Strange Logarithmic Coincidences

I was planning to write a post about the Rule of 72 for computing the doubling time of an investment, but got sidetracked when I saw this post from John D. Cook‘s blog about logarithmic coincidences. Apparently, Donald Knuth, in The Art of Computer Programming, notes that the three most commonly-used logarithms share the following relationship:


So, the base-2 logarithm can be approximated by the base-e and base-10 logarithms. Here are the corresponding graphs:


The difference barely registers on this scale. Using a simple change-of-base formula, we see that


which means that the real source of the approximate values can be expressed with the natural logarithm only:


The relative error here is about 0.00582. Naturally (pun intended), I was curious: are there other trios of bases that give even better approximations? (Though since this is a genuine mathematical coincidence, maybe it’s better to call them near-equalities.) After messing around in Sage for a while, I found that the answer is clearly yes. Here are a few more logarithmic coincidences for you…


The first one isn’t really a coincidence at all, since (4, 8, 64) = (22, 23, 26) means that the equation reduces to a decomposition of unit fractions: 1/2 = 1/3 + 1/6. So that’s a for-real equality, not just a near-equality. The others appear genuine, though. I’ve discovered dozens of these so far, but the third one is the most accurate near-equality that I’ve found.

I’m not sure what all of this means. Maybe these logarithms closely approximate some certain fraction decompositions, but maybe they are just a collection of coincidences drawn from the set {ln(n) | n > 1} of transcendental numbers. Either way, I’ll keep you updated on any new discoveries.

The Evolution of Arithmetic

This post is the second in a series; if you haven’t read the first post, on the evolution of English counting words, I’d recommend reading that one first.

As promised, this post looks at the origins of the English words for arithmetic operations. Read on, friend!

  • Plus and Minus. These two are fairly straightforward—they’re the Latin words for “more” and “less”, respectively. The symbols, though, are less clear. It appears that the letters p and m were used (sometimes appearing as p and m) during the 1400s—Wikipedia claims that these first appeared in Luca Pacioli’s Summa de Arithmetica, though I’ve been unable to find a satisfactory example. In the 1500s, the modern + and – signs began to appear; Schwartzman attributes the + to an abbreviation of the Latin “et” (taking the t only) and the – to the bar from m.
  • Multiply. This word comes from the Latin multiplicare, meaning “to increase.” Breaking it down a little further, we have the prefix multi– (“many”) and the suffix -plex (“fold”) so that the compound word multiplex means “many folds.” (We still use “fold” language today—when we speak of a “threefold increase,” we mean that something had been multiplied by three.) The x symbol for multiplication is attributed to William Oughtred, while Schwartzman gives credit for the dot • to Gottfried Wilhelm Leibniz.
  • Divide. This word comes from Latin as well, with the origin being dividere, meaning “to separate.” (As a side note, the root videre means “to see” and gives us the modern word video, which means “I see”.) Putting di– and videre together, I suppose this means that division is literally “to see in two.”

Notice that all four of these words originate in a description of the operation itself. It turns out that exponents and roots are a little more metaphorical in their meaning:

  • Exponent. Once again, we have a Latin origin: the prefix ex– and the verb ponere, roughly meaning “to put out.” Unlike the four arithmetic operations, though, the original meaning is typographical—the exponent is the number that is “put out” above and to the right of the base. In part, it’s because the exponent is a relatively new development; Schwartzman attributes the notation to Descartes, specifically La Géométrie (1637).
  • Root. Finally, a non-Latin word! The word rot means “cause” or “origin”, which makes sense when you consider that since 8 = 23, its “origin” is 2. If you trace the word further back, the Proto-Indo-European root (see what I did there?) is wrad-. Thus, the Latin-based words radical and radish come from a source similar to root.

And there you have it! In the next installment, I’ll get a little more geometric and explore some words we’ve come to use for algebraic curves.

Extreme Musical Venues

No, I don’t mean this: But now that I’ve got your attention, I want to return to a topic I wrote about last month: the Earth’s gravity and its effect on mechanical metronomes (or, more abstractly, its effect on pendulums). That post boiled down this fact:

Pendulums near the poles will swing more quickly than pendulums of the same length near the equator.

Of course, that begs the question—how much more quickly? At the time of my previous post, I hadn’t researched the mathematics required to do this. But now I’ve crunched some numbers and have some results to share. Before that, though, let me set up some of the basic ideas behind the calculations.

1. The Earth isn’t a sphere. We know today that the Earth is an oblate spheroid, and is flatter at the poles than at the equator. The verification of this fact dates back to the 1730s, and is credited to Pierre Louis Maupertuis. It’s worth mentioning that the oblateness follows from Newton’s mechanics, of which Maupertuis was a notable advocate. [Maupertuis did many other things, and you can read about them in this book.]

2. Gravity varies according to latitude. This follows quickly from point (1) above, though the first attempt at a precise formula was made by Alexis Claude de Clairaut in his 1743 treatise, Théorie de la figure de la terre. W. W. R. Ball has a short summary of Clairaut’s work. Today, we use a more precise formula, as part of the World Geodetic System:

Screen Shot 2013-09-21 at 7.11.24 PM

Here, GE ≈ 9.78033 is gravity at the equator, e ≈ 0.0818192 is the Earth’s eccentricity (which measures the “sphereness” of the ellipsoid, ranging from  0 to 1), and k ≈ 0.00193185 is a parameter that depends on gravity at the poles. (Details can be found here on page 4-2). 

3. Pendulum formula requires calculus. I’m going to skip the derivation, and point you to Wikipedia on this one (grain of salt and all that). Here’s the formula for one “swoop” of a pendulum:

Screen Shot 2013-09-21 at 7.25.12 PM

with l being the length of the pendulum and θ0 being its initial angle.

4. Gravity decreases with altitude. The experts use something called the free-air correction to compensate for this. For example, Reynolds’ An Introduction to Applied and Environmental Geophysics states that the proper way to adjust the formula in part (2) above is by subtracting (3.086×10-7)h, where h is your elevation above sea level (I’ve converted the units from Reynolds’ version).

Now we’re ready for some results! For all calculations, I took θ0 = π/4, and l ≈ 0.229060. This way, a pendulum at sea level at the equator will beat exactly 120 times per minute. Then, I picked five extreme points on the surface of the Earth:  the summits of Chimborazo and Everest, and the Antarctic stations McMurdo, Russkaya, and Amundsen-Scott. And here’s what we have:

Location Elevation Latitude Beats per Minute
Reference Point 0 m. 120
Chimborazo 6268 m. 1.469° S 119.881
Everest 8848 m. 27.988° N 119.902
Russkaya 0 m. 74.766° S 120.296
McMurdo 0 m. 77.850° S 120.304
Amundsen-Scott 2835 m. 90° S 120.264

I’ve ignored the fact that the mass of the terrain that you’re standing on will affect the gravity at its summit (the Bouguer anomaly)—so the values for Everest and Chimborazo are probably a bit off. But, taking the numbers at their word, our hypothetical metronome will move the most quickly at McMurdo Station—which has the double-benefit of being both close to the pole (and thus closer to the center of the Earth) and having a low elevation. (In fact, McMurdo has the world’s southernmost port; you can go see what they’re up to right now.)

To conclude, it appears that a piece of music set to 120 beats per minute, that takes 5 minutes to perform, will finish almost exactly 1 beat sooner at McMurdo station than it would at the summit of Mount Everest. Not enough to mess anything up musically, but enough to notice. Extreme, right? (Not extreme enough for you? Then go read this comic.)