In my previous article I talked about how Pi was evolved and still today how we tend to reach to precise calculation of this irrational number. After receiving a great response about this article I decided to research about e (Euler’s number). There is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one for π. The number e is, compared to π, a relative newcomer on the mathematics scene.
The symbol e for the base of natural logarithms (2.71828 . . . ) was first used by the Swiss mathematician Leonhard Euler(170783) in a 1727 or 1728 manuscript called Meditatio in Experimenta explosione tormentorum nuper instituta (Meditation on experiments made recently on the firing of cannon). He was all of 21 at the time. Euler also used the symbol in a letter written in 1731, and e made it into print in 1736, in Euler’s Mechanica. Some assume e was meant to stand for “exponential”; others have pointed out that Euler could have been working his way through the alphabet, and the letters a, b, c, and d already had common mathematical uses. What seems highly unlikely is that Euler was thinking of his own name, even though e is sometimes called Euler’s number.
Euler was not the inventor of the number e, even though he gave mathematicians the symbol e. The number e first comes into mathematics in a very minor way. This was in 1618 when, in an appendix to Napier’s work on logarithms, a table appeared giving the natural logarithms of various numbers. However, that these were logarithms to base e was not recognized since the base to which logarithms are computed did not arise in the way that logarithms were thought about at this time. Although we now think of logarithms as the exponents to which one must raise the base to get the required number, this is a modern way of thinking. This table in the appendix, although carrying no author’s name, was almost certainly written by Oughtred. A few years later, in 1624, again e almost made it into the mathematical literature, but not quite. In that year Briggs gave a numerical approximation to the base 10 logarithm of e but did not mention e itself in his work.
The next possible occurrence of e is again dubious. In 1647 SaintVincent computed the area under a rectangular hyperbola. Whether he recognized the connection with logarithms is open to debate, and even if he did there was little reason for him to come across the number e explicitly. Certainly by 1661 Huygens understood the relation between the rectangular hyperbola and the logarithm. He examined explicitly the relation between the area under the rectangular hyperbola xy = 1 and the logarithm. Of course, the number e is such that the area under the rectangular hyperbola from 1 to e is equal to 1. This is the property that makes e the base of natural logarithms, but this was not understood by mathematicians at this time, although they were slowly approaching such an understanding.
Huygens made another advance in 1661. He defined a curve which he calls “logarithmic” but in our terminology we would refer to it as an exponential curve, having the form y = ka^{x}. Again out of this comes the logarithm to base 10 of e, which Huygens calculated to 17 decimal places. However, it appears as the calculation of a constant in his work and is not recognized as the logarithm of a number (so again it is a close call but e remains unrecognized). Further work on logarithms followed which still does not see the number e appear as such, but the work does contribute to the development of logarithms. In 1668 Nicolaus Mercator published Logarithmotechnia which contains the series expansion of log(1+x). In this work Mercator uses the term “natural logarithm” for the first time for logarithms to base e. The number e itself again fails to appear as such and again remains elusively just round the corner.
Perhaps surprisingly, since this work on logarithms had come so close to recognizing the number e, when e is first “discovered” it is not through the notion of logarithm at all but rather through a study of compound interest. In 1683 Jacob Bernoulli looked at the problem of compound interest and, in examining continuous compound interest, he tried to find the limit of (1 + ^{1}/n)^{n} as n tends to infinity. He used the binomial theorem to show that the limit had to lie between 2 and 3 so we could consider this to be the first approximation found to e. Also if we accept this as a definition of e, it is the first time that a number was defined by a limiting process. He certainly did not recognize any connection between his work and that on logarithms.
Of course from the equation x = a^{t}, we deduce that t = log x where the log is to base a, but this involves a much later way of thinking. Here we are really thinking of log as a function, while early workers in logarithms thought purely of the log as a number which aided calculation. It may have been Jacob Bernoulli who first understood the way that the log function is the inverse of the exponential function. On the other hand the first person to make the connection between logarithms and exponents may well have been James Gregory. In 1684 he certainly recognized the connection between logarithms and exponents, but he may not have been the first.
In 1690 Leibniz wrote a letter to Huygens and in this he used the notation b for what we now call e. At last the number e had a name (even if not its present one) and it was recognized. Retrospectively, the early developments on the logarithm became part of an understanding of the number e. Johann Bernoulli began the study of the calculus of the exponential function in 1697 when he published Principia calculi exponentialium seu percurrentium. The work involves the calculation of various exponential series and many results are achieved with term by term integration.
