Did You Know: The History of e (Euler's Number)
Did You Know: The History of e (Euler’s Number)

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(1707-83) 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 Saint-Vincent 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 = kax. 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 = at, 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 ii = √(eπ). 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 ee is algebraic, although of course all that is lacking is a proof – no mathematician would seriously believe that ee 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 ee and e to the powere2 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, Formula: (a + b^n)/n=x, 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
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905198704230017946553679
Note: These are arranged 75 decimal places on each line.

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