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|
|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 =ax, y=ex is special. To understand why, we have to quantify exactly how fast the function grows.
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:
The fact that the instantaneous rate of change of y = ex (1.000…) = ex 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 2x and 3x , 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 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 ey is itself (see end of blog for both derivational approaches.)
(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 + 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.
1ST APPROACH DESCRIBED
2ND APPROACH DESCRIBED
Single Variable Calculus. 7E. James Stewart. Brooks Cole
Differential Equations With Applications. Ritger and Rose. McGraw-Hill
The Catenary. David Maslanka