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The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom - Historical Background

Sarah Glaz (University of Connecticut)

John Napier (1550-1617)The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17th and 18th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617), shown at right, who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and at the same time also simplify division and exponentiation), by finding a model that transforms multiplication into addition. He proceeded to construct such a model using two lines, each of which contains one moving point, and figuring out a correspondence between various line segments generated by the moving points. This model could be translated into a two-column numerical table where the numbers in one column are in one-to-one correspondence with the numbers in the second column, and multiplication of two numbers in one column is equivalent to addition of the corresponding numbers in the second column (modulo a factor of 107). In other words, Napier came up with a model that shares with today's logarithms the property of transforming multiplication into addition. This model is almost equivalent to what we call the logarithm today, namely: y = logb x if and only if by = x.

Napier's definition did not use bases or algebraic equations. Algebra was not advanced enough in Napier's time to allow for our modern definition. Logarithmic tables were constructed, even tables very close to natural logarithmic tables, but the base e did not make a direct appearance until about a hundred years later. Gottfried Leibniz (1646-1716), in his work on the Calculus, identified e as a constant, but labeled it b. As with many other concepts, it was Leonhard Euler (1707-1783), shown at right, who gave the constant its modern letter designation, e, and discovered many of its remarkable properties. Euler's discoveries cast new light on the previous work, bringing out e's relevance to a host of results and applications. More details on the history of e may be found in the next section and in references [1, 4, 7, 12, 13, 14, 26, 27, 28, 30, 31, 33, 34]. In particular, it is interesting to note that a model simplifying multiplication by transforming it into addition was discovered at about the same time as Napier's logarithm by another mathematician, Jost Bürgi (1552-1632). See, for example, [30] for some of the controversies regarding priority for the discovery and the early calculations of logarithms.

Compared with the number π, which appeared in print as early as 550 BC (Hebrew Bible, I Kings vii.23) and whose accuracy of digits following the decimal point traces the history of ancient mathematics [3], e seems to have little to boast about. In addition, the geometric definition of π is easily accessible to any literate person, while e's abstract meaning requires more advanced mathematical knowledge even for basic comprehension. For these reasons, π became part of the popular culture: songs, poems, movies, cartoons, websites, and even an annual holiday (Pi Day, on March 14) are dedicated to it (see, for example, [15]), while e still awaits public recognition.

Among mathematicians, however, e is considered to be one of the most important numbers in mathematics [14], along with π, i, 0 and 1, all of which are linked in the famous and mysterious Euler Identity, eiπ + 1 = 0. Moreover, there is a special fascination with e's varied and unexpected appearances at the core of several important areas of modern mathematics. If π's long history traces the ancient development of mathematics, e's shorter history traces the birth of modern mathematics. And it is a captivating history, complete with eccentric personalities, spectacular mathematical results, and still unsolved conjectures—a history worth celebrating and sharing with students.

Sarah Glaz (University of Connecticut), "The Enigmatic Number [i]e:[/i] A History in Verse and Its Uses in the Mathematics Classroom - Historical Background," Convergence (November 2010), DOI:10.4169/loci003482