More than you wanted to know about e – part 1

If you ever owned a scientific calculator or used one on your computer, it is still likely that you have never used the button that reads ex. One of the most interesting secrets of the universe is evoked by pressing that button, but I have never heard it explained well. This post will likely add little to help that situation.

Just as pi (the Greek letter 𝛑) was a strange number that kept showing up whenever the early Greek mathematicians (and the Egyptians and Babylonians before them) tried to measure the circumference or area of circles, e is an even stranger number that keeps showing up in nature whenever something is growing or accelerating at a proportional rate, say, 5% of its size or speed in any given period. And that kind of growth, it turns out, is central to much of the natural world around you.

Sometimes called “Euler’s number,” named after the Swiss mathematician Leonhard Euler (1707– 1783) who made a lot of contributions to understanding it and first used this notation, e is usually written as an italicized letter “e”, and it has a value of approximately 2.71828…

The ellipsis at the end of the number indicates that, like 𝛑, this is a transcendental number, one that never resolves itself in our decimal numbering system no matter how many digits you go out to the right of the decimal point. I have decided to leave the math and physics definitions of e for Part Two of this post and focus on where you might more likely see the effects of e in the biology around you or in business finance. This order runs the risk of the mathematicians and physicists saying, “That’s not exactly right…” but bear with me until Part Two.

The biology definition

For starters, recognize that the “base 10” number system and the “integer” (1, 2, 3, etc.) counting system humans usually use are themselves human inventions. Biology was here some 3.5 billion years before us, and you can think of e as the base of “nature’s numbering system,” which is a continuous process rather than a discrete one (that is, a finite or countable set of values). [1]

I also have to introduce here the idea of a logarithm. Logarithms are a way to turn multiplication problems into addition problems, and exponential problems into multiplication problems. For example, in our common base 10 math, the logarithm of 10 is 1, while the logarithm of 100 is 2, and for 1000 it is 3 (101, 102, and 103, respectively). If you have ever used a slide rule (a fast-vanishing skill), you used logarithms, likely without knowing it, to turn a multiplication problem into the physical addition of two sliding logarithmic scales. When numbers are represented by their “exponential power” (their logarithm), as on the sliding scales, then positioning them against one another to “add” them yields the product of their multiplication.

Slide Rule

It turns out that biology figured this out first, but to see it you need to use what are called “natural logarithms,” using e (2.71828…) as a base instead of 10 (and abbreviated as “ln”). The natural logarithm (ln) of e is 1. the “ln” of e2 (approximately 7.389) is 2, and so on. As we will see, these values keep showing up whenever we attempt to translate natural biological growth into our decimal numbering system.

Take for example the beautiful chambered nautilus, shown in a cross-sectioned cutaway below with its “logarithmic spiral” This spiral is the result of a near-constant and continuous growth rate, and simultaneously at a near-constant growth angle.

Chambered nautilus

Source: Wikipedia

The aforementioned natural logarithm (ln) of e shows up in the determination of the size of the nautilus at any given angle around the center (represented by the Greek letter theta, or θ). The constants “a” and “b” will determine the angle of growth and eventual size of the nautilus if it maintains its growth rate:

Nautilus angle

Another biological example that can be expressed using e or its natural logarithm is found in how we hear both the frequency and the loudness of sounds. As I noted in an earlier post, we perceive the western 12-tone musical scale as a set of linear (constant) increments, but instead they have logarithmic relationships, as demonstrated on any guitar fretboard. The audio frequency “distance” between the notes changes as you go up the musical scale.

DAngelico EXL fretboard

The same goes for our perception of “loudness.” While we usually imagine increasing sound intensity as a linear scale on a volume knob, the acoustic “power” behind the sound is actually increasing at an exponential rate that can be described in nature using the natural logarithm (called the Weber-Fechner Law), although this is usually translated to a base 10 logarithm on the common “decibel” scale used in audio production.

volume knob

According to the Weber-Fechner Law, a sound’s loudness (and a light’s brightness as well) that we perceive is proportional to a logarithm of the actual intensity measured with a nonhuman instrument, say a sound meter or light meter. And the “natural logarithm” (ln) based on e shows up again:

The financial definition

It was actually in the study of finance that the value for e was first uncovered. In 1683, Swiss mathematician Jacob Bernoulli was experimenting with the mathematics of compound interest. This is the common form of interest on borrowed or invested funds where you pay or earn “interest on your interest.” For instance, if you are charged a 100% interest rate annually on a $100 debt (not uncommon in today’s “payday loan” industry) you would owe two times the original amount, or $200 after one year if this were a simple interest loan. On the other hand, if the interest were compounded every six months, you would owe $50 in interest for the first six months (1/2 of the annual interest), but that amount would be added to the loan principal. Thus, you would pay $75 in interest for the second six months, bringing the total amount required to payback the loan to $225.

Bernoulli then tried quarterly compounding (four times per year), which will result in $244.14 owed on that $100 debt. With daily compounding (365 periods charging 1/365th of the rate every day), this payback rises to $271.46, or $2.7146 per dollar borrowed. But this number has a limit if the compounding period gets smaller and smaller, and that limit is $2.71828 (with an infinite number of decimal digits behind this). Note that this number is exactly the value of e.

This is not some magic, but rather it has to do with the mathematical nature of continuous growth of any kind, physical, biological or financial, which will be explored in Part Two of this post.

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Part Two of this series has now been posted.


Notes:

  1. The contention that our integer counting system (1..2..3, etc.) is a human invention is perhaps more controversial than is the base 10 decimal system. The issue here is a philosophical one as well as a physical one. Is “one of something, two of something” a fundamental property of the universe, or is it the result of humans making arbitrary aggregations into this thing versus that thing? For instance, consider “one day, two days,” etc. There is no single physical point differentiating one “Earth day” from another. The movement of Earth through the Cosmos and its rotation are continuous processes, and so our integer counting of days and years are human simplifications of a continuous process. I will leave it to you to parse through more difficult examples like “one person, two persons,” etc.

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