# What are the uses and the origin of the constant $e$?

It was to my understanding that the constant $e$ came about as a result of simplifying the differentiation of an exponential.

For example, the derivative of $2^x$ is $2^x \cdot \ln 2$; for $3^x$ it's $3^x \cdot \ln 3$ etc. and essentially a number was required such that the natural logarithm at the end would equal to 1. Such a number would be called $e$. At least, that's how I thought it originated.

I didn't realize that the expression $\ln$ required $e$ as a base. Furthermore, to differentiate exponentials like $2^x$ and $3^x$ you need to have some knowledge that $\frac {d(e^{cx})} {dx} = ce^{cx}$ (as you bring $2^x$ in terms of $e$), and bringing it in terms of any other base like $10$, would essentially be putting you back to square one.

So essentially, it seems like $e$ goes back much further than I thought. I was just wondering on where the idea of natural logarithms came to be and why, and whether the knowledge of the derivative of $e$ dates back further than the derivative of other exponents.

## 2 Answers

Constant $e$ (but not the name) appeared long before any differentiation or exponential function were invented. It probably appears for the first time in the work on Napier on the logarithms. His logarithms are (equivalent to, more or less) the natural logarithms, and the old name of the constant was the "base of the natural logarithms". For practical use, natural logarithms were soon displaced with logarithms based on $10$.

Formal definition, and modern notation of the constant and of the function $e^x$ are due to Euler, and it was called sometimes "Euler's constant", but now this usage is out of date because there is another famous constant which is called the "Euler (Masceroni) constant" and denoted by $\gamma$.

The constant $e$ was discovered because it is indeed the base of "natural" logarithms. You may look here Which came first, the natural logarithm or the base of the natural logarithm? for more detail on Napier and natural logarithms.

Number $e$ had several "near discoveries" by Napier (1614), St. Vincent (1647) and Huygens (1661) in connection with natural logarithms before it was finally explicitly introduced... without any connection to them. Napier's was led to (natural) logarithms by considering motion of a point, whose velocity is equal to its distance to the origin (in modern notation $dx/dt=x$). St. Vincent considered areas under a rectangular hyperbola ($y=1/x$), which as Huygens observed later have the same multiplicative property as logarithms. Huygens even introduced general exponential function $ka^x$, which he called "logarithmic", and calculated what we call "logarithm to base 10 of $e$" to 17 decimal places. But he did not "see" $e$.

As MacTutor describes it, "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... if we accept this as a definition of e, it is the first time that a number was defined by a limiting process." It was also Jacob Bernoulli who started treating logarithms as functions, rather than numerical aids, and realized that they were the inverses of exponential functions (at the level of numbers Gregory might have known it before 1683). The first notation for $e$ appears in Leibniz's 1690 letter to Huygens, and it was $b$. Thus, $e$ got a proper name 16 years earlier than $\pi$ when Jones christened it.

Calculus for exponential functions first appears in Johann Bernoulli's (Jacob's brother) 1697 Principia Calculi Exponentialium seu Percurrentium, only then derivatives come into play. But only Euler pulled logarithmic, area under hyperbola, compound interest, exponential function, approximation by series, etc. threads together into a systematic treatment in his He also gave $e$ its modern name in a letter to Goldbach in 1731, not because of his own name, and possibly not even because of "exponential", but because the preceding vowel $a$ was already taken.

For more on Napier and St. Vincent see Fauvel's and Katz's articles in Learn from the Masters.

• why one had to use vowels? – Anixx Sep 2 '15 at 4:42