Part One of this post looked at the mysterious number called e, also called Euler’s number, a transcendental number (never resolving in our decimal numbering system) with a value of approximately 2.71828. That prior post showed examples of this value occurring repeatedly in nature and in the world of finance. This post gets a bit more technical to present perspectives from mathematics and physics. I am writing this for a largely non-math audience, so please excuse any too-far simplifications.

The mathematics definitions

There are multiple ways to derive the value of e mathematically, with each hinting as to “what is going on.” The one I’ll show here requires a basic understanding of calculus, which I wrote about in this recent post as the math which describes something that moves or changes continuously. This discipline was historically called “infinitesimal calculus” because central to it is the idea that you need to conceptually divide curves and rates of change into smaller and smaller amounts approaching the infinitely-small, the so-called “infinitesimal” (and subject of this earlier post), but never quite getting to zero.

The formula that Swiss mathematician Jacob Bernoulli discovered in 1683, as introduced in Part One, was the limit of the financial interest compounding formula, which is (1 + r/n)ⁿ  where r is the annual interest rate and n is the number of interest compounding periods annually. If r has the value of 1 (or 100% annually), then as the value of n approaches infinity, this computation approaches the value of e (2.71828…). What this calculation is showing us is that “growth,” whether in biological or financial terms, happens in very tiny infinitesimal increments, never quite reaching zero. “Something is happening” at this infinitesimal level, and that “something” reflects the value of e. [1]

e becomes important in calculus because working in this number base, rather than the more familiar base 10, simplifies the calculation of curves and the areas under curves (say, the parabolic arc of a ball thrown into the air). This is because e is the “natural base” of curvature itself if the curve is built on some constant rate change (or two in this case, the velocity from the initial throw counteracted by gravity, which is a “downward” acceleration rate if we are standing on the Earth).

The parabolic arc of a ball thrown in the air. Source: University of Regina

Graphing the simple function of y = e^x (e to the x power) on a Cartesian (x,y) grid yields a curve with interesting properties:

Source: Wyzant

Note that for any point along the x axis, the y value of this formula (e to the x power) is the also slope of the curve itself (the change in the vertical distance y divided by the change in the horizontal distance x), which is also called the tangent line. When x equals 1, for instance, both the y value (e¹) and the slope/tangent are 2.72… At the horizontal point x = 0, the slope is 1 (45 degrees), because e⁰ is equal to 1.

In calculus terms, this means that, e^x is its own derivative, the only number that behaves this way. This greatly simplifies the calculation of derivatives and integrals, essential to evaluating any curved line or area using calculus.

The physics definitions

Because calculus is the primary mathematical language of physics, the value of e continually pops up in mathematical descriptions of both the biggest things in the universe as well as the smallest things. On the cosmic scale, as an example, the “arms” of spiral galaxies such as our own Milky Way follow the form of the logarithmic spiral, which was noted in Part One as the same math behind the familiar spiral growth pattern of the chambered nautilus. And yet, each “arm” of the Milky Way galaxy, consisting of billions of stars, stretches out some 50,000 light years away from the center of the galaxy.

A Hubble telescope image of spiral galaxy UGC 12158, which is thought to resemble the Milky Way in appearance. Source: Wikipedia

On the smallest scale, the value of e is central to the computation of the behavior of subatomic particles. Erwin Schrödinger (1887–1961), who first defined many of the “wave vs. particle” formulations underlying quantum physics, determined that the value e itself likely comes from the behavior of these smallest things in the universe, curved “waves” of energy, the forces making up atoms.

Euler’s formula – tying it all together

Also called Euler’s identity and discovered by Leonhard Euler (1707–1783), Euler’s formula is a mysteriously-simple relationship between the values of e, 𝛑, and a something called i, the imaginary unit, which is the square root of -1. The relationship is this:

e^i𝛑 + 1 = 0

I won’t get into why this relationship is the way it is, frankly because it hurts my brain. All I can say is that, just as in my small-town home of many years in rural Iowa, “Everyone is related” in this universe of ours.

Notes:

1. Quarterly interest compounding, where n = 4 and r = 1, would thus be calculated as (1 + 1/4)⁴  times the principal value (approximately 2.4414), while daily compounding would be (1 + 1/365)³⁶⁵ (approximately 2.7146). If we take n up to 10,000 compounding periods per year (or about every 52 minutes), we calculate approximately 2.71815, getting ever closer, but never quite reaching, e‘s value of 2.7182818284…