The constant e is one of the most widely important numbers in mathematics. It was in use many years before it was formally recognized. It’s current notation, e, comes from Euler in the early 1700s, but logarithms to base e were first seen in the work of Napier in 1618 [7]. Several other mathematicians made connections to this irrational constant in between Euler and Napier, and there exist many applications in different fields of modern mathematics.
John Napier, a Scottish scholar born in 1550, began the story of the number e. Though he did not uncover the number specifically, he did give a table of natural logarithms [7]. The algebra of the time did not permit use of logarithms how we think of them in the modern day. In fact, Napier did not think of them algebraically at all, but rather as individual numbers that were to be calculated [6]. The process he used is called dynamical analogy. He invented a way of transforming multiplication into addition as modern logarithms do by using two lines each with one moving point. He recognized the property:
if and only if
One large issue with Napier’s logarithms were that the logarithm of 1 was not equal to zero [6].
Henry Briggs was drawn to Napier’s work and its potential towards the field of astronomy [5]. He studied and improved upon Napier’s logarithms. During a scheduled meeting in the summer of 1615, Briggs suggested to Napier that new logarithm tables should be made with base 10 with the logarithm of 1 being equal to zero [6]. Napier agreed but his health issues prevented him from recreating the tables, so Briggs did so and gave due credit to Napier [5]. He is known to have given credit to others when appropriate and was viewed by his peers as very honorable. His publications sparked the use of logarithms by scientists [5].
Another friend of Briggs, Edmund Gunter, published seven figure tables of logarithms of sines and tangents in 1620. The English translation of the work is Canon of Triangles: or Tables of Artificial Sines and Tangents, and it is the first publication of logarithms of trigonometric functions. He is also credited with the invention of the words cosine and cotangent. Specifically regarding logarithms, he came up with a physical device that could multiply numbers based on logarithms using a scale and a pair of dividers, an important foundation for the slide rule [3].
Ten years later, William Oughtred used two Gunter rulers, getting rid of the dividers, and invented a circular slide rule. This device is described in Oughtred’s Circles of Proportion, published in 1632. Oughtred treated logarithms slightly different than his contemporaries. He viewed logarithms as numbers and numbers in general as being continuous, rather than discrete. He also placed great emphasis on the understanding of logarithms instead of procedural calculation [8]. This remains an essential idea in math education.
In the decade after Oughtred’s work with logarithms, Gregorius Saint-Vincent worked with the geometric area under a rectangular hyperbola, which may have had an impact on Huygens’ connection between this and the logarithm [7]. Though the focus was still primarily on logarithms with no explicit recognition of the number e this marks a significant connection and resulting movement towards uncovering e.
The first known approximation of the number e comes from Jacob Bernoulli’s exploration of compound interest. He used the binomial theorem to show that the limit of
as n approaches infinity lies between 2 and 3. This is also the first time in mathematical history that a number was defined by a limit. Though this approximation was done in 1683, Bernoulli did not make the connection between this number and logarithms [7].
The year 1690 marks the first official and proper use of the number e by Gottfried Leibniz in a letter he sent to Huygens. Though he used the notation b, he was the first to recognize the number itself enough to give it a name [7]. This followed a three-year stint of travel where, though it was not his primary goal, he spent time visiting other elite scholars [4].
Perhaps most important in the development in the number e, and in fact how it gained its current notation was the work of Leonhard Euler [7]. In 1731, in correspondence with Goldbach, Euler writes [15]:
Though a translation is not provided, one can see the very first use of the letter e and statements involving calculus with mention of logarithm.
Euler’s notation stuck, but more significantly, his exploration of the properties of this number led to great discoveries in the fields of number theory and complex analysis. In 1748, he published Indroductio in Analysin infinitorum, in which he showed that
as well as an approximation of the number to 18 decimal places (e=2.718281828459045235), likely by calculating the first 20 terms of the above series [7]. He also gives the continued fraction expansion of the expressions (e-1)/2 and e-1and notes a pattern in each [7]. He related the complex exponential e^(it) to the trigonometric functions cos(t) and sin(t) with the definition e^(it)=cos(t)+isin(t). Of course, he is also responsible for the notation of the notation i for the square root of negative 1 in this equation [11]. Taking this equation at the angle pi gives Euler’s Identity, which remains one of the single most important facts in complex analysis today.
Finding increasing decimal places of the number e was not nearly as popular as it was for the number pi. Still, some mathematicians who tackled it were Shanks, Glaisher, Boorman, and Adams. These calculations occurred in the mid to late 1800s [7]. Applications and connections using the number seem to be of more importance than the number itself.
Euler proved with his fraction expansion that the number e is in fact irrational. In 1873, Charles Hermite became the first to prove that the number e is not algebraic and therefore transcendental [7]. A similar proof, which was adapted from one by Herstein’s Topics in Algebra, is given by [12]:
One important example of the number e in the financial world today is continually compounding interest, stemming from Bernoulli’s work. In this case, the number e is the factor by which a bank account will increase if it otherwise would have doubled. This factor also applies in the same way to reproducing populations and, more generally, to any quantity that grows at a rate proportionally to its current value at any given time [10].
Since the topic of derivative investigates instantaneous rate, it comes as no surprise that the number e also brought new developments to the topic of calculus. Due to nature of its derivative, the number e is preferred as the base in exponentials and logarithms [10]. Not only is the derivative of e^x equal to itself, its integral works the same with addition of a constant.
