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An appearance of the number e in nature (Graph and photo from brilliant.org)

The number e, also called the natural number and Euler’s number, starts with the six numbers 2.71828 and extends into infinite more decimal places. Mathematicians indicate the various appearances of in natural patterns, but its more practical usage appears in calculations of exponential growth, including the continuous compound interest formula APert

We may credit Scottish scientist John Napier (1550-1617) for introducing the idea of e in his efforts to create a model to simplify multiplication by transforming multiplication into addition. This idea would eventually promulgate in the logarithmic functions of today’s mathematics, and their addition property, whereby logb(xy)logb(x) + logb(y).

Back to e, Napier himself did not define or explicitly acknowledge the number. This task passed to Gottfried Leibniz (1646-1716), one of the founders of calculus, who called it b. Its present designation as e comes from Leonhard Euler (1707-1783), who also played with the imaginary number i. [1]

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A WordCloud of e from gogeometry.com

Today computers can calculate e to millions if not billions of decimal places, saving mathematicians much time and agony. Nevertheless in the beginning the value of e was hand-computed.

A friend of Leibniz, Jacob Bernoulli (1655-1705) drew close to its discovery when he tried to solve \lim_{n \rightarrow \infty} ({1} + \frac{1}{n})^{n} during his investigation of compound interest, and determined that the answer lay between 2 and 3. [2] Euler, in his Introductio in Analysin infinitorum, found the real value of e using this limiting process. He also used summation to prove its value, in the equation \sum\limits_{n=0}^{\infty} \frac{1}{n!} .

Later calculations by Euler and French mathematician Charles Hermite (1822-1901) confirmed e as an irrational and transcendental number, meaning that one cannot state e as the quotient of non-zero integers p and q nor as the root of a single variable, non-zero polynomial equation with integer coefficients.

e ranks as one of the five most fundamental numbers of mathematics, falling in beside i, π, 1, and 0. In a previous post, I discussed the history and meaning of the number i. A future one will deal with π. In a final post, we will bring these five numbers together in the remarkable equation known as Euler’s identity.

[1] A history in verse of the number e
[2] The number e