# e: The Story of a Number

This is a paperback edition in the *New Princeton Science Library* of a 1994 classic. The series brings reprints in cheap paperback and eBook format of classics, written by major scientists and makes them available for a new generation of the broad public. The series includes not only math books but covers a broader area, although there are several mathematics classics in the catalog written by J. Havil, P. Nahin, and R. Rucker, but also J. Napier, A. Einstein, O. Toeplitz, R. Feynman, S. Hawking, R. Penrose, W. Heisenberg, etc. So if you missed out on some of the original editions, or were not even born at that time, this is a chance to get one of these more recent reprints. Another recent reprint is for example Nahin's *An imaginary tale: the story of v-1* that is also reviewed here.

This book with the shortest possible title: e, needs a subtitle to make clear what it is about. This is of course the e in the exponential function $e^x$ and it is the number that is the basis of the natural logarithm. However the logarithm function came earlier and the exponential was just a way of inverting the logarithm and it was only accepted as a full-bred function later in history. The idea of multiplying numbers by adding the exponents when they are represented by powers was proposed by John Napier (1550-1617). To make the steps of the successive powers as small as possible, he decided to take as a base a number close to 1, and he defined $L$ to be the logarithm of a number $N$ when they were related by $N=10^7(1-10^{-7})^L$. In a time when all computations (for example in astronomy) were done by hand, the idea and the use of the first logarithm tables (1614) caught on very quickly. After a meeting in 1615 with Briggs (1561-1630), it was decided that a base 10 was a better choice, which corresponds to what is now called the common (or Briggs) logarithm. And that introduced the logarithm function much appreciated by mathematicians, astronomers and engineers of the 17th century.

So far, no number e is involved, but its roots involve another "power story" about compound interest. With an interest rate of $r$, that is reinvested after the *n*-th part of the period (e.g. every month or even every day instead of at the end of a year) will give a return $(1+r/n)^n$ per unit. What happens if *n* tends to infinity? The result, as we know is *e**r*, but it took a very long time to come to this result. It needs binomial powers, the concept of infinity and of a limit. This came only after calculus was introduced by Newton and Leibniz, Jacob Bernoulli linked compound interest and the exponential, and it became only fully explored by Euler, the master of them all. Of course there are many precursors before one arrived at calculus. There are Zeno's paradoxes that relate to the concept of infinity. And the Greek approximated the circle by an *n*-gon with *n* indefinitely increasing, which lead to this other magic number we know as *p*. This polygon approximation allowed to approximate the circumference but also the area of the circle, and similar techniques applied to other conic sections. The integral under the rectangular hyperbola $y=1/x$ investigated by Fermat and Descartes is another road to the logarithm. But the proper machinery to compute this integral as an anti-derivative was only provided by Newton and Leibniz. With calculus and infinite sequences available, it became possible to define $y(x)=a^x$ for a number $x$ that was the limit of a sequence of rational numbers, and one could obtain its derivative which has the form $ky$. A natural question is when $k=1$ so that $y$ becomes its own derivative and the answer is $y(x)=e^x$. And there appears the number e like magic.

Once the exponential and logarithmic functions are known, they show up in all kinds of applications like solutions of differential equations, music scales, spirals, catenary and other curves, hyperbolic functions, and of course the most magic formula showing a family picture of the most famous actors of our number system: $e^{i\pi}+1=0$. Of course the latter directly relates to complex numbers too. The spreading of Leibniz's ideas throughout Europe was mainly due to the Bernoulli family. Leibniz himself died at the age of seventy almost completely forgotten. Jacob was particularly fond of the logarithmic spiral (*spira mirabilis*) and he had it carved on his tomb. The formula $e^{i\pi}+1=0$ is due to Euler, who, besides the notation for e, is responsible for other standard notations too like $\pi$, $i$ and $f(x)$. Euler was the first to considered the exponential function in its own right, next to the logarithm, and not as just as an inverse by-product. He also derived a continued fraction expansion for e and a series expansion for $e^x$ and so arrived at $e^{ix}=\cos(x)+i\sin(x)$ and expressed the cosine and sine functions with complex exponentials.

The acceptance of negative and complex numbers is another interesting story that Maor takes the opportunity to tell. Mathematics had developed well with only positive numbers (basically only rationals). Negative numbers were known to Hindus, but were neglected by Europeans. It was only when Bombelli used a number line to represent numbers that a meaning could be given to negative numbers. Similarly, it was the geometric interpretation of complex numbers as points in the complex plane that finally began to make sense. That was the merit of several mathematicians among which Gauss. Only later Hamilton gave a formal definition as couples of real numbers with appropriate addition and multiplication, which he later generalized to quaternions. Maor even discusses complex functions and complex calculus of course including the complex exponential and logarithm. Much more on complex numbers and complex functions, although much more mathematically advanced, can be found in Nahin's book telling the story of $\sqrt{-1}$ mentioned above. Another topic that could not have been missed is in the trailing chapter about the transcendence of e (proved by Hermite in 1873). Hermite's proof inspired Lindemann for his proof of the transcendence of $\pi$ published in 1882.

Maor has written with this book the first (his)story of e as a counterpart for the well documented history of $\pi$. Since the original publication in 1994, J. Havil has compiled a biography on Napier: *John Napier: Life, Logarithms, and Legacy* (Princeton University Press, 2014) and there were of course also several books that had chapters on the logarithm or the exponential, which emerged in a period where mathematics experienced a boost in Europe. This story has e as the central star, but e has many strings attached to it Thus many other issues of mathematics are also wonderfully told by the author, much to the liking of the public who made it a bestseller. It reads smoothly, is well illustrated, with some more technical material moved to appendices although the main text is not avoiding formulas, it remains quite accessible for a general interested reader. Also the brief interleaving sections on several topics (e.g. how to work with logarithm tables, a list of numbers related to or derived from e, a fictitious meeting between J.S. Bach and Johann Bernoulli, the logarithmic spiral in nature and art,...) are most interesting. So this reprint in this series of classics is most appropriate. It is of course the original publication and this means that no updates are done, no additional comments or recent references are added. For example the MacTuror biography of Briggs (and other sources too) tells us that he died 26 January 1630, while Maor mentions 1631, which could have been corrected. Another example is that it mentions that the largest Mersenne prime mentioned is $M_{2976221}$. That was in 1997 (this is a reprint of the first paperback edition of 1998), but of course there were several more found since then.

**Submitted by Adhemar Bultheel |

**27 / Apr / 2016