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

The Physics of Snowflakes & the Spirit of Christmas

We shovel snow from our door steps because although no two snowflakes are alike, far more than two land in the same place. Rigorous shovelling helps beat the cold. and once warmed up, we can think about falling snowflakes. And if we want to be even more captivated, we can observe them.

First the thinking part. If we want to merely predict the velocity of falling snowflakes, we already run into a complication. At least raindrops begin as spheres, and then as they grow larger, their shape approximates that of a burger bun. That affects their area and drag coefficient —numbers needed in assessing to what extent air slows down the rate of falling drops. But snowflakes are formed in a countless variety of shapes and sizes. There is far more averaging out to do.

So assume that it’s been done. We subsequently write an expression for the product of air density, the flakes’ average area, their average total drag coefficient and square of their velocity. Then we subtract that expression from the force of gravity. The difference will equal to the so-called net force, which is the product of mass and rate of change of velocity with respect to time—Newton’s Second Law.

In our differential equation, velocity appears on the equation’s two sides, one of which also has the variable of time. Isolating the variables and using appropriate substitutions allows us to integrate and solve for velocity. As the time that the snowflake falls increases, exponential terms drop out of the equation, and the flake’s terminal velocity seems to depend only on the  the snowflake’s mass and the shape -influenced and gravitational constants we mentioned earlier.

8051109Now we observe. As we stated at the onset, many snowflakes land in the same place. But only a few meters above any given spot, it is apparent that many paths lead to a common destination. Some flakes tumble; some abandon the terminal velocity we took so long to calculate, and they yield themselves to whimsical eddies. How they arrive is influenced not only by shape, mass and gravity but by sheer luck—luck due to the random, pinpoint fluctuations in temperature and pressure that affect their air space.

And these unpredictable*, forgotten, dance-like movements of deviant snowflakes open our eyes and widen our mouths. They drain our minds of thoughts of shovelling and of future slush and social conflicts. For a few moments the destinies of snowflakes is all that matters, and then we are reminded of a beautiful, non-mathematical expression in which snow is equated with Christmas.

*N.B. In reality the larger snowflakes may behave like sheets of falling paper which experience aerodynamic lift, a lift dominated by the product of linear and angular velocities. Those of you interested in computer simulations of falling snow might find this link interesting: https://www.cs.rpi.edu/~cutler/classes/advancedgraphics/S08/final_projects/fermeglia_willmore.pdf

WordPress.com.

Up ↑