If you want to, you can read Part I: Prelude first.Mathematical bases sure are a convenient shorthand when adding, multiplying and subtracting, and their invention was necessary for the kind of recordkeeping that allowed ancient civilizations to blossom — amphoras of wine shipped, monies owed to the emperor, sacrifices made to the gods — let’s hear it for property rights and central planning.

But bases are arbitrary. The Babylonians used base 60. The Mayans used base 20. The Greeks and others used base 10, while the geeks have adopted binary notation, base 2, as their own.

All these number systems have quirks, too. Modern systems use place to determine the value of a number (i.e the 5 in 350 stands for 5 groups of 10); the Romans used different numeric symbols (XXX stood for three tens, CCC stood for three hundreds). Without prior access to such rules, a stranger would need a good number of examples from which to glean patterns. It wasn’t until the 1820s that we deciphered the Mayan number system, for example, and it took all the lateral thinking the splendidly named Constantine Samuel Rafinesque-Schmaltz could muster.

Number systems have another weakness: Division. They have no neat way of precisely representing all rational numbers — numbers that can be represented by a fraction, such as 6/2, 7683/99746 or 1/7. All bases can represent some rational numbers precisely: In base 10, for example, 6/2 is precisely 3 and 1/4 is precisely 0.25. But 1/7 is 0.142857142857… the 142857 repeating forever, the actual number never quite managing to get nailed. In base 7, meanwhile, the number equivalent to 1/7 in base 10 can be precisely represented as 0.1, but 1/4 becomes 0.151515… ad infinitum.

Base 2 merits some special consideration. The received wisdom is that binary notation is somehow more natural, because it is the simplest system, requiring the fewest symbols (2) without having to resort to pebbles or prison wall notches as a counting tool. Base 2 is special, it is argued or assumed, because its ones and zeros (“bits”) lend themself perfectly to representing the trues and falses of Boolean logic, which in turn can be physically embodied in the presence (“on”) or absence (“off”) of electromagnetic charges in our computers’ transistors and circuits.

That amounts to attaching too much importance to historical accident, however. Alternative logic systems do exist. In ternary logic, for example, base 3’s zeros, ones and twos map to “unknown”, “true” and “false”. And base 3 is more efficient than base 2 at representing numbers, as this great article points outHow to measure this efficiency? Take a number, any number. Depending on your choice of base, you will require different amounts of digits to write out the number. For example, the number 66 requires two digits in base 10; the base 2 equivalent of that number, 1000010, requires seven digits; the base 3 equivalent, 2110, requires four digits. Clearly, the higher the base, the fewer digits you need. Using a higher base means having to differentiate between more number symbols, which uses up bandwidth or computing resources — though it also means needing fewer digits, which saves resources. You can measure the total resources needed by multiplying the base a number is written in by its length in digits. In our case, 66 requires 10 x 2 = 20 “units” in base 10, 2 x 7 = 14 units in base 2, and 3 x 4 = 12 units in base 3. It turns out that for almost all numbers, using base 3 provides us with the notation that is the most economical in terms of these “units”. (The same argument applies to using ternary logic instead of Boolean logic. Often, fewer steps are needed to obtain the same result.). Computers using “trits” instead of bits were developed in the 1950s, notably by the Russians, but these efforts never caught on. According to the article, which you still haven’t read (it’s worth printing out the PDF and unplugging for this one), the likely reason why base 2 became the Microsoft of information theory notation is that in those days we didn’t have the technology to make transistors that could reliably represent three states. We could do two states, and that was good enough, so people like Claude Shannon ran with the idea and we never looked back. The cost of switching would now be too great. (Or would it? When it comes to data transmission, the Swedes are all over this.)

