Origin and Properties of e

This story will not begin with how the irrational side of human nature and money are indeed at the root of much evil. Instead we will look at how a 17th century financially related-insight by Jacob Bernouilli eventually led us to the discovery of the irrational nature and properties of the number e.

Imagine the other extreme of today’s artificially-low interest rates, an annual rate of 100% = 1 ,  compounded twice a year:

A = Ao (1 + 1/2 )2 = 2.25 Ao   

This equation reveals that after a year, the original investment, Ao,  becomes 2.25 times larger than the original. By applying the same interest rate but with twice the frequency, an original investment of $1000  grows to $2250 as opposed to $2000. However, what Bernouilli noticed is that although further increases in compounding keep increasing the factor, the gains become progressively more miniscule. (See table. Needless to say, Bernouilli did not have a computer and did not use a googol in his calculations. 🙂 )

Compounding Type Frequency of Compounding Factor By Which Ao Increases Additional Amount Gained Over Previous Frequency ($1000 invested)
 monthly  12 2.613… =$1000 [ (1+ 1/12)12 – (1+1/2)2= $363.03
 daily  365 2.7145… $101.53
 every second  365(24)(3600)  2.7182… $3.71
a billion times a year  1  000 000 000 2.71828182709… $0.018
a quadrillion times a year 1015  2.718281828459043…  < $0.00000001…
a googol times a year  10100  2.71828182845904523… none, even with all the world’s $

The limiting factor of  2.718281828… is an irrational number like π; it cannot be expressed as a fraction and consequently its decimals are like some staff meetings, going on forever without a pattern.  The number was eventually called e.  When it was used as a base for an exponential function, it became even more interesting as it surfaced not only in financial formulas but in those of chemistry, engineering, biology and physics.

To see why e surfaces in the representation of many natural phenomena we will first express Bernouilli’s insight as a formula—it’s essentially what we have been using all along, but the number of times the interest has been compounded is n, and as n approaches infinity, we get closer to the value of e:e_limit

Next, we will arrive at this same formula by a completely different and far more bumpy route, but an important one which meanders through several key concepts. Among all exponential functions of the form y =ax,  y=ex is special. To understand why, we have to quantify exactly how fast the function grows.

tangent to e
The exponential curve was obtained by plotting only straight lines, a common trick used in both modern sculpture, architecture and in 1960s”hippies-string-art”. Knowing its rate of change and applying y =mx +b, a computer can easily plot various lines corresponding to different values of x1,  using y=ex1*x+(1-x1)*ex1

From the steepness of the tangents at various values of x, we can see that the rate of change for any exponential function (with a base >1) keeps increasing. How do we quantify it? The mathematical details for those interested are shown at the end of this blog entry. It’s a question of deriving an expression for the rate of change of a function y =ax , which in turn is based on the idea that if we zoom in enough on any continuous curve, we can represent it as a sequence of tiny and gradually steeper segments. If we use variables for points that are extremely close to each other, the rate of change- expression will hold for any point on that particular curve. The point’s coordinates will be the only necessary input needed to yield the instantaneous rate at that spot on the curve. For y = 2x, the rate of change is approximately 0.693(2x). For y=3x , it’s about 1.0986(3x). If we try bases bigger than 2 and smaller than 3, we see that it’s possible to have a base that yields an instantaneous rate of change that comes pretty close to exactly 1 times itself.  If we use the base e that  Bernouilli “stumbled upon” along with a small h-value like 10-6, we obtain a value of 1.0000000:

definition ofe

The fact that the instantaneous rate of change of y = ex (1.000…) = ex has many interesting consequences:

(1) For beginners it’s tied in to the coefficients of 0.693….. and 1.0986 for the derivatives of  2 and  3, since those are the exponents required by e  in order to become either 2 or 3, respectively.

(2)When we invert the x and y coordinates for y = ex  and end up with the function y = ln x (which is what we were doing when we obtained 0.693 for 2) we get a reflection of the function, as if the y=x line was acting as a mirror. The rate of change of that new function is simply y’= 1/x.

(3) And if we use limits to get the instantaneous rate of change for the inverse of the general exponential functions, and use the discovered fact that y’= 1/x when y = ln x, we can travel along a different path to reveal again thatCapture(see end of blog for a mental map)

(4) The reason that some form of y = ex is the solution to many differential equations is tied into the fact that many instantaneous changes are proportional to their own instantaneous amounts like a growing, compounded investment; or a multiplying bacteria colony with adequate resources or a decaying radioactive nucleus. In each of those cases when we isolate the variable of time, on the other side of the equation we find the incremental amount of money, bacteria or atoms as dx multiplied by 1/x. Taking the antiderivative of that product, on our way to isolating the variable of time, leads us to ln x and eventually to an expression of its inverse, a function of e.