So much of our mathematical notation is due to Euler that it will come as no surprise to find that the notation e for this number is due to him. Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731. He made various discoveries regarding e in the following years, but it was not until 1748 when Euler published Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding e. He showed that
e = 1 + ^{1}/_{1!} + ^{1}/_{2!} + ^{1}/_{3!} + …
and that e is the limit of (1 + ^{1}/_{n})^{n} as n tends to infinity. Euler gave an approximation for e to 18 decimal places,
e = 2.718281828459045235
without saying where this came from. It is likely that he calculated the value himself, but if so there is no indication of how this was done. In fact taking about 20 terms of 1 + ^{1}/_{1!} + ^{1}/_{2!} + ^{1}/_{3!} + … will give the accuracy which Euler gave. Among other interesting results in this work is the connection between the sine and cosine functions and the complex exponential function, which Euler deduced using De Moivre’s formula.
Interestingly Euler also gave the continued fraction expansion of e and noted a pattern in the expansion. In particular he gave
and
Euler did not give a proof that the patterns he spotted continue (which they do) but he knew that if such a proof were given it would prove that e is irrational. For, if the continued fraction for (e – 1)/2 were to follow the pattern shown in the first few terms, 6, 10, 14, 18, 22, 26, … (add 4 each time) then it will never terminate so (e – 1)/2 (and so e) cannot be rational. One could certainly see this as the first attempt to prove that e is not rational.
The same passion that drove people to calculate to more and more decimal places of π never seemed to take hold in quite the same way for e. There were those who did calculate its decimal expansion, however, and the first to give e to a large number of decimal places was Shanks in 1854. It is worth noting that Shanks was an even more enthusiastic calculator of the decimal expansion of π. Glaisher showed that the first 137 places of Shanks calculations for e were correct but found an error which, after correction by Shanks, gave e to 205 places. In fact one needs about 120 terms of 1 + ^{1}/_{1!} + ^{1}/_{2!} + ^{1}/_{3!} + … to obtain e correct to 200 places.
In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula i^{–i} = √(e^{π}). In his lectures he would say to his students:
Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.
Most people accept Euler as the first to prove that e is irrational. Certainly it was Hermite who proved that e is not an algebraic number in 1873. It is still an open question whether e^{e} is algebraic, although of course all that is lacking is a proof – no mathematician would seriously believe that e^{e} is algebraic! As far as we are aware, the closest that mathematicians have come to proving this is a recent result that at least one of e^{e} and e to the powere^{2} is transcendental.
Further calculations of decimal expansions followed. In 1884 Boorman calculated e to 346 places and found that his calculation agreed with that of Shanks as far as place 187 but then became different. In 1887 Adams calculated the logarithm of e to the base 10 to 272 places.
Euler was probably one of the most brilliant and important mathematicians of all time. He published more than 500 books and papers on mathematics, many of which were composed after Euler became completely blind. He was said to have been able to perform vast calculations in his head, and he wrote many of his mathematical treatises while holding one of this thirteen children on his lap. Amongst his other contributions, we owe to Euler the mathematical symbols i for √1, f(x), and Ʃ, as well as the generalized acceptance of the symbol π.
In his own words:
For the number whose logarithm is unity, let e be written, which is 2,7182817… [sic] whose logarithm according to Vlacq is 0,4342944… [translated from Latin by Florian Cajori]. (source)
“Sir, , hence God exists; reply!”
–Euler to the French philosophe Diderot, who was trying to convert the court of Catherine the Great of Russia to atheism. Diderot had no idea what Euler was talking about and left the court to a chorus of laughter. (source)
[upon losing the use of his right eye]
Now I will have less distraction.
Quoted in H Eves In Mathematical Circles (Boston 1969). (source)
Hope you’ve enjoyed the article. If you have anything else to say, please share it here. More interesting articles are coming. Till then stay tight.
