mathematical constant number important mathematical constant base natural approximately equal limit approaches infinity expression arises study compound interest calculated sum infinite series formula constant defined ways example defined unique positive number function unit slope function called function natural base natural positive number defined directly area curve case number natural alternative called number swiss mathematician confused constant called simply constant number known constant choice symbol said retained honor constant discovered swiss mathematician studying compound interest number eminent importance mathematics alongside numbers play important recurring roles mathematics appearing identity like constant ratio like root non zero rational numerical value truncated places sequence history references constant published table appendix work contain constant simply list calculated constant assumed table written discovery constant credited attempted find value following expression fact formula known use constant represented letter correspondence introduced letter base natural writing letter november started use letter constant unpublished paper explosive forces cannons appearance publication subsequent years researchers letter common eventually standard applications compound interest discovered constant studying question compound interest account starts pays percent interest year interest credited end year value account year end happens interest credited frequently year interest credited twice year interest rate months initial multiplied twice yielding end year quarterly yields monthly yields intervals interest interval value end year noticed sequence approaches limit force interest larger smaller intervals weekly yields daily yields cents limit grows large number came known continuous account value reach generally account starts offers annual interest rate years yield dollars continuous equivalent rate interest expressed percentage interest trials number applications probability theory arises way obviously related growth suppose plays slot machine pays probability plays times large million probability lose bet approximately approximately example trials process time plays slots million chance winning playing million times modelled distribution closely related probability winning times million trials formula particular probability winning zero times formula close following limit formula application discovered problem known hat check problem guests invited party door guest checks hat butler places boxes labelled guest butler know identities guests puts hats boxes selected random problem find probability hats gets right box answer formula number guests tends infinity approaches furthermore number ways hats placed boxes hats right box rounded nearest positive number occurs naturally connection problems involving prominent example formula function numbers enter formula particular consequence formula standard normal distribution simplest case normal distribution known standard normal distribution described probability density function formula factor formula expression ensures total area curve equal proof ensures distribution unit unit standard function maximum value formula points principal motivation introducing number particularly perform differential integral functions general function derivative given limit formula limit far right independent variable depends base base limit equal symbolically defined equation formula function base particularly suited choosing opposed number base function makes calculations involving derivative simpler motivation comes considering base considering definition derivative limit formula step limit appearing calculation limit depends base base limit equal symbolically formula special base called natural represented limit carry calculations ways select special number way set derivative function solve way set derivative base solve case arrives convenient choice base fact solutions actually number alternative possible limit sequence sum infinite series rely integral far following equivalent properties introduced number unique positive real number formula number unique positive real number formula following proven equivalent number limit formula similarly formula number sum infinite series formula number unique positive real number formula properties motivation function important unique function constant derivative formula formula number unique real number formula positive inequality formula real equality furthermore unique base inequality holds like functions problem asks find global maximum function formula maximum occurs precisely proof inequality formula evaluated formula gives formula formula positive similarly global minimum occurs function formula defined positive generally function formula global maximum positive occurs global minimum occurs infinite formula formula approximately number theory real number proved showing simple continued fraction expansion infinite proof furthermore meaning solution non constant equation rational number proved having specifically constructed purpose compare number proof given normal meaning expressed base possible base uniformly distributed occur equal probability sequence given length complex numbers function written taylor series formula series keeps important properties complex commonly extend definition complex numbers taylor series sin allows derive formula formula holds special case identity formula follows principal branch formula furthermore laws formula formula expression formula referred differential general function formula solution differential equation formula representations number represented real number variety ways infinite series infinite product continued fraction limit sequence chief representations particularly introductory courses limit formula given series formula given evaluating power series common continued fraction sequence formula written looks like formula continued fraction times quickly formula written looks like formula series sequence continued fraction infinite product representations developed representations addition exact expressions representation techniques estimating approach begins infinite sequence independent random drawn uniform distribution let number sum observations exceeds formula expected value known number known increased substantially decades increased performance computers improvements time modern high speed computers possible computer culture contemporary internet culture individuals organizations frequently pay homage number instance filing google typical round number money company announced intention raise billion dollars rounded nearest dollar google responsible billboard appeared heart later cambridge massachusetts seattle washington austin texas read digit prime found consecutive solving problem visiting advertised defunct web site led difficult problem solve turn led google visitor invited submit resume digit prime starts digit instance computer scientist let version numbers program approach versions forth
E (mathematical constant)