Worldbuilding Number Systems

One of the most fun aspects of worldbuilding is constructing new aspects of every society, to make the world new again, as I have enjoyed recently in my first and second posts on worldbuilding my calendar. One aspect of worldbuilding that is often neglected is number systems, perhaps understandably so given the complexity involved.

Beyond Base-10

Today the standard number system is base-10, but even today it’s not all-dominant; for everyday timekeeping we use base-60 or base-12, and for computing we famously use “binary”, which is another name for base-2. Many other bases have been used by various cultures and for various applications. An obvious what-if that adds some exotic flavor to any world is what if the culture you’re constructing used a different system from base 10 as its standard?

Base-10 is perhaps the most obvious system for human beings, with our 10 fingers, to use, but 10 is not a particularly convenient number. Other than itself and 1, 10 only has two factors: 2 and 5. The nearby number 12, by contrast, has four such factors: 2, 3, 4, and 6. The historical prevalence of dozens and twelfths as measurements (e.g. 12 inches in a foot) is because 12 is more convenient to divide than 10 is (this is also a key obstacle to decimal months; 12 divides neatly into 4 quarters or seasons with whole numbers of months, whereas 10 does not), which has prompted advocacy for switching from the decimal (base-10) to the duodecimal (base-12) system.

Duodecimal Numbers

A dozen dozen, 12 to the power of 2, 144, is already known as a gross, and a “great gross” is the base-12 equivalent to a thousand, 12 to the power of 3, amounting to 1,728. Higher numbers can be named according to the worldbuilder’s imagination, such as “grosand” (by analogy with “thousand”) for the next level, “overgrosand” for the next highest, “twovergrosand”, “throvergrosand”, etc.

As for how to represent these numbers, new symbols for 11 and 12 might be desirable, or even a whole new set of numerals, which usually are developed over the course of constructing a language for worldbuilding anyway. Making up two more numerals isn’t so hard, but adds a lot of exotic flavor to the setting. Alternatively 11 and 12 might just be used as-is, perhaps joined together as one character to emphasize their unity or, as in base-60 clock times, separated by colons or whatever.

Sexagesimal Numbers

Curiously, the Babylonians used a base-60, or sexagesimal, system that essentially did this, with their sub-base being 10 like ours. Minutes and seconds still use base-60, as do angles, and special symbols are attached to these figures to communicate that they are in base-60 rather than base-10. This isn’t terribly different from the approach they used!

Base-60 is rather interesting, as it’s the highest base of all the historically prominent number systems. 60 is a convenient number to select as a base. Aside from itself and 1, 60 has 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 as factors, 10 factors! Makes even base-12’s 4 look like peanuts. The only major disadvantage of base-60 is that it’s rather unwieldy, requiring 60 distinct numerals, unless you kinda cheat like the Babylonians did. Still, logographic scripts like Chinese contain thousands of characters and aren’t that hard for people to learn, so 60 numerals seem to me to be at least viable, even if it’s not exactly the most likely number system to develop.

Its mathematical properties make base-60 attractive even to humans, so imagine how obvious it would be for a species that had 60 digits instead of our 20; a squid-like alien that had the ten limbs our squid have but with 6 digits per limb to aid in finer manipulation would find finger-counting by 60 so easy and convenient for mathematical calculations it would be a bit odd if they didn’t adopt base-60 as their standard.

Large Bases for large Numbers?

Large bases like base-60 have another key advantage; rendering larger numbers with a much smaller number of digits than base-10. 60 to the second power, its version of the hundred, is 3600; instead of decimal’s four digits only three digits would be needed: 1, 0, and 0. For example, 3659 would be represented as the numeral for 59 plus another numeral for 59, 59 sixties plus 59, analogous to how 99 in decimal is 9 tens plus 9.

The thousand in base-60 is 60 to the third power, or 216,000 in our system; numbers up to 215,999 would be represented by only four digits, whereas they would need six in our system! The difference of course compounds as you move higher up the number scale.

With it being more convenient to handle large numbers and more easily divided into fractions, I wouldn’t be shocked if sexagesimal made a comeback at some point in the distant future. Duodecimal as well is another strong candidate for an alternative base.

