2.6 The number e: A story of mathematical discovery

The logarithm function (or simply, the logarithm) was discovered by Scotsman John Napier (1550-1617) in the sixteenth century and soon thereafter simplified by the English mathematician Henry Briggs (1561-1631) who introduced base 10. In his "Arithmetica Logarithmica" (1625), he introduced the first logarithm tables. The scientific impact of this work cannot be understated, leading quickly to the invention of the slide rule (a calculating device used in schools well into the 1960's) and Briggs' tables were instrumental to the work of astronomers like Johannes Kepler (1571-1630).

Since logarithms allow one to reduce multiplication to addition and division to subtraction, namely

for positive A and B, difficult calculations with very large numbers become possible.

One should also note that, at the time, the Indian system of numeration using 0,1,2,..,9 (known also as the Arabic system) had not yet really taken hold in Europe.

Leonhard Euler (1707-1783) was the greatest mathematician of the eighteenth century.

He was born in Basel, Switzerland, and studied under Johann Bernoulli. Euler contributed to many fields of mathematics and engineering, including calculus, number theory and was a founder of the fields of topology and the calculus of variations.

In the year 1728, when Euler was only 21, he wrote a scientific paper titled "Meditation Upon Experiments Made Recently on the Firing of a Canon." This paper was only published in 1862 in Euler's Opera Posthuma II. In this paper he introduces, for the first time, the number . At a later time he introduces the logarithm to the base e.

Euler defines e as the limiting value of the numbers as n gets larger and larger, but did not prove that this limiting value existed (perhaps it was obvious to a genius of his magnitude.) Moreover, it is not clear to us how he actually arrived at this number. We present here a likely scenario.

As a student of the Swiss mathematician Johann Bernoulli, who was the author of the first ever calculus book, Euler quickly became a master of this new discipline. Clearly, the early masters would have wanted to calculate the more difficult derivatives of the transcendental functions, particularly the derivative of , which had been around for over one hundred years. So let us look at a likely birth of e. The important point we wish to make is that the existence of this incredible number is most assuredly not the result of a sudden burst of inspiration, but the result of seeking the answer to a natural question, for example like finding the derivative of .

So let us, as Euler likely did, find this derivative from the definition:

Here we must consider taking h small, but either positive or negative. Let us set or . Then the expression becomes either or .

It turns out that, as n gets larger and larger (), both of these expressions get closer and closer to the same number, which Euler called e (for Euler?). We show first that the numbers get closer to some number (called e) as n gets larger. This is the claim that Euler makes in his 1728 paper, when he introduces the expression . From the binomial theorem one can show two facts:

• if , then
• for all n.

Thus, as n gets larger, the numbers are moving to the right on the real number line (see figure) yet must always stay to the left of the number 3.

A very deep property of the real numbers asserts that there must be a real number e such that gets arbitrarily close to e as n gets larger. We write this as:

Thus in the process of finding the derivative of , we (or Euler) discovered a totally new number e. Before we proceed, we should show that

Here is a video of a proof of this fact: (Ed. note: this will later be replaced by "For a video of this fact, click here.")

Thus we may conclude that if , then

Now, by the continuity of (not shown here), this equals

which seems like a very clumsy formula. We use ten digits because we have ten fingers, but mathematics is telling us that the number ten is not a "natural" base for logarithms. If we used instead e as a base, then since , the same argument as above shows then that for ,

which is a much simpler formula; and we are witnessing the birth of the natural logarithm

Having discovered e, we now move on to the function

We sketch a proof (arguing loosely as Euler often did) that

that is, the function is indestructible under differentiation. We proceed as follows:

and from the definition of e this equals

We now argue loosely that we can interchange limits and parentheses (this may not have bothered mathematicians in the eighteenth century) to conclude that this is equal to

and if these limits exist, it does not matter on how h goes to zero. So let us take . Then as and we can simply write the above expression as

Thus

We will give another derivation of this important formula in section ??. Euler subsequently showed that e is not a rational number and discovered an amzing connection between the numbers e, pi, and i. To see a discussion of this, check here (Enrichment lecture to be created/link to be inserted).