 (3), london (3), same (3), large (3), estimating (3), who (3), dover (3), great (3), 1998 (3), whose (3), hyperbolic (3), logarithmus (3), continuous (3), led (3), 1690 (3), 1685 (3), solution (3), chance (3), question (3), were (3), let (3), annual (3), geometric (3), larger (3), illustrated (3), solve (3), during (3), culture (3), made (3), independent (3), many (3), product (3), been (3), converges (3), citation (3), represented (3), list (3), because (3), over (3), find (3), graph (3), own (3), motivation (3), called (3), growth (3), equivalent (3), way (3), set (3), but (3), planning (3), infinity (3), approaches (3), boxes (3), discovered (3), derangements (3), zero (3), winning (3), gambler (3), plays (3), itself (3), rate (3), 71828 (3), view (2), contact (2), privacy (2), policy (2), text (2), available (2), creative (2), last (2), june (2), norsk (2), bahasa (2), wikiquote (2), version (2), download (2), information (2), link (2), pages (2), upload (2), file (2), changes (2), what (2), tools (2), community (2), events (2), contents (2), read (2), talk (2), contributions (2), personal (2), categories (2), 2018 (2), 2017 (2), short (2), description (2), language (2), german (2), https (2), germany (2), liouville (2), billion (2), wilson (2), story (2), uses (2), wolfram (2), mathworld (2), million (2), external (2), 2006 (2), doi (2), book (2), further (2), reading (2), math (2), billboard (2), archived (2), 2013 (2), original (2), tex (2), metafont (2), donald (2), wozniak (2), steve (2), 116 (2), daniel (2), wrench (2), computed (2), program (2), 2003813 (2), calculation (2), decimals (2), boorman (2), 1884 (2), 1871 (2), royal (2), society (2), numerical (2), 1853 (2), 1748 (2), volume (2), roger (2), cotes (2), 1714 (2), 718281828459 (2), especially (2), bottom (2), gourdon (2), socr (2), simulation (2), feb (2), oeis (2), books (2), online (2), 2010 (2), how (2), did (2), serie (2), acta (2), dorrie (2), heinrich (2), 1965 (2), elementary (2), mean (2), kline (2), physical (2), 2003 (2), cambridge (2), university (2), press (2), published (2), introduction (2), petersburg (2), russia (2), 1736 (2), speed (2), denotes (2), denotat (2), numerum (2), hyperbolicus (2), mechanica (2), 7182817 (2), cujus (2), goldbach (2), november (2), 1731 (2), correspondence (2), leibniz (2), gottfried (2), anno (2), lender (2), money (2), much (2), owed (2), power (2), less (2), appears (2), journal (2), des (2), quam (2), eruditorum (2), 166 (2), company (2), notes (2), 7427466391 (2), solving (2), consisted (2), fifth (2), term (2), invited (2), 718 (2), rounded (2), nearest (2), dollars (2), since (2), around (2), dec (2), computers (2), increased (2), due (2), both (2), min (2), expressions (2), ddots (2), these (2), equations (2), follows (2), principal (2), holds (2), taylor (2), meaning (2), when (2), expressed (2), possible (2), length (2), polynomial (2), rational (2), coefficients (2), lindemann (2), weierstrass (2), minimum (2), defined (2), generally (2), similarly (2), precisely (2), steiner (2), inequalities (2), well (2), important (2), quantity (2), factor (2), tau (2), decay (2), factorial (2), introduced (2), integral (2), five (2), define (2), along (2), alternative (2), arrives (2), doing (2), turns (2), denoted (2), through (2), calculations (2), begin (2), particularly (2), makes (2), involving (2), graphs (2), slope (2), appear (2), event (2), lfloor (2), rfloor (2), tends (2), none (2), box (2), check (2), butler (2), montmort (2), constraint (2), deviation (2), results (2), curve (2), textstyle (2), phi (2), binomial (2), machine (2), pays (2), years (2), intervals (2), twice (2), initial (2), 1683 (2), studying (2), various (2), although (2), table (2), swiss (2), mathematician (2), redirects (2), hexadecimal (2), jump (2), web (2), cookie, statement, statistics, developers, mobile, disclaimers, additional, apply, site, you, agree, registered, trademark, profit, organization, foundation, inc, attribution, sharealike, license, edited, utc, tiếng, việt, اردو, українська, türkçe, тоҷикӣ, ไทย, татарча, tatarça, taclḥit, தமிழ், svenska, suomi, srpskohrvatski, српскохрватски, српски, srpski, کوردی, slovenščina, slovenčina, සිංහල, sicilianu, shqip, scots, русский, română, português, polski, patois, occitan, nynorsk, bokmål, nordfriisk, 