An Ode to e

Mathematics is a fascinating subject to some people, but a horror to most. Formulas and rules abound to govern purely abstract relationships that appear alien to ordinary life. Yet, mathematical laws govern our entire world, and the Universe. Physicist Max Tegmark believes that the Universe is itself entirely mathematics, i.e., we are all elaborate formulas in some metaverse.

Embedded within the mathematical laws are inscrutable constants, such as pi and e, where e is the so-called base of the natural logarithm. e is roughly 2.718281828…. Although Pi (3.14159…) has an official day, 3-14, or March 14, e has yet to acquire this honor. Last year, I wrote in honor of World Pi Day on this site and also below.  This year, I propose making 2-7, or February 7, National or World e Day.

The Wonders of e

The constant e is found primarily in mathematical theories and physics computations, but it also turns up in finance. Just like the mysterious pi, the constant e has a lot of stories and mysteries of its own and is also related to ordinary life. For example, e is found in the study of compound interest in banking, as well as probability theory. But e is of considerable value in the entire field of calculus, where the use of e reduces computational complexities.  Another explanation of e:

Before calculators and computers, e and natural logarithms were a mainstay in slide rules, which were graduated scales of numbers along two slidably-arranged pieces of wood. Multiplication and division were easy using a slide rule, simply lining up numbers on the appropriate scales. But these devices included considerably more functionality with the usage of logarithmic techniques, using the base e, and exploiting the properties of these functions to simplify complex calculations using log scales.

The Renaissance was a time of great intellectual exploration, and science required precise measurements and instruments for calculations. Just after Galileo, John Napier invented logarithms, which in essence is a simplified mathematical reformulation of numbers to make them easier to calculate. Such concepts were, however, generally known by the Babylonians (2000 BC) and Indians (800 AD).

To the Moon

Using logarithms simplifies the math, e.g., the process of multiplication is simplified to addition, and large or small numbers could be calculated by first simplifying the number, e.g., 4,567 could be reduced to finding the log of .4567 (from a precomputed table) and adding the exponent value (10,000) afterward. During the Industrial Revolution, slide rule design and usage went into overdrive with the rise of science, technology and engineering. We went to the Moon using slide rules and e.

The constant e, actually hinted at by Napier and others, was first calculated by Jacob Bernouli, one of the famous Bernouli Brothers of mathematics, in 1683. Gottfried Leibnitz first used the letter b for the constant in 1690, but Leonhard Euler coined the letter e for this constant (for Euler?) in 1727 or so, and that coinage later took.

A Complicated, but Constant Cousin

Just as with pi, e is both an irrational number, i.e., it is not a ratio of integers, and a transcendental number, i.e., not a root of any non-zero polynomial with rational coefficients, which means that e’s digits, like pi’s, continue unrepeating to infinity. The constant e is also prevalent in mathematical formulas that involve a series going to infinity, e.g., the Taylor series in calculus and many others.

e does, however, crop up in unexpected places. For example, Google’s IPO valuation was for $2,718,281,828, or e billion dollars. More obscurely, famous computer scientist Donald Knuth labeled the versions of his Metafont program, as 2, 2.7, 2.71, 2.718, and so on.  See here for a Simpsons take on transcendentals pi and e:

Unlike pi, which is much better known, and which is even relatable to many, e is a more complicated and more distant constant cousin. Still, this marvelous constant is of immense value to science and society at large and should be commemorated accordingly. Hence, I proclaim February 7 as World, National, or International e Day.

Raymond Van Dyke is an IP consultant, strategist and educator; he has an undergraduate degree in mathematics, and was on the math team in high school.

The mystery and transcendence of pi

Today is National and World Pi Day because the numbers of the day (3-14) match the first three digits for pi or π, the Greek letter, 3.1415926535897… Although most people think that π is relegated to just geometry and trigonometry, the number pervades all of mathematics and the natural sciences, even statistics.

Several thousand years ago the Egyptians, the Babylonians, the Chinese and the Ancient Greeks tried to make sense of the world through mathematics, an abstract way to envision and explain the operations of Nature, not as the activities of the gods. Over time geometry developed, which could explain much of the world. For example, Euclid and his various axioms were employed to describe much of the natural world. However, when it came to circles and non-linear lines, there remained a mystery among all the Ancients, which was π.

It had long been recognized (and still taught to reluctant students in high school geometry) that the ratio of the circumference of any circle to its diameter is a constant. The Ancients knew this, but the value of that constant eluded them. They realized, however, that there were approximations, e.g., the fractions 25/8, 22/7, 256/81, etc., that were close, and these fractions were employed for centuries as substitutes for pi.

Over two thousand years ago Archimedes carried this approximation technique to its logical limit, using techniques akin to calculus infinities approaches, and was able to obtain very close estimates of π to whatever tolerance was needed, e.g., through circumscribing and inscribing large numbers of polygons, e.g., an algorithm employing up to 96 such polygons for an accuracy between 3.1408 and 3.14285, about 99.9% accuracy. But, around the year 480 A.D., Chinese mathematician Zu Chongzhi used this approach with 12,288 polygons, and created a far more accurate fractional approximation, 355/113, roughly 99.99999% accurate, which was the best approximation for π for the next 800 years.

As a side note, through recent discoveries, Archimedes is also credited with understanding aspects of calculus long before Newton and Leibnitz, who developed differential and integral calculus just over three hundred years ago. Had the Roman soldier not killed Archimedes in the siege of Syracuse, our world may have been very different. But, I digress.