Even larger Bases

With 60 being the least common multiple of 1, 2, 3, 4, 5, and 6, what about subsequent numbers? If we want to add 7 to the list we need to use base-420. Adding 8 means base-840, 9 and 10 means base-2520. 2520, being the least common multiple of all the finger-counting numbers, 1 through 10, might be of interest to people in the future, but imagine having to keep 2520 numerals straight. Yikes.

Though I guess one could cheat Babylonian-style, 2520 is really far out there as far as bases go. But large numbers would be a breeze. The hundred in this system would be 6,350,400, represented by 1, 0, 0; the next lowest number, 6,350,399, would be represented by the numeral for 2519 plus another numeral for 2519, 2519 groups of 2520, plus 2519, once again analogous to how 99 in our system is 9 tens plus 9.

The thousand in base-2520 would be 16,003,008,000. The million would be 40,327,580,160,000, just over 40 trillion in our system! Even numbers as big as the United States federal debt or the number of stars in the observable universe would still be in the thousands; it’s amazing people aren’t going for it already!

I joke, but as an aside it’s interesting to note that from the Babylonian era to the age of computers the most cutting-edge system seems to have actually contracted from base-60 to base-2 (binary), suggestive of another possibility: that we’ll all make like the Bynars from “Star Trek: The Next Generation” and use base-2 for everything in the future. Who knows?

Number Systems in my far-future Space Opera

Personally in my space opera over a thousand years from now I envision the human mainstream still using base-10; indeed, my worldbuilding is suggestive of base-10 actually increasing its dominance, as the standard calendar by then divides the day into hundredths, hundredths, and then hundredths again instead of twenty-fourths, sixtieths, and sixtieths again.

Though this is more an expansion of the metric principle rather than base-10 as such. Our metric system famously uses powers of ten, but this is no doubt inspired by our number system being base-10; in a culture that used e.g. base-12 the metric system would use powers of 12 instead of 10.

In my far-future space opera I envision base-10 remaining the standard among the human mainstream; by now it’s deeply entrenched, making change difficult and costly, and even if it wasn’t it’s arguably the most natural system for beings with 10 fingers anyway. But that doesn’t mean everyone will be using it; there’s no shortage of isolated colonies and cultures in my universe that seek to begin the world anew, and they might easily adopt new bases, most likely base-12 or base-60, and might adapt the metric system to better suit their own number systems.

As for the metric system itself, I envision the metric principle becoming even more entrenched and pervasive with time for everyday measurements; it just makes too much sense to resist. That doesn’t mean the base units will be the same, though.

Light-Time Distances: the Meter of Spacefarers?

Already for long distances, for example, parsecs and light-years predominate, so working backwards from the light-year (courtesy of the ever-helpful Wolfram Alpha), I find that dividing a light-year into 10 quadrillion pieces (100 atto-light-years) yields a nice convenient distance of 0.9461 meters, close to the present-day meter, and even closer to the imperial yard!

Working backwards from the light-day, dividing it into 100 trillion pieces (10 femto-light-days) yields a unit equal to 0.259 meters (10 inches; close to the imperial foot!); 10 trillion divisions (100 femto-light-days) yields 2.59 meters. Helpfully, my cosmic standard calendar uses metric time; the smallest unit, the trice, is one millionth of a day, making the 0.259-meter unit equal to a hundred-millionth of a light-trice. In metric this could be expressed as 100 light-nanotrices, the 2.59-meter unit as 1 light-microtrice. Cool, eh?

Such units might be more natural for a spacefaring civilization, and it would conveniently enable metric conversion (just move decimal places!) between shorter distances and light-days or light-years, depending on which is adopted as the baseline. Every civilization gravitates to the units that make the most sense to them; future civilizations will be no different.

Conclusion

Measurement systems in worldbuilding could take up another post all by itself, so I’ll stop there; the point is that using alternative bases for the number system can add a lot of flavor to one’s world; base-10 is (understandably) very common, so using a different base is a way to instantly set your setting apart from the crowd, either the whole world or, as I intend to do, different parts of a wider world.

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