(c) 2017 Barry Scott Davies
In their book "enough is ENOUGH" (Berrett-Koehler 2013), Rob Dietz & Dan O'Neill (on page 75) quote a physicist named Albert Bartlet as having said the following:
"The greatest failing of the human race is our inability to understand the exponential function."
At first this may seem outrageous, but it (said function) is arguably causing a mass extinction on this earth.
And a mathematician once said to me "Perpetual growth is cancer and it's always fatal."
Following the "needs" link on sphinx-muse.com, you will find detailed calculations illustrating rolled over growth,
where p is the growth rate in percentage points and
and n is time with integral values corresponding to compoundings.
On this page, I would like to convey how quickly the exponential function increases.
First, look at the right hand, 'yellow', part of the image below. 'Log' is the natural logarithm function and 'Exp' is its (Log's) inverse function. Note: The natural logarithm function is commonly represented by 'ln', (l as in Lima & n as in November), but I use the notation, Log, of my version (4.1 VERY OLD saves $s and works fine for me) of Wolfram's software, Mathematica. The integral gives the area under the shown plot from 1 to x. Note that the three shaded rectangles each have area 1/2, and the sum of these are less than Log, which is the area under the 1/t curve from t=1 to t=8. Since there is no largest integer, and every integer, n, has a multiplicative inverse, 1/n, we can create under the curve an unboundedly great number of rectangles with area equal to 1/2. Therefore the area under the curve is unbounded, and that means that Log[x] tends to infinity (VERY SLOWLY) as t tends to infinity:
Log[x] is unbounded.
The exponential function, Exp, is defined to be the inverse function of Log. With Log, we start with a value of x on the t-axis and compute the area. With Exp, we start with a value of the area and compute the corresponding value on the t-axis. Note that the curve gets very close to the t-axis, and this means that increasing values for the area will force us out very far on the t-axis. This is what makes the exponential function increase so rapidly.
In the left hand part of the image, we have three parts:
Beginning : The distance across the observable universe was found by google given in kilometers (km), and since there are 1000 meters in a kilometer, and 1000 millimeters (mm) in a meter, we have 10^6 = 1,000,000 mm in a km. We get then about 10^30 mm as the diameter of the observable universe.
Middle : e is the value of the exponential function at unity: Exp, but the written expression is just shorthand notation for
where n is the number of compoundings and where p is the growth rate in percentage points. It is the fact that Exp and Log are inverses which makes the equation true.
End : We see here that Exp is, if we use the millimeter as a plotting unit,
the distance across the observable universe
above the origin of coordinates on the y-axis.
That's staggering: 10 centimeters on one axis, and more universes than you can shake a stick at on the other axis.