Short answers, pulled from the story.
The number e is a mathematical constant approximately equal to 2.71828, and it is the base of the natural logarithm and the exponential function. It is irrational, meaning it cannot be written as a ratio of integers, and transcendental, meaning it is not a root of any non-zero polynomial with rational coefficients.
The Swiss mathematician Jacob Bernoulli discovered the constant e in 1683 while studying the problem of continuous compounding of interest. He introduced it as the limit a climbing sequence of compound-interest values approached.
The number e is called Euler's number after the Swiss mathematician Leonhard Euler, who began using the letter e for it in 1727 or 1728. It can also be called Napier's constant after John Napier. The name Euler's number can invite confusion with Euler numbers or with Euler's constant, which are different.
The number e arises when interest is compounded continuously. An account starting at $1.00 with 100 percent annual interest settles at $2.718281828 under continuous compounding, which is the value of e. Compounding weekly yields about $2.692596 and compounding daily about $2.714567.
The number e is important in calculus because the exponential function with base e is its own derivative, which makes differential and integral calculus with exponentials and logarithms much simpler. The logarithm with base e is the natural logarithm, and it behaves well under differentiation because no undetermined limit lingers.
Charles Hermite proved that e is transcendental in 1873. It was the first number proved transcendental that had not been specifically constructed for that purpose, and the result follows from the Lindemann-Weierstrass theorem.
On the 24th of December 2023, Jordan Ranous set a record by computing e to 35 trillion digits. Earlier milestones include John von Neumann reaching 2,010 digits on the ENIAC in 1949 and Steve Wozniak reaching 116,000 digits on the Apple II in 1981.