If aliens assume anything, why shouldn’t they assume that since base 3 is the most efficient notation, this must obviously be the base to use for interstellar communication? Since radio transmissions use phase shifts variable signal strength to encode “on” and “off” bits, it would be trivial to encode multiple states. Why didn’t we do so in our data transmissions? I’m guessing at more anthropocentrism, and because we forgot that sometimes simplicity is not the same as efficiencyWay off topic: When quantum computing happens, and it will, there is every reason to consider using the opportunity to swith to ternary logic, and using qutrits instead of qubits. This paper [PDF] and this paper make the case. This one [PDF] even has pretty pictures!.

But I want to hold off on the aliens for a while longer. I’m trying to make a case here for jetissoning bases in favor of a more rigorous approach to describing the numbers that lie between whole numbers. That’s because no number system is precise when it comes to representing arbitrary rational numbers, and downright hopeless when it comes to irrational numbers — and I think it is some of those numbers that we should consider beaming to aliens. (But I’m getting ahead of myself.)

The solution to the division problem is ingenious, and therefore first discovered by the Greeks. I’d come across continued fractions before, in high school, but their deeper significance completely passed me by until Roger Penrose, of course, rubbed my nose in them in his The Road to Reality (Chapter 3)You might remember my commitment to blog each chapter of that book. It’s not feasible, and frankly boring to impose my half-baked solutions to his problems on you. Perhaps a wiki is in order. Later. Instead, there’ll be posts like this one, inspired by the eye-opening (to me) stuff from the book..

It turns out that every number on the number line is representable by a unique continued fraction that looks like one of these:

It’s not as bad as it looks, so bear with me. You can make these continued fractions yourself: Use a calculator to divide 13 by 11. The answer is 1.181818… Subtract the bit to the left of the decimal point (it’s a 1) and put it aside. Divide the remainder into 1, and you get 5.5. Again subtract the bit to the left (the 5), then again divide the remainder into 1, and you get 2, this time with no remainder (if your calculator is any good). The numbers you took away from the running total, 1, 5 and 2, are the terms along the left edge in the continued fraction, above, which uniquely describes 13/11. This is the “basic” way of constructing a continued fraction — there are other ways [PDF].

There is only one continued fraction of this kindBy “kind” I mean continued fraction where all the numerators are 1. for each number because if you changed any of the denominators you would (obviously) get a different number. Another way of saying this is that every real number (whole, rational, irrational, transcendental) has a one-to-one correspondence to a specific sequence of whole numbers. In the case of the number 13/11, this sequence is short and finite: 1,5,2. In the case of the square root of 3, the sequence is 1,1,2,1,2,1,2… with the 1 and 2 alternating ad infinitum.

You might be wondering why this is any better than the repeating “181818…” we got in the decimal notation of 13/11. I have two reasons. First, the whole numbers that make up these sequences do not change depending on the base we use to write them. The sequence is base-neutral. Second, these sequences are more elegant: They are always finite when it comes to describing rational numbers exactly, and they are far better at divulging patterns in irrational numbers than any number system you care to use.

Here are some examples showing off the elegance of continued fractions when describing some irrational numbers — the square root of 41, and Phi, the Golden Mean.

In decimal notation, and in any other base, the numbers following the decimal point convey no order; they might as well be random. Not so when it comes to the representation of these numbers via continued fractions; for the square root of 41, the sequence is [6,2,2,12,2,2,12,2,2,12,2,2,12,…] ad infinitum. The pattern is clear.

What about that special class of “really” irrational numbers, the transcendental numbers, such as pi and eTranscendental numbers are irrational numbers that cannot be the solution to a polynomial equation. There are in fact far more transcendental numbers than any other kind of number — pi and e are just the most famous ones.?

For e, there is pattern visible, though it is one that changes in a set way with each repetition. As for pi, there is no discernible pattern via this kind of continued fraction, though much has been [Swedish PDF] made of the sequence of numbers that corresponds to it.

In sum, what we now have is a means of representing any real number in a base-independent manner using only sequences of whole numbers. This is good for communicating with aliens: No bases means no opportunity for anthropomorphising, while whole numbers are about as close to made-for-radio blips as you can get.

So, now that we can unambiguously send real numbers to aliens, which numbers shall we send them? Stay tuned for part III.