日本語, nederlands, монгол, melayu, मराठी, മലയാളം, македонски, magyar, lombard, lingua, franca, nova, lietuvių, latviešu, latina, kriyòl, gwiyannen, қазақша, ქართული, ಕನ್ನಡ, עברית, italiano, íslenska, interlingua, indonesia, hrvatski, हिन्दी, հայերեն, 한국어, galego, français, فارسی, euskara, esperanto, español, ελληνικά, eesti, deutsch, dansk, čeština, чӑвашла, català, brezhoneg, bosanski, български, беларуская, башҡортса, বাংলা, azərbaycanca, asturianu, aragonés, العربية, አማርኛ, afrikaans, languages, projects, printable, print, export, item, cite, permanent, recent, portal, learn, help, contribute, donate, current, views, namespaces, create, logged, menu, hidden, good, lccn, j9u, gnd, category, september, august, needing, citations, january, indefinitely, move, protected, different, containing, latin, cs1, sources, index, php, title, mathematical_constant, oldid, 1092990332, united, states, israel, authority, control, national, libraries, trigonometric, schizophrenic, silver, golden, erdős, borwein, supergolden, plastic, apéry, twelfth, gauss, chaitin, searchable, engine, robin, february, audio, video, gresham, college, jan, 2008, earliest, symbols, approximations, nasa, gov, quotations, media, mccartin, brian, 123033482, s2cid, 1007, bf02987150, intelligencer, master, obsession, commentary, endnote, maor, eli, 691, 05854, shea, andrea, npr, entices, job, searchers, puzzle, explore, portland, kazmierczak, mkaz, 2012, brain, tags, 1990, 145, mag, future, alexander, yee, 1981, 392, byte, impossible, dream, computing, 1962, 7090, 265d, obvious, jstor, 2307, 204, magazine, naperian, proceedings, values, modulus, system, england, bell, lausanne, switzerland, marc, michel, bousquet, introductio, analysin, infinitorum, logometria, 338, means, porro, eadem, inter, philosophical, transactions, reported, computations, pifast, sebah, dinov, hands, activities, december, russell, american, statistician, a003417, hofstadter, fluid, concepts, analogies, models, fundamental, mechanisms, thought, basic, 1995, 7139, 9155, sandifer, maa, 2014, lambertina, plurimisque, eius, insignibus, proprietatibus, 1783, reprinted, leipzig, teubner, 350, 369, 1921, facsimile, opera, omnia, prima, commentationes, algebraicae, acad, scient, petropol, 359, exercise, apostol, 1967, taken, section, 337, courier, publications, 486, 40453, derived, logarithmic, intuitive, steven, finch, 521, 81805, 1997, addison, wesley, 183, 201, 03801, art, programming, grinstead, snell, gfdl, petropoli, academy, sciences, chapter, corollary, paragraph, 171, cuius, erit, enim, seu, ubi, sive, motus, scientia, analytice, exposita, scribatur, pro, numero, unitas, qui, meditatio, experimenta, explosione, tormentorum, nuper, instituta, 136, 387, 97195, springer, verlag, remmert, reinhold, lettre, dated, fuss, famous, geometers, 18th, century, 1843, hic, correspondance, mathématique, physique, quelques, célèbres, géomètres, xviiieme, siècle, wilhelm, look, sämliche, schriften, und, briefe, carl, boyer, 2nd, wiley, 471, 54397, 419, uta, merzbach, considered, quæstiones, nonnullæ, usuris, cum, solutione, problematis, sorte, alearum, propositi, ephem, gall, questions, games, proposed, 219, poses, kind, invest, accumulate, moment, receive, proportional, would, constructs, calculate, answer, writes, our, reduces, reference, posed, 314, sçavans, minus, debebitur, plu, quæ, nostra, major, alterius, naturæ, hoc, problema, quæritur, creditor, aliquis, pecuniæ, summam, fænori, exponat, lege, singulis, momentis, pars, proportionalis, usuræ, annuæ, sorti, annumeretur, quantum, ipsi, finito, debeatur, 222, ephemerides, gallicanæ, savants, posamentier, alfred, lehmann, ingmar, prometheus, 59102, 200, biography, world, mysterious, robinn, oxford, preface, 251405, pioneering, beautiful, howard, whitley, eves, 1969, holt, rinehart, winston, 029558, connor, robertson, mactutor, extract, pickover, clifford, 2009, sterling, publishing, 4027, 5796, pythagoras, 57th, dimension, 250, milestones, weisstein, eric, responsible, appeared, heart, later, 99th, visiting, advertised, now, defunct, website, even, difficult, finding, 7182818284, 8182845904, 8747135266, turned, summed, 5966290435, 127th, second, finally, webpage, visitor, submit, résumé, labs, austin, texas, seattle, washington, massachusetts, silicon, valley, instance, filing, rather, typical, round, amount, announced, intention, raise, 281, 828, dollar, usd, ipo, early, his, versions, forth, scientist, emergence, individuals, organizations, paid, homage, internet, proliferation, modern, high, feasible, amateurs, compute, trillions, within, acceptable, amounts, record, setting, giving, 