\(e\): The Story of a Number

Although the number \(e\) is known as the second most important mathematical constant, after the number \(\pi\), there are fewer written sources about it and its history. Perhaps this is because \(e\) is much younger than \(\pi\). The book under review aims to fill the gap by tracing \(e\) in the history of mathematics. It starts with the history of logarithms and the amazing work of John Napier. The author explains the importance and significance of the discovery of logarithms, focusing on the very fast acceptance of logarithms as a computational tool, particularly for their applications in astronomy. This is what Laplace highlighted by the phrase “By shortening the labors, the invention of logarithms doubled the life of the astronomer.”

Maor discusses several topics from the history of mathematics in which the number \(e\) appears naturally, including “squaring the hyperbola” and “discovering and developing calculus.” Then he studies the exponential function as the function that equals its own derivative, and considers some related topics, particularly differential equations. In continuation, the author considers the “logarithmic spiral,” which was Jakob Bernoulli’s favorite curve, and compares the Bernoulli family with the Bach family. The author reports an imaginary meeting between Johann Sebastian Bach and Johann Bernoulli (Johann I), both eminent exponents of their families, and explains an application of the logarithmic spiral to musical harmony. He also writes about the appearance of the logarithmic spiral in nature and art, giving several examples, including one from the works of Escher. As another matter from calculus related to the number \(e\), the author studies the “hanging chain” problem, and its solution in terms of the exponential function, giving the curve known as the “catenary.”

The author lists some interesting formulas involving \(e\), and highlights Euler’s formula connecting the numbers \(e\), \(\pi\), \(i\) and \(-1\), and then considers analytic properties of the exponential as a complex-valued function. He writes about the “prime number theorem,” a result about prime numbers involving the natural logarithm, and finally studies the irrationality and transcendence of \(e\). The book ends with some appendixes giving details of some results.

I think this book tells a very interesting story about the history of \(e\), logarithms, and related matters, especially the history of calculus. It seems a useful complement to a course in calculus and analysis, shedding light on some fundamental topics. A reader with knowledge in calculus will be able to follow all parts of the book well.

Mehdi Hassani is associate professor at the Department of Mathematics, University of Zanjan, Iran. His main fields of interest are Elementary, Analytic and Probabilistic Number Theory.