e: Α number who can teach

The following article is republished after our recent communication we had with Russel Tarr, lead organizer of Lead organizer, Practical Pedagogies Conference which accommodated our proposal about number e.

We are going to discuss today about the enigmatic number e, its secret origins and its huge contribution to large data calculation, logistics and science. Everyone knows number e as a famous irrational number, one of the most important number in Mathematics, with value equal to 2.718281… But it is only an arithmetic value? What purposes does it serve? Where and how did it come from?

In the Beginning was the Word (Logos)

The Holly Bible starts with the phrase… ”In the Beginning was the word”. It is something absolutely real and obvious, if you consider the word “word” as ”logos”, which means that using logic and simplicity, everyone is able to understand and explains the every day life situations with numbers and mathematics in general. Obviously, the mathematical alphabet was created from the first 10 digits 0, 1, 2, …, 9. Moreover, the set of natural numbers, the set of rational ones and the set of integers were made from these digits in the 10th century.

But there are also numbers with “exceptional personality”, that have a special place to mathematical ecosystem and calculus formulation. If you asked someone to name his favorite mathematical constant, you would probably get some quizzical looks. After a while, someone may volunteer that the best constant is pi. Pi typically denoted by the Greek letter Π, is the mathematical constant defined as the ratio of a circle’s circumference to its diameter.

There have been many extraordinary attempts in the past to approach number pi with the greatest possible accuracy. In 400 BC the Hebrew Bible describes the construction of the Temple of Solomon, saying that the ceremonial pool’s dimensions include a diameter of 10 cubits and circumference of 30 cubits. This would leave to assumption that pi would equal around 3, if the pool is circular. However, first recorded algorithm for calculating pi’s value was done by Archimedes, by using polygons. In order to determine pi value, he had to know the circumference of a circle with given radius. He drawed a circle between two squares and he calculated their perimeters. Consequently, he computed an upper and lower bound of circle’s circumference & so an upper and lower bound of pi. In order to achieve greater accuracy to his calculations, he started gradually increasing the number of polygons’ sides, until he reached a 96 - sided polygon. By doing so, he proved that pi is between 3,1408 and 3,1429.

The Birth of Logarithms

But pi is not the only important mathematical constant. A close second, if not contender for the crown of most ubiquitous constant, is number e. Like pi, e is an irrational real number that cannot be written as a fraction and its decimal expansion goes on forever with no repeating block of numbers that continually repeats e = 2,718281… It shows up in mathematics and is the base of the Natural Logarithms, invented by John Napier. It was in 16th, 17th century, when the Age of Renaissance was being completed, the traditional trade was being developed and the scientific revolution took place. Then a huge explosion in scientific developments’ area took place, including the Copernical sun – centered solar system, Maggellan’s circum navigation, Keppler’s laws of planets’ motion. However, these and still much more scientific achievements were greatly impeded by the continually increasing complexity and labor of numerical calculation.

The solution was given, in 1614, when John Napier a Scottish mathematician, started working to speed up and simplify such calculations. Based on this trigonometry relation, he observed that any multiplication could be expressed as addition or substraction between numbers and that to multiply any two powers of a fixed number?r, say?rm?and?rn, one need only add the exponents to obtain?rm+n.? This is the key – idea behind Napier’s discovery: ?if any positive number X could be showed as the power of a given constant number r, then multiplication (or division) between numbers is equivalent to addition or substraction between their corresponding exponents”. According to his calculations, any number n is the logarithm of an X number to a given base r, if and only if the following is true: X = rn.

The base r can be any number, but Napier after 20 years’ calculations, he came to choose a value approximately equal to 2,71828, which simplified some of the calculation he was working on. Using powers of 2,71828, Napier tabulated his work in logarithms under the title Mirifici logarithmorum canonis descriptio,?which translates literally as A Description of the Wonderful Table of Logarithms.

Note that then Sine was not defined as the ratio of opposite over hypotenuse, as we know it today. It was defined as the length of that semi-chord of a circle with given radius R, which subtends the angle at the center. In modeρn notation it says that: Sineθ = R*sinθ.

So, Sine0ο = 0 and Sine90o = 10^7. Refer to the diagram above, AB is the chord and AC is the semi-chord & we have the relation AC = sin (θ/2). As he had also astronomical calculations in his mind, Napier took the radius to be R = 10^7 units. He arranged his table by taking increments of arcθ,?minute by minute, then listing the sine of each minute of arc, and then its corresponding logarithm.

The last row says that?10^7*sin(9o 15′) = 1607426107, that?l(1607426)=182795071 and reading from the right, it gives the Sine of the complement angle?80o 45′. To be precise, Napier's table gave the "logarithms" of sines of angles from?0ο?to?90ο i.e., of numbers between?0?and?10^7, positive and so as to contain a considerable integral part. And so with the help of logarithms, sines of angles & as a result large distances could be expressed as powers of 2,71 and so it make easier for the astronomers and navigators to make calculations with large numbers.