415, 926, 535, 897, desktop, apple, 1978, 265, 1961, eniac, von, neumann, 010, 1949, 346, 137, performed, date, substantially, decades, performance, algorithmic, improvements, expected, mid, addition, exact, analytical, techniques, begins, variables, drawn, least, observations, exceeds, uniform, three, quickly, looks, obtained, evaluating, variety, often, introductory, courses, family, qquad, deduced, referred, cis, moivre, inx, laws, exponentiation, branch, commonly, extend, definition, allows, derive, convergent, conjectured, uniformly, distributed, occur, without, having, specifically, constructed, purpose, compare, 1873, charles, hermite, showing, expansion, fourier, 0660, 4447, tetration, evaluated, simplifying, gives, 4446, 6786, 1009, 7661, 3365, accurate, asks, equality, limiting, intersect, respectively, intersects, infer, lies, therefore, antiderivative, nontrivial, multiplication, reciprocal, takes, increase, four, proven, still, others, rely, far, colored, regions, units, hyperbola, angle, selecting, convenient, choice, solutions, actually, behaves, differentiation, undetermined, carry, substitution, symbolically, comes, considering, consequently, suited, opposed, derivatives, simpler, choosing, parenthesized, variable, introducing, perform, general, argument, shown, dotted, blue, dashed, they, pass, point, red, line, tangent, consequence, sim, naturally, connection, stirling, stated, result, discussed, measure, gleaned, occurring, essentially, division, secretary, below, stick, broken, parts, maximizes, lengths, either, placed, integer, application, party, door, their, turn, labelled, name, guest, asked, identities, puts, selected, gets, put, hat, pierre, remond, derangement, variance, exponent, total, symmetric, attains, inflection, points, density, particular, binom, process, slots, playing, modeled, closely, pascal, triangle, trial, obviously, suppose, slot, lose, bet, already, observing, observed, increases, never, appearing, tries, rapidly, note, percentage, offers, yield, noticed, smaller, weekly, 692597, daily, 714567, cents, grows, came, reach, 718281828, 365, force, months, multiplied, yielding, quarterly, monthly, interval, 613035, 4414, percent, per, once, happens, frequently, effect, earning, investment, frequencies, typographical, convention, typeset, italics, roman, hand, 2019, recommends, typesetting, upright, style, iso, 80000, 1691, writing, started, 1727, 1728, unpublished, paper, explosive, forces, cannons, appearance, publication, researchers, subsequent, eventually, became, christian, christiaan, huygens, discovery, attempted, references, 1618, appendix, work, however, contain, simply, assumed, oughtred, a001113, 69995, 47093, 77572, 66249, 71352, 02874, 23536, 59045, 18284, importance, alongside, formulation, play, recurring, roles, across, cannot, integers, confused, gamma, satisfies, hence, directly, image, inverse, limits, characterized, arises, study, calculated, schanuel, conjecture, topics, people, defining, half, lives, shaded, codes, food, additives, things, named, redirected, free, encyclopedia, wayback, http, archive, 20220615170635, wiki, timestamps, capture, fail, success, 2023, 2021, jul, jun, 2005, mar, 2026, captures,

Text of the page (random words):
arance of e in a publication was in euler s mechanica 1736 13 although some researchers used the letter c in the subsequent years the letter e was more common and eventually became standard citation needed in mathematics the most common typographical convention is to typeset the constant as e in italics although sometimes e in roman is used on the other hand the iso 80000 2 2019 standard recommends typesetting constants in an upright style citation needed applications edit compound interest edit the effect of earning 20 annual interest on an initial 1 000 investment at various compounding frequencies jacob bernoulli discovered this constant in 1683 while studying a question about compound interest 3 an account starts with 1 00 and pays 100 percent interest per year if the interest is credited once at the end of the year the value of the account at year end will be 2 00 what happens if the interest is computed and credited more frequently during the year 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 5 2 2 25 at the end of the year compounding quarterly yields 1 00 1 25 4 2 4414 and