The early life of e

Lynn Gui

e is well-known for being the base of natural logarithms. In fact, e first appeared in 1618 in the appendix to a work on logarithms by John Napier. The number was not defined specifically, but the table in the appendix gave the natural logarithms to various numbers. Though later mathematicians’ works on logarithms came close to recognising e, surprisingly, in 1683, Jacob Bernoulli discovered e by studying another topic – compound interest (the same gist as the Sheldon mitosis!) He defined e using this limit, and it required further insight from a combination of mathematicians to link it back to the earlier work on logarithms.

Jacob Bernoulli LOVED the logarithmic spiral. There’s even a funny story behind it. He was so fascinated by the spiral’s equiangular property that he asked to have it engraved on his tombstone with the phrase “eadem mutata resurgo” (“I shall arise the same, though changed”). How romantic! But through some erroneous stone carving, the spiral that ended up on his tombstone was not the logarithmic spiral, but instead the Archimedean spiral. The angles on the Archimedean spiral do not stay the same—they change each time the curve spirals around!

The number e was first represented using the letter b in the letters between Gottfried Leibniz and Christiaan Huygens in 1690 and 1691. The name “e” started to gain popularity when Leonhard Euler used the letter e for this number in 1736 in his publication Mechanica. Some think that Euler picked the letter e because it was the first letter to both “Euler” and “exponential” (which is also spelled exponential in German, Euler’s native language). In fact, it was most likely that Euler only picked it because it was the next vowel after a, and a was already used to represent something else in his work. In 1748, Euler published his results on the famous Euler’s formula, eix = cos(x) + i sin(x) which, when x = π, directly evaluates to the probably even more famous Euler’s identity, e + 1 = 0.

A lot of mathematicians are astonished by this equation. It contains the five most important numbers in mathematics – e, i, π, 1, 0 – and it relates them all together! Benjamin Pierce (1809-1880), a professor of mathematics at Harvard College, was deeply inspired by it. He believed the values of π and e were much more closely related than people normally think, and part of the reason is that the notations for π and e are so different. Pierce thought that the special connection between these two numbers ought to be reflected in their notation. He recommended the following symbols – which he claimed to have used with success in his lectures –

where “Naperian base” means the number e (because, remember, it first appeared in Napier’s work). Although Pierce claimed that these new notations worked just fine, people weren’t too keen on his idea. Take a look at the Euler’s identity written this new notation:

which literally means eπ = (-1)-i, but it’s just another version of the Euler’s identity. It requires… a bit of skill to tell those two apart.

Pierce’s students actually preferred the traditional π and e as well – perhaps they didn’t tell Professor Pierce because they could do this in his pop quizzes:

Jokes aside, I’m sure that despite having to remember those almost indistinguishable notations, Professor Pierce’s students were able to gain a better understanding of the numbers π and e and the close relation between them.