The size of man’s greed… in the limit of time

However, John Napier’s work did not contain the constant e itself, but simply a list of logarithms calculated form that constant. As we see in this slide, he first appearance of e… took place through the field of loans & interest, a field that was quite familiar to the Hebrews in the period before Christ. The apocalypse of the constant itself is credited to Jacob Bernoulli in 1683, by studying a question about compound interest. An account starts with $1 and pays 100% interest per year. If the interest is credited once, at the end of the year, the value of the account at the year-end will be $2.

If the interest is credited twice a year, the interest rate for each six months will be 50%, to the initial $1 yields $ 2,25 at the end of the year. Compounding quarterly yields $ 2,4414… and compounding monthly yields $ 2,613035… If there are n compounding intervals, the interest for each interval will be 100%/n and so the value will be $ 1*(1+1/n)n.

Compounding weekly (n = 52) yields $ 2,714567, while compounding daily (n = 365) yields $ 2,714567. The limit as n grows large, is the number that came to be known as e; with continuous compounding, the account value will reach $ 2,7182818.

The Apocalypse on Number e, by Leonard Euler

Although this existence was more or less implied both in the work of John Napier and Jacob Bernoulli, Euler was also the first to use the letter “e” for it in 1727 (the fact that is the first letter of his surname is coincidental). Please note that the period when Euler lived, the Enlightment period, the Industrial Revolution and other critical chronological points of 18th century, came to be a historical mark to social and scientific frame’s formulation.

A remarkable point where scientific explosion of developments took place in several fields of philosophy, mechanics, science and calculus. So, Euler came up with an effective way to approach the one what we know today as “number e”, by using the following infinite sum:

It is a fact, proved by Euler, that e is an irrational number, so its decimal expansion never terminates, nor is it eventually periodic. Thus no matter how many digits in the expansion of e you know, the only way to predict the next one is to compute e using the method above using more accuracy. The Euler’s Identity combines five of the most important numbers in mathematics. These are:

• 1 – the basis of all other numbers
• 0 – the concept of nothingness
• pi – the number that defines a circle
• e?– the number that underlies exponential growth
• i?– the "imaginary" square root of -1

The numbers all have many practical applications, including communication, navigation, energy, manufacturing, finance, meteorology and medicine. But that's not all. Euler's identity also contains the three most basic mathematical operations: addition, multiplication and exponentiation.

The personality of number e, has the following main characteristics:

1. The relationship between the logarithm to base e and number 1, which is the starting point of natural numbers. lne = 1
2. One of the special properties of number e is that its arithmetic value from the one hand and its rate of change from the other, is one and the same. That property is vitally important when it comes to physics, where the speed of a body’s change is rational to the mass of this body. d(e^x)/dx = e^x????? ? ????d(f(x))/dx = α f(x)
3. The possibility that number e can be expressed as an infinite sum, showing a special harmony between power of numbers and factorial to each fraction. We could describe it as a special hyper-polyonyme consists of infinite terms, that it remains the same, no matter if we refer to its first, second or third… etc derivative.

1. The unique property that combines the imaginary unit with sinus and cosinus of each angle. This combination is expressed through a mathematic formula, which is the key factor not only for Euler’s infinite series, but also for creating a passage from the mathematical concept of the past to the correspondent one of the present. e^ix = cosx + i sinx (Analysis of Complex Numbers)
2. Solving differential equations with constant coefficients, is taking place exactly through the exponential function.
3. Have we ever thought about, how could we describe possibility theory distributions (Νormal, Poisson, Exponential), without the exponential function, or in other words… how this magnificient Normal Distribution curve could be plotted without the exponential function’s magical ?rate of change?

Call of Cooperation to the Open Teachers’ Community

With our participation to Practical Pedagogies 2016, organized by IST School of Toulouse and Mr. Turr, we took the “boat of thinking & knowledge”… the boat with number e as passenger, which we led to the open sea of educational challenges. As people of different age groups, we have traveled through the teaching of Mathematics, from Primary Education to University. But we have also traveled through history, when the human concept started to develop and the common logical language of science and improvement was created... the Mathematics! We created the Open Teachers’ Community in order to give real meaning to technology, to share educational practices and finally create collaboration teams, without time or space boundaries.

Dear fellow teachers, we invite you to sign up for free, to our website teachers.arnos.gr and it we will be our pleasure to determine our next steps of action and creativity.

Do not forget that our thoughts are the ones to create boundaries... and that boundaries are captured from our inaction.

Krokos P. Yannis: Mathematician – Civil Engineer

Tsilivis K. Vasilis: Mathematician

Toulouse 3-11-2016, International School IST