compounding monthly yields 1 00 1 1 12 12 2 613035 if there are n compounding intervals the interest for each interval will be 100 n and the value at the end of the year will be 1 00 1 1 n n bernoulli noticed that this sequence approaches a limit the force of interest with larger n and thus smaller compounding intervals compounding weekly n 52 yields 2 692597 while compounding daily n 365 yields 2 714567 approximately two cents more the limit as n grows large is the number that came to be known as e that is with continuous compounding the account value will reach 2 718281828 more generally an account that starts at 1 and offers an annual interest rate of r will after t years yield e rt dollars with continuous compounding note here that r is the decimal equivalent of the rate of interest expressed as a percentage so for 5 interest r 5 100 0 05 bernoulli trials edit graphs of probability p of not observing independent events each of probability 1 n after n bernoulli trials and 1 p vs n it can be observed that as n increases the probability of a 1 n chance event never appearing after n tries rapidly converges to 1 e 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 then for large n the probability that the gambler will lose every bet is approximately 1 e for n 20 this is already approximately 1 2 79 this is an example of a bernoulli trial process each time the gambler plays the slots there is a one in n chance of winning playing n times is modeled by the binomial distribution which is closely related to the binomial theorem and pascal s triangle the probability of winning k times out of n trials is n k 1 n k 1 1 n n k displaystyle binom n k left frac 1 n right k left 1 frac 1 n right n k in particular the probability of winning zero times k 0 is 1 1 n n displaystyle left 1 frac 1 n right n the limit of the above expression as n tends to infinity is precisely 1 e standard normal distribution edit main article normal distribution the normal distribution with zero mean and unit standard deviation is known as the standard normal distribution given by the probability density function ϕ x 1 2 π e 1 2 x 2 displaystyle phi x frac 1 sqrt 2 pi e frac 1 2 x 2 the constraint of unit variance and thus also unit standard deviation results in the 1 2 in the exponent and the constraint of unit total area under the curve ϕ x displaystyle phi x results in the factor 1 2 π displaystyle textstyle 1 sqrt 2 pi proof this function is symmetric around x 0 where it attains its maximum value 1 2 π displaystyle textstyle 1 sqrt 2 pi and has inflection points at x 1 derangements edit main article derangement another application of e also discovered in part by jacob bernoulli along with pierre remond de montmort is in the problem of derangements also known as the hat check problem 14 n guests are invited to a party and at the door the guests all check their hats with the butler who in turn places the hats into n boxes each labelled with the name of one guest but the butler has not asked the identities of the guests and so he puts the hats into boxes selected at random the problem of de montmort is to find the probability that none of the hats gets put into the right box this probability denoted by p n displaystyle p_ n is p n 1 1 1 1 2 1 3 1 n n k 0 n 1 k k displaystyle p_ n 1 frac 1 1 frac 1 2 frac 1 3 cdots frac 1 n n sum _ k 0 n frac 1 k k as the number n of guests tends to infinity p n approaches 1 e furthermore the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is n e rounded to the nearest integer for every positive n 15 optimal planning problems edit a stick of length l is broken into n equal parts the value of n that maximizes the product of the lengths is then either 16 n l e displaystyle n left lfloor frac l e right rfloor or l e 1 displaystyle left lfloor frac l e right rfloor 1 the stated result follows because the maximum value of x 1 ln x displaystyle x 1 ln x occurs at x e displaystyle x e steiner s problem discussed below the quantity x 1 ln x displaystyle x 1 ln x is a measure of information gleaned from an event occurring with probability 1 x displaystyle 1 x so that essentially the same optimal division appears in optimal planning problems like the secretary problem asymptotics edit the number e occurs naturally in connection with many problems involving asymptotics an example is stirling s formula for the asymptotics of the factorial function in which both the numbers e and