Note: If anyone is interested in first 1000 digits of e after decimal point, here it is only for them:
e = 2.718281828459045235360287471352662497757247093699959574966967627724076630353
547594571382178525166427427466391932003059921817413596629043572900334295260
595630738132328627943490763233829880753195251019011573834187930702154089149
934884167509244761460668082264800168477411853742345442437107539077744992069
551702761838606261331384583000752044933826560297606737113200709328709127443
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770785449969967946864454905987931636889230098793127736178215424999229576351
482208269895193668033182528869398496465105820939239829488793320362509443117
301238197068416140397019837679320683282376464804295311802328782509819455815
301756717361332069811250996181881593041690351598888519345807273866738589422
879228499892086805825749279610484198444363463244968487560233624827041978623
209002160990235304369941849146314093431738143640546253152096183690888707016
768396424378140592714563549061303107208510383750510115747704171898610687396
965521267154688957035035402123407849819334321068170121005627880235193033224
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648326277933386566481627725164019105900491644998289315056604725802778631864
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102374432970935547798262961459144293645142861715858733974679189757121195618
738578364475844842355558105002561149239151889309946342841393608038309166281
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469329011574414756313999722170380461709289457909627166226074071874997535921
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573221459784328264142168487872167336701061509424345698440187331281010794512
722373788612605816566805371439612788873252737389039289050686532413806279602
593038772769778379286840932536588073398845721874602100531148335132385004782
716937621800490479559795929059165547050577751430817511269898518840871856402
603530558373783242292418562564425502267215598027401261797192804713960068916
382866527700975276706977703643926022437284184088325184877047263844037953016
690546593746161932384036389313136432713768884102681121989127522305625675625
470172508634976536728860596675274086862740791285657699631378975303466061666
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321560367482860837865680307306265763346977429563464371670939719306087696349
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651483752620956534671329002599439766311454590268589897911583709341937044115
512192011716488056694593813118384376562062784631049034629395002945834116482
411496975832601180073169943739350696629571241027323913874175492307186245454
322203955273529524024590380574450289224688628533654221381572213116328811205
214648980518009202471939171055539011394331668151582884368760696110250517100
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573653822353499293582283685100781088463434998351840445170427018938199424341
009057537625776757111809008816418331920196262341628816652137471732547772778
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210133617299628569489919336918472947858072915608851039678195942983318648075
608367955149663644896559294818785178403877332624705194505041984774201418394
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875200959345360731622611872817392807462309468536782310609792159936001994623
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177452751318680998228473086076653686685551646770291133682756310722334672611
370549079536583453863719623585631261838715677411873852772292259474337378569
553845624680101390572787101651296663676445187246565373040244368414081448873
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810419044804724794929134228495197002260131043006241071797150279343326340799
596053144605323048852897291765987601666781193793237245385720960758227717848
336161358261289622611812945592746276713779448758675365754486140761193112595
851265575973457301533364263076798544338576171533346232527057200530398828949
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172067478541788798227680653665064191097343452887833862172615626958265447820
567298775642632532159429441803994321700009054265076309558846589517170914760
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378762491885286568660760056602560544571133728684020557441603083705231224258
722343885412317948138855007568938112493538631863528708379984569261998179452
336408742959118074745341955142035172618420084550917084568236820089773945584
267921427347756087964427920270831215015640634134161716644806981548376449157
390012121704154787259199894382536495051477137939914720521952907939613762110
723849429061635760459623125350606853765142311534966568371511660422079639446
662116325515772907097847315627827759878813649195125748332879377157145909106
484164267830994972367442017586226940215940792448054125536043131799269673915
754241929660731239376354213923061787675395871143610408940996608947141834069
836299367536262154524729846421375289107988438130609555262272083751862983706
678722443019579379378607210725427728907173285487437435578196651171661833088
112912024520404868220007234403502544820283425418788465360259150644527165770
004452109773558589762265548494162171498953238342160011406295071849042778925
855274303522139683567901807640604213830730877446017084268827226117718084266
433365178000217190344923426426629226145600433738386833555534345300426481847
398921562708609565062934040526494324426144566592129122564889356965500915430
642613425266847259491431423939884543248632746184284665598533231221046625989
014171210344608427161661900125719587079321756969854401339762209674945418540
711844643394699016269835160784892451405894094639526780735457970030705116368
251948770118976400282764841416058720618418529718915401968825328930914966534
575357142731848201638464483249903788606900807270932767312758196656394114896
171683298045513972950668760474091542042842999354102582911350224169076943166
857424252250902693903481485645130306992519959043638402842926741257342244776
558417788617173726546208549829449894678735092958165263207225899236876845701
782303809656788311228930580914057261086588484587310165815116753332767488701
482916741970151255978257270740643180860142814902414678047232759768426963393
577354293018673943971638861176420900406866339885684168100387238921448317607
011668450388721236436704331409115573328018297798873659091665961240202177855
885487617616198937079438005666336488436508914480557103976521469602766258359
905198704230017946553679
Note: These are arranged 75 decimal places on each line.
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Arindam Bose
Scientific History Blog Writer • Art enthusiast and Illustrator • Amateur Photographer • Biker and Hiker • Beer Enthusiast • Electrical Engineer • Chicago

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