Rubbing Shoulders with Newton: An Improbable Story

Harlan Brothers

Both Shawn Carlson and Forrest Mims agreed that I should share the unusual story of my mathematical endeavors with my fellow SAS members. That story follows. A link to my most recent article on one of the fundamental constants of nature, e, appears at Improving the Convergence of Newton's Series Approximation to e with permission of The College Mathematics Journal. A link to a presentation I made in January 2005 at the SAS Conference at the University of Nevada at Las Vegas is at "Rubbing Shoulders with Newton: A New Look at a Fundamental Constant of Nature". This offers an overview and explanation of the paper along with some historical context.

I was never a math genius. I new kids who were really good at math, and I was not one of them.  I did OK.  I had to work at it, but I was never afraid of the subject.

About ten years ago, I was lying in bed late one night with a pad and pencil in hand. I started wondering how I might express each counting number in an alternative and unique way.

Roots of numbers were too hard to calculate by hand, so I tried raising each positive integer to its own power, nⁿ. I noticed that this sequence of numbers grows very rapidly, as shown here:

I wondered whether I could find a function that described this rapid rate of growth.  When I divided each value of nⁿ by the previous such value, the sequence seemed to grow at a relatively constant rate.

What I discovered was that as n grew larger, the value of



approached that of the universal constant e, which is equal to approximately 2.71828182818459… .

Like its more famous cousin Π, e is a transcendental number that appears in a wide range of formulas in math and physics.  The values of both of these constants take the form of a never-ending, never-repeating decimal that can be approximated to an arbitrary number of places.

I dubbed my new formula the Power Ratio Method. I compared it to the classical formula, which dates back to the turn of the 17th century and is known as the limit definition of e:



Not only did my new formula converge much faster than the classical approximation, but the comparison led me to a second new formula.

I was cautiously excited. The only problem was that I had studied music composition and jazz guitar in college; I had no formal college-level mathematics education.  I knew very little about logarithms or calculus and knew of no mathematicians I could turn to.

I decided to write a brief description of my results that included some tables and graphs. Unsure of how to proceed, I sent this short paper to Ira Flatow, host of the National Public Radio show Science Friday, and also to a well-known mathematician at Scientific American. I never heard back from Scientific American.

However, my communication with Science Friday led to a fruitful collaboration with meteorologist John Knox, who came up with theoretical validation for my formulas. We became best friends and together discovered over two dozen new formulas. Over the next two and a half years, we published two papers on our methods. The work became somewhat popular; we received e-mail from around the world and write-ups in the journal Science and in Science News Online.  Our methods subsequently found their way into the standard college calculus curriculum by way of a popular textbook on the subject, Larson Calculus (7th edition).

I decided to go back to school to study calculus and differential equations.  Around the same time, I had been communicating with a German graduate student, Sebastian Wedeniwski, who had recently set the world record for computing the digits of  e. He expressed interest in our “closed-form” approximations and explained to me exactly the kind of “series” formula that would be more useful from a computational standpoint.

I soon came up with a family of such expressions based on the work of Isaac Newton. I went on to publish this new approach to deriving infinite series for e in The College Mathematics Journal. These expressions include the fastest known formulas for approximating this fascinating constant of nature.

I should add that as exciting as it is to get published in a peer-reviewed journal, the process is not designed for those who are either impatient or thin-skinned. Although every journal has its own character, in general, the world of peer-review is highly competitive, very demanding, and at times surprisingly unimaginative.

Even if your paper is accepted, you will usually have to “defend” it by responding to the criticism of one or more referees whose comments can be anywhere from indispensably helpful to frighteningly off-base. Be advised that by the time,

1)  the editor responds to your submission,

2)  you successfully and politely convince him/her that you are indeed correct,

3)  the journal implements any changes you've made and sends you the proof copy,

4)  you double-check their work and make any final corrections, and

5)  your article finally appears in print,

the process can take 12 to 18 months.

On the other hand, if you are inspired by your results and have the time and energy to present them in a clear, concise, and convincing fashion, it is well worth the effort. The intellectual rigor of the process helps you to indeed become an expert in your subject.