π appear n 2 π n n e n displaystyle n sim sqrt 2 pi n left frac n e right n as a consequence e lim n n n n displaystyle e lim _ n to infty frac n sqrt n n in calculus edit the graphs of the functions x a x are shown for a 2 dotted a e blue and a 4 dashed they all pass through the point 0 1 but the red line which has slope 1 is tangent to only e x there the value of the natural log function for argument e i e ln e equals 1 the principal motivation for introducing the number e particularly in calculus is to perform differential and integral calculus with exponential functions and logarithms 17 a general exponential function y a x has a derivative given by a limit 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 cdot left lim _ h to 0 frac a h 1 h right end aligned the parenthesized limit on the right is independent of the variable x its value turns out to be the logarithm of a to base e thus when the value of a is set to e this limit is equal to 1 and so one arrives at the following simple identity d d x e x e x displaystyle frac d dx e x e x consequently the exponential function with base e is particularly suited to doing calculus choosing e as opposed to some other number as the base of the exponential function makes calculations involving the derivatives much simpler another motivation comes from considering the derivative of the base a logarithm i e log a x 18 for x 0 d d x log a x lim h 0 log a x h log a x h lim h 0 log a 1 h x x h x 1 x log a lim u 0 1 u 1 u 1 x log a e displaystyle begin aligned frac d dx log _ a x lim _ h to 0 frac log _ a x h log _ a x h lim _ h to 0 frac log _ a 1 h x x cdot h x frac 1 x log _ a left lim _ u to 0 1 u frac 1 u right frac 1 x log _ a e end aligned where the substitution u h x was made the base a logarithm of e is 1 if a equals e so symbolically d d x log e x 1 x displaystyle frac d dx log _ e x frac 1 x the logarithm with this special base is called the natural logarithm and is denoted as ln it behaves well under differentiation since there is no undetermined limit to carry through the calculations thus there are two ways of selecting such special numbers a one way is to set the derivative of the exponential function a x equal to a x and solve for a the other way is to set the derivative of the base a logarithm to 1 x and solve for a in each case one arrives at a convenient choice of base for doing calculus it turns out that these two solutions for a are actually the same the number e alternative characterizations edit the five colored regions are of equal area and define units of hyperbolic angle along the hyperbola x y 1 displaystyle xy 1 see also representations and characterizations of the exponential function other characterizations of e are also possible one is as the limit of a sequence another is as the sum of an infinite series and still others rely on integral calculus so far the following two equivalent properties have been introduced the number e is the unique positive real number such that d d t e t e t displaystyle frac d dt e t e t the number e is the unique positive real number such that d d t log e t 1 t displaystyle frac d dt log _ e t frac 1 t the following four characterizations can be proven to be equivalent the number e is the limit e lim n 1 1 n n displaystyle e lim _ n to infty left 1 frac 1 n right n similarly e lim t 0 1 t 1 t displaystyle e lim _ t to 0 left 1 t right frac 1 t the number e is the sum of the infinite series e n 0 1 n 1 0 1 1 1 2 1 3 1 4 displaystyle e sum _ n 0 infty frac 1 n frac 1 0 frac 1 1 frac 1 2 frac 1 3 frac 1 4 cdots where n is the factorial of n the number e is the unique positive real number such that 1 e 1 t d t 1 displaystyle int _ 1 e frac 1 t dt 1 if f t is an exponential function then the quantity τ f t f t displaystyle tau f t f t is a constant sometimes called the time constant it is the reciprocal of the exponential growth constant or decay constant the time constant is the time it takes for the exponential function to increase by a factor of e f t τ e f t displaystyle f t tau ef t properties edit calculus edit as in the motivation the exponential function e x is important in part because it is the unique nontrivial function that is its own derivative up to multiplication by a constant d d x e x e x displaystyle frac d dx e x e x and therefore its own antiderivative as well e x d x e x c displaystyle int e x dx e x c inequalities edit exponential functions y 2 