Clearly, these fractional representations of π were all approximations and not a pure answer, which galled the Ancients at their inability to solve the conundrum. Indeed, the purity in mathematics was at the heart of Euclidian geometry’s goals of solving problems. For example, in their effort to solve the π enigma, the Greeks were famous in their efforts to “square the circle,” i.e., geometrically constructing a square having the same area as a given circle, and asking whether Euclid’s axioms posit the existence of such a number. However, the Greeks and many others later could not do it, which had profound implications to Plato regarding the usefulness of Euclid’s theorems or even mathematics to actually describe the real world. In short, the quest was impossible. But why?

With Euclid and the pre-Socratics trying to explain the world in physical ways, e.g., Democritus postulating atoms in a very logical way 2,500 years ago, it is sad that the mystery of π seems to have derailed the very influential thinkers Socrates and Plato to fully trust mathematics. Accordingly, Plato looked to another realm to describe the world: using his forms or abstractions. For example, the concepts of a circle and π were perfect, idealized forms, but every attempt to depict them in the real world would, by definition, be imperfect. This philosophical view held sway until the Renaissance started new ways of thinking.

But, back to π. We now know that pi is both Irrational and Transcendental. An irrational number is defined as a number that is not a ratio of two whole numbers, i.e., fractions. This irrationality of pi is strongly suggested by Archimedes’ and others’ succession of better and better fractional approximations, without a final answer. Also, with computerization it has been found that the digits of pi have no pattern, and for several trillion digits pass the mathematical test of normality, i.e., all of the digits appear equally often in the series. The irrational nature of pi was formally proven in 1761.

A transcendent number is defined as a number that is not the root of any non-zero polynomial with rational coefficients, which is a modern way of saying you cannot square the circle.   The transcendence of π was proven in 1882. The staggering notion that the digits go on and on, without repeating or in any pattern to infinity, was (and remains) hard to grasp, the immensity of which was something well understood to Aristotle and others. Over a hundred years ago, however, mathematician George Cantor tackled the mathematical problem of infinity and actually demonstrated the nuances between infinities. π is also computed by various techniques, e.g., equations and trigonometric series, that have terms that go to infinity.

The use of the Greek letter π in this context dates from about three hundred years ago when the great mathematician Leonhard Euler started popularizing it. Mathematician William Jones in 1706 is accredited with being the first to symbolize the circle circumference-to-diameter ratio as π, which is also attributed to the Greek word for perimeter. Prior to computers, pi calculation was a laborious and very error-prone endeavor. With the advent of computing, the mere six or seven hundred digit manual calculations not too many decades ago have jumped to many trillions of digits.

Despite all of the mathematical rigor of the modern era, π remains a mystery, a constant that in a way is inconstant. Of course, there are many other such enigmatic irrational and transcendent numbers out there, e.g., e (2.71828182845…), but π is the oldest of these cosmic constants for us humans. On a related note, this is the 50th anniversary of Stanley Kubrik’s 2001: A Space Odyssey, an inscrutable movie that still contains innumerable mysteries. It is also the 20th anniversary of π, the movie, a psychological thriller about the irrationality of π and the human mind. In Star Trek, Mr. Spock crashed a hostile computer making it calculate pi precisely. π also pops up once and a while in TV shows, such as the Simpsons.

This magical number is everywhere, and is part of our lives – even if you hated high school geometry and math. Indeed, we are all still trying to understand the meanings of π.

 

Raymond Van Dyke is an intellectual property/patent attorney, educator and a science and technology enthusiast. He has a B.S. in mathematics/computer science and was admitted Pi Mu Epsilon, an honorary mathematics society, has an M.S. in Computer Science, and a J.D. from the University of North Carolina at Chapel Hill. He is the Chair of several organizations and teaches IP, technology law, the history of technology and IP. His website is:  www.rayvandyke.com.  A version of this article was published on ipwatchdog.com.

The Importance of Pi

Today is National Pi Day because the numbers of the day (3-14) match the first three digits for Pi, which is both an irrational and a transcendental number, i.e., the number is not a ratio or a root of any other numbers and the digits in Pi have no pattern (at least in the first few trillion digits).  Mathematicians have spent their lives trying to discern meaning in Pi.  Carl Sagan’s book (not the movie) Contact dealt with this theme at length where Pi contains messages from ancient beings, and where the main character Ellie gleans some preliminary meanings.

The earliest approximations of Pi by the Egyptians, Babylonians and Indians were fractions, e.g., 22/7, which is close to Pi, 3.14159265358….  The Greeks were famous in their efforts to “square the circle,” i.e., geometrically constructing a square having the same area as a given circle, and asking whether Euclid’s axioms posit the existence of such a number.  The Greeks and many others could not do it, which had profound implications to Plato regarding the usefulness of Euclid’s theorems to describe the real world.

To this day, the importance of Pi remains high since the number pervades much of Nature, and mathematicians are still calculating it.  Also, people love to recite the digits.  Piphilology is study of techniques for people to memorize the digits.  The record now is over 100,000.  As a mathematics/computer science major, I also memorized Pi – to about 20 decimals.

On a different note, today is also National Potato Chip Day, commemorating the 1853 (or so) creation of this snack.  Apparently, there are over 1,500 such “National” days for various things, in addition to national Weeks and Months. Happy Pi Day!

Raymond Van Dyke, Pi enthusiast