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 = A_o (1 + 1/2 )² = 2.25 A_o   

This equation reveals that after a year, the original investment, A_o,  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:

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 =a^x,  y=e^x is special. To understand why, we have to quantify exactly how fast the function grows.

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 x_1,  using y=e^x₁*x+(1-x₁)*e^x₁

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 =a^x , 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 = 2^x, the rate of change is approximately 0.693(2^x). For y=3^x , it’s about 1.0986(3^x). 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:

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

(1) To start with, it’s tied in to the coefficients of 0.693….. and 1.0986 for the derivatives of  2^x  and  3^x , 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 = e^x  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 and of course, conversely, the area under the curve y= 1/x from 1 to x equals ln(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 that

Conversely, if you accept the above to be a trial and error discovery from Bernouilli, you can use the fundamentals of calculus to derive that y’= 1/x when y = ln x and from this fact and from the Bernouilli definition of e, it will follow that the derivative of e^y is itself (see end of blog for both derivational approaches.)

(4) The reason that some form of y = e^x 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(e^x + e^–x ).    (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:

1ST APPROACH DESCRIBED

2ND APPROACH DESCRIBED

CATENARY DERIVATION

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

The Causes of Heart Disease: Mostly Not Genetic

Some diseases such as chicken pox, TB, maple syrup urine disease and sleeping sickness are rooted in a single cause of a viral, bacterial, genetic and parasitic nature, respectively. But the two most common cardiovascular diseases, atherosclerosis and stroke, which combine to be the number-1 killer of humans worldwide, result from several factors, most of which are not genetic. Yet in too many minds the simplistic view that heart disease results from some pipe-clogging-like agent persists. The excessive focus has shifted from cholesterol to saturated fat and trans fats, while the potential of preventative medicine based on a fuller understanding of the causes of heart disease has been compromised.

Four completely different diseases, each brought on by a single cause. Heart diseases , unfortunately, do not fall under such a neat and tidy category.

As early as in the 1970s some medical researchers realised that dietary cholesterol was not cause of atherosclerosis and that the multifactorial nature of the disease was evident. In a 1977 Scientific American article on atherosclerosis, Earl P. Benditt wrote:

Moss and I fed cholesterol and administered estrogen to chickens and examined their vessels with the electron microscope. We found, first of all, that the lesions that cholesterol induced in the chickens did not resemble human atherosclerotic plaques. They appeared to be composed entirely of fat-filled cells derived from blood macrophages; there was no evidence of significant smooth-muscle-cell proliferation. And none of the lesions evolved into the raised plaques characteristic of the human disease.

Poring through recent review articles of the  medical and scientific literature, we observe that the current consensus on atherosclerosis is that’s an inflammatory disease characterised by intense immunological activity.  Moreover, as a November 2016 Nature Reviews Cardiology paper, Nature versus nurture in coronary atherosclerosis, points out:

“Lifestyle factors may powerfully modify risk of coronary artery disease regardless of the patient’s genetic risk profile.”

These findings are consistent with the modified response to injury theory of atherosclerosis. It identifies a number of “insults”, most of which are environmental or linked to lifestyle choices. The following factors trigger the intense immunological activities responsible for heart disease.

(1) Reactive oxygen species from smoking and/or air pollution

(2) High blood concentrations of “low density lipoprotein”(LDL) or “very low density lipoprotein” (VLDL), which contains more triglycerides.

(3) Chronically elevated levels of glucose

(4) Turbulent blood flow from arterial branching

(5) Stress from hypertension or elevated blood pressure

(6) Inherited metabolic defect that raises homocysteine levels, affecting the endothelium of tissue.

Until recently, it was not clear how air pollution led to heart disease. But Mark Miller and his team from Edinburg University and three Dutch institutes wondered whether tiny particles of soot migrate from the lung’s air sacs to the walls of blood vessels of the heart and brain.  Since they used human subjects in the study,  for ethical reasons, the researchers relied on gold, which is biochemically inert. Their patients were asked to inhale soot-sized particles of the precious metal. Within 15 minutes to 24 hours the gold showed up in the blood and urine, and three months later it was still in the body. When mice inhaled gold nanoparticles twice a week for 5 weeks, diseased arteries contained five times as much gold as healthy ones. Three human patients who were already scheduled for heart surgery and who inhaled the fine gold particles ended up concentrating them in their arterial plaques, just like the mice had done.

Small particles tend to get into the blood stream more easily than larger ones.  From ACS Open access

Upon examining the data, I also found it interesting that small gold particles (less than 10 nm) got into the blood stream far more efficiently than larger ones. This helps supports the current emphasis that environmental scientists place on PM 10 and PM 2.5 pollution, which are not only linked to heart disease but to lung cancer. Also, in the arterial plaque of mice, the gold particles were found within macrophages, suggesting that these may play a role in the translocation of the particles. How the gold is carried around is still not understood, but the authors point out that such knowledge is needed to assess the full impact that inhaled nanoparticles have on health.

Inhaled gold particles were found in immune system cells within arterial plaques of mice. From ACS open access

Sources:

• World Health Organization Cardiovascular diseases (CVDs) Fact Sheet
• The Origin of Atherosclerosis
• Pathogenesis of Atherosclerosis
• Nature Reviews of Immunology 
• Nature versus nurture in coronary atherosclerosis 
• Inhaled Nanoparticles Accumulate at Sites of Vascular Disease