x and y 4 x intersect the graph of y x 1 respectively at x 1 and x 1 2 the number e is the unique base such that y e x intersects only at x 0 we may infer that e lies between 2 and 4 the number e is the unique real number such that 1 1 x x e 1 1 x x 1 displaystyle left 1 frac 1 x right x e left 1 frac 1 x right x 1 for all positive x 19 also we have the inequality e x x 1 displaystyle e x geq x 1 for all real x with equality if and only if x 0 furthermore e is the unique base of the exponential for which the inequality a x x 1 holds for all x 20 this is a limiting case of bernoulli s inequality exponential like functions edit the global maximum of x x occurs at x e steiner s problem asks to find the global maximum for the function f x x 1 x displaystyle f x x frac 1 x this maximum occurs precisely at x e the value of this maximum is 1 4446 6786 1009 7661 3365 accurate to 20 decimal places for proof the inequality e y y 1 displaystyle e y geq y 1 from above evaluated at y x e e displaystyle y x e e and simplifying gives e x e x displaystyle e x e geq x so e 1 e x 1 x displaystyle e 1 e geq x 1 x for all positive x 21 similarly x 1 e is where the global minimum occurs for the function f x x x displaystyle f x x x defined for positive x more generally for the function f x x x n displaystyle f x x x n the global maximum for positive x occurs at x 1 e for any n 0 and the global minimum occurs at x e 1 n for any n 0 the infinite tetration x x x displaystyle x x x cdot cdot cdot or x displaystyle infty x converges if and only if e e x e 1 e or approximately between 0 0660 and 1 4447 due to a theorem of leonhard euler 22 number theory edit the real number e is irrational euler proved this by showing that its simple continued fraction expansion is infinite 23 see also fourier s proof that e is irrational furthermore by the lindemann weierstrass theorem e is transcendental meaning that it is not a solution of any non constant polynomial equation with rational coefficients it was the first number to be proved transcendental without having been specifically constructed for this purpose compare with liouville number the proof was given by charles hermite in 1873 it is conjectured that e is normal meaning that when e is expressed in any base the possible digits in that base are uniformly distributed occur with equal probability in any sequence of given length complex numbers edit the exponential function e x may be written as a taylor series e x 1 x 1 x 2 2 x 3 3 n 0 x n n displaystyle e x 1 x over 1 x 2 over 2 x 3 over 3 cdots sum _ n 0 infty frac x n n because this series is convergent for every complex value of x it is commonly used to extend the definition of e x to the complex numbers this with the taylor series for sin and cos x allows one to derive euler s formula e i x cos x i sin x displaystyle e ix cos x i sin x which holds for every complex x the special case with x π is euler s identity e i π 1 0 displaystyle e i pi 1 0 from which it follows that in the principal branch of the logarithm ln 1 i π displaystyle ln 1 i pi furthermore using the laws for exponentiation cos x i sin x n e i x n e i n x cos n x i sin n x displaystyle cos x i sin x n left e ix right n e inx cos nx i sin nx which is de moivre s formula the expression cos x i sin x displaystyle cos x i sin x is sometimes referred to as cis x the expressions of sin x and cos x in terms of the exponential function can be deduced sin x e i x e i x 2 i cos x e i x e i x 2 displaystyle sin x frac e ix e ix 2i qquad cos x frac e ix e ix 2 differential equations edit the family of functions y x c e x displaystyle y x ce x where c is any real number is the solution to the differential equation y y displaystyle y y representations edit main article list of representations of e 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 two of these representations often used in introductory calculus courses are the limit e lim n 1 1 n n displaystyle e lim _ n to infty left 1 frac 1 n right n given above and the series e n 0 1 n displaystyle e sum _ n 0 infty frac 1 n obtained by evaluating at x 1 the above power series representation of e x less common is the continued fraction 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 6 1 1 2n 1 24 25 which written out looks like e 2 1 1 1 2 1 1 1 1 1 4 1 1 1 1 displaystyle e 2 cfrac 1 1 cfrac 1 2 cfrac 1 1 cfrac 1 1 cfrac 1 4 cfrac 1 1 cfrac 1 1 ddots this continued fraction for...