The number e also finds its place in the study of probability and statistics. For example, rolling a n-sided dice n times approaches 1/e as n gets increasingly large, applied from the Bernoulli trial process [14]. This ratio is found other places in probability such as the topic of derangement of items. The number e also appears in probability mass functions and density functions of Poisson distributions, exponential distributions, gamma distributions, normal distributions, and chi-squared distributions. A few more probability examples involving the number e are given in [1]:
Example 1. Each of two people is given a shuffled deck of playing cards. Simultaneously they expose their first cards. If these cards do not match (for example, two “four of clubs” would be considered a “match”), they proceed to expose their second cards and so forth through the decks. What is the probability of getting through the decks without a single match? […] the answer is given by the sum,
which is the initial portion of a series for 1/e (based on the Maclaurin series below):
This problem has appeared in the literature under the name “Hat-Check Problem,” wherein the following question is asked: If men have their hats randomly returned, what is the probability that none of the men winds up with his own hat?
Example 2. […] If numbers are randomly selected from the interval what is the expected number of selections necessary until the sum of the chosen numbers first exceeds 1? The answer is e. […]
Example 3. The “Secretary Problem” concerns an employer who is about to interview applicants for a secretarial position. At the end of each interview he must decide whether or not this is the applicant he wishes to hire. Should he pass over an interviewee, this person cannot be hired thereafter. If he gets to the last applicant, this person gets the job by default. The goal is to maximize the probability that the person hired is the one most qualified. His strategy will be to decide upon a number to interview the first k applicants, and then to continue interviewing until an applicant more qualified than each of those first k is found. […] the probability of hiring the most qualified applicant is greatest when k/n is approximately 1/e. Moreover, this number 1/e, is in fact the approximate maximum probability. For example, if there are n=30 applicants, the employer should interview 11 (which is approximately 30/e) and then select the first thereafter who is more qualified than all of the first eleven. The probability of obtaining the most qualified applicant is approximately 1/e.
Example 4. With each purchase, a certain fast-food restaurant chain gives away a coin with a picture of a state capitol on it. The object is to collect the entire set of 50 coins. Question: After 50 purchases, what fraction of the set of 50 coins would one expect to have accumulated? […] this fraction is
which is approximately 1-(1/e).
Example 5. A sequence of numbers is generated randomly from the interval [0,1]. The process is continued as long as the sequence is monotonically increasing or monotonically decreasing. What is the expected length of the monotonic sequence? For example, for the sequence beginning .91, .7896, .20132, .41, the length of the monotonic sequence is three. For the sequence beginning .134, .15, .3546, .75, .895, .276, the length of the monotonic sequence is five. The probability that the length L of the monotonic sequence is greater than k is given by
Example 6. A slight revision of the previous example gives a more pleasing answer. First, we require the sequence to be monotonically increasing and, second, in computing the length of the sequence we include the first number that reverses the increasing direction of the sequence. Hence, the sequence beginning .154, .3245, .58, .432 is assigned a score of four and the sequence beginning .6754, .239 is assigned a score of two. Using an argument similar to that of the [preceding] example, it can be shown that the expected score is that ubiquitous and fascinating number, e.
Following the solution of the catenary problem explained in [2], Euler was the first to publish the hyperbolic sine and hyperbolic sine expressions
and
respectively, though he does not use the word hyperbolic or give any special notation for them. Lambert later made connections with these and the circular trigonometric functions [2].
An important application in science is Newton’s Law or Cooling/Heating, given by
The example below, found in [9], shows the graph of the temperature of an object cooling over time. It starts at 120F, cools according to the law equation as it approaches the surrounding temperature of 60F.
The number e indeed has a unique history and immense presence all over mathematics. It was founded through the use of logarithms, recognized and named, then applied and connected to countless mathematical ideas onward.
References
[1] H. Shultz & B. Leonard, Unexpected occurrences of the number e, Math. Mag. 62 (1989) 269—271.
[2] J. Barnett, Enter, stage center: The early drama of the hyperbolic functions, Math. Mag. 77 (2004) 15—30.
[3] J. O’Connor & E. Robertson, “Edmund Gunter,” February 2017, https://mathshistory.st-andrews.ac.uk/Biographies/Gunter/.
[4] J. O’Connor & E. Robertson, “Gottfried Leibniz,” October 1998, https://mathshistory.st-andrews.ac.uk/Biographies/Leibniz/.
[5] J. O’Connor & E. Robertson, “Henry Briggs,” July 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Briggs/.
[6] J. O’Connor & E. Robertson, “John Napier,” April 1998, https://mathshistory.st-andrews.ac.uk/Biographies/Napier/.
[7] J. O’Connor & E. Robertson, “The number e,” September 2001, https://mathshistory.st-andrews.ac.uk/HistTopics/e/.
[8] J. O’Connor & E. Robertson, “William Oughtred,” February 2017, https://mathshistory.st-andrews.ac.uk/Biographies/Oughtred/.
[9] “Newton’s Law of Cooling,” 2011, http://vlab.amrita.edu/index.php?sub=1&brch=194&sim=354&cnt=1.
[10] P. Spencer, “Simple and Compound Interest,” September 1997, https://www.math.toronto.edu/mathnet/answers/answers_10.html.
[11] R. Sachs, “Euler’s Formula for Complex Exponentials,” March 2011, http://math.gmu.edu/~rsachs/m116/.
[12] R. Schwartz, “Transcendence of e,” https://www.math.brown.edu/~res/M154/.
[13] S. Glaz, “The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom,” May 2010, https://www2.math.uconn.edu/~glaz/My_Articles/TheEnigmaticNumberE.Convergence10.pdf.
[14] S. Reichert, “e is everywhere,” September 2019, https://www.nature.com/articles/s41567-019-0655-9.
[15] The Euler Archive, http://eulerarchive.maa.org/.
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