To an uncritical eye, an outdoor telephone wire or chain sagging from its own weight may seem like a parabolic curve. But it is not. If we balance a chain’s horizontal components of tension and do likewise for its weight and vertical tension- component, upon dividing we get an expression for the tangent-ratio of the angle between a horizontal component and the chain. The former can be expressed as a rate of change between the y and x coordinates. Its derivative of second-order ends up being related to the rate of change of the chain’s arc length. After some sneaky substitutions, the second degree differential equation can be solved and we reveal that the shape of the chain is a function of cosh (δx/H + c), the so-called catenary derived from the Latin word catena for chain. But a cosh x function is simply defined as  0.5(ex + ex ).    (Again see below for details.)

Even as I do the laundry and hang it out to dry (consistent use of the electrical dryer is a waste of energy and removes too much lint from clothes), I cannot escape the beauty of e.

Mathematical Details:

deriv of e

derivof a^x
deriv of ePt2

 

Sources:

Single Variable Calculus. 7E.  James Stewart. Brooks Cole

Differential Equations With Applications. Ritger and Rose. McGraw-Hill

The Number e  School of Mathematics and Statistics
University of St Andrews, Scotland  

The Catenary. David Maslanka

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Wine from a Radioactive Age

There’s little doubt that a fair amount of good can come out of the synthesis of radioactive isotopes in the fields of medicine and pure research. The most common radioisotope used in diagnosis is technetium-99, with about 40-45 million procedures performed annually. Without carbon-14 as a tracer, the pathway of photosynthesis’ dark reactions would never have been elucidated. Without ratios such as that of ²H to ¹H in ice cores, we would not have been able to verify that for tens of thousands of years temperature has correlated well with levels of carbon dioxide. (Of course given that CO2 molecules absorb heat, we have both causation and correlation between the two variables).

philippe_hubert_slide-f9313655df965457a08776d5cbc4b4cf156b94fa-s1100-c85
Image of French physicist Philippe Hubert from http://goo.gl/I121Z4

The nuclear industry does not keep such facts secret. Another truth is that these isotopes can be produced in fairly small reactors, and even at that, they are not incident-free; maintenance and regulation are essential, and eventually old reactors have to be shut down, such as will occur with the one at Chalk River in Canada in 2018.

Cs137
from http://i.stack.imgur.com/5cziZ.jpg

Radioisotopes can also serve as interesting probes in the wine industry. Shortly before and after various countries banned nuclear testing (mostly in the 1960s; although France persisted until the 70s and China, 1980s) the level of cesium-137 was much higher in the environment. Before the nuclear age, that particular isotope did not exist in nature. It was first formed from the fission of 235U. Vines absorbed 137Cs that had spread in the environment, and it ended up in wines.  By measuring the amount of gamma radiation, which goes through the wine’s glass bottle, scientists can authenticate the dates on the label and expose fraud. Thanks to the testing-bans and to its half-life of 30 years,the concentration 137Cs has steadily declined in the environment and Bordeaux wines produced since 1990 should emit near-zero levels. It is true that without the 1990s development of low background germanium (Ge) semiconductor detectors, the very low levels of radioactivity found in wine  (0.01 to 1 Bq– 1 disintegration per second) would never have been detected. In contrast, a banana has 15 Bq due to its small percent of beta-emitting potassium-40. One kilogram of low-level radioactive waste accounts for 1 million Bq. Multiply that by 107, and that’s what’s found in 50-year-old high-level waste!

ChelyabinskCesium-137 can also come from nuclear accidents. The little-known Kyshtym Disaster of 1957 in Russia released 7.4 X 1016 Bq, which killed 200 people, evacuated 10 000 and affected about a quarter of a million others. Another source claims that in the same Chelyabinsk province region, about half a million people were irradiated in three separate incidents, exposing them to as much as 20 times the radiation suffered by the Chernobyl victims in 1986.

For fear of ringing alarm bells over levels in old wines,  comparison to natural background radiation is made and laradiaoctivite.com points out that whatever gamma released by those wines is insignificantly smaller.  Even if we consider that cesium-137 wasn’t the only isotope released—beta-releasing carbon-14 in Australian wines shows a similar pattern— the combined amount of radiation originating from those 1960s wines is still miniscule. But for people living at the time, they were also ingesting other foods, so needless to say, for that reason and others, the nuclear test ban was more than necessary. Any man-made radiation from radionuclides is added on to other sources of beta, gamma or alpha and to that of medical X-rays,  all of which lead to more of an assault on our DNA.

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