It all depends on the use to which you put your numbers.
If your primary use is counting things, and adding quantities, then a system like the Romans used makes a lot more sense. It’s very easy to teach, it’s much more robust against small errors in writing, placement, or style, and both addition and subtraction are generally much easier with it. (In fact, if you look at the way math is taught in K and 1st grade, it is essentially through Roman numbers, only they use markers like pennies and dimes instead of I and X. Experience tells us that children learn the concepts of abstract number (number dissociated from an actual number of objects), addition and subtraction faster if they learn it using a system where position carries no information.
The only time positional (i.e. Arabic) numbers come into their own is when you need to do lots of multiplication and division. Both are much easier to do with a positional system. The tradeoff is that addition and subtraction become more complicated – you need all that weirdness with carrying and borrowing that has 2nd graders scratching their heads, and very reasonably so. So you only go to a positional system when the value of making multiplication and dvision easier is worth the cost in addition and subtraction. Say, when you’ve got a government doing big accounting, or astrologers doing fancy calculations, and when the system is driven less by the needs of small business or the household, which rarely need to multiply and divide, and for whom the ability to do addition and subtraction in their heads would be valuable indeed.
The question of base is similar. The base you choose has something to do with the ease of counting, which is what (and only what) points to base 10, what with the ten fingers business. A somewhat more sophisticated understanding will emphasize the bases that lend themselves to fractions, because working with fractions in your head is much easier than working with decimals. (All woodworkers know this: it’s much easier to divide a 5 foot board in 18 pieces with 1/4 inch allowance for the cut between each IN YOUR HEAD if you are using a base, e.g. 12 in this case, that allows for lots of even fractions: 5 x 12in = 60 in of board, 16 gaps x 1/4 in each = 4 in waste, leaving 56 inches for 18 pieces, or 56/18 = 3 and 2/18 = 3 and 1/9 in each. Try doing that in your head with decimals, e.g. starting with a 5 m board and wanting 18 pieces with 0.15 cm allowance for the cut.)
Often you have a balance between number of available fractions and the size of the number, i.e. whether you can work up enough fingers, toes, knuckles and symbols to readily count. The general winners are 12 (gives an integer for 1/2,1/3,1/4, and 1/6) and 16 (1/2,1/4,1/8), which accounts for our inches in a foot and ounces in a pound. Base 60 is certainly remarkable (1/2,1/3,1/4,1/5,1/6,1/10,1/12,1/15,1/20,1/30) but it’s not easy to count on fingers and toes, and devising 60 separate symbols would be nuts. Still, it’s useful enough to have gone into minutes per hour, which in turn led to degrees per standard watch arclength (4 hours of the Sun’s passage across the sky).
Still another often overlooked advantage of bases related to 2, e.g. base 16, is the ease of making good instruments in an era when they all had to be made by hand. For example, people frequently mock Daniel Fahrenheit’s temperature scale, as compared to Anders Celsius, but this is simply historical ignorance. Fahrenheit’s scale was ingenious because you could make a reliable thermometer with just your bare hands, a piece of string, some freezing water, a tube of glass, a liquid (alcohol works fine) and a scribe.
You thrust the tube into a bucket of ice and water, and scribe a mark on the tube. That’s 32. Now you put the tube under your arm until it heats to body temperature and make another mark. That’s 96. Now you need to make 64 equally spaced marks between 32 and 96, which is easy to do with your string and scribe, since you can just divide the distance in half again and again.
And as a special bonus, you can extend your marks downward from 32 to 0, and, amazingly enough, that will turn out to be essentially the lowest temperature seawater can reach and still be liquid, the temperature of an equilibrium mix of salt, ice and water. (Fahrenheit was a meteorologist on the coast of the Baltic.)
Compare that to Anders Celsius" scale: you put your tube into a bucket of ice and water and mark 0. Now you put it into some boiling water…er wait a minute, you need to pick the right place and day, since (as Fahrenheit knew well) the boiling point of water varies nontrivially with air pressure, and hence with altitude and weather. (The freezing point varies, too, but much less.) Hmm, OK, so you’ve gone to some canonical altitude and waited for what you hope is the same kind of weather as your scientific colleagues across the sea, and you mark 100 on your thermometer. Now you just need to divide that distance into 100 perfectly equal divisions. Which…hmmm…is going to be kind of difficulty, without some first-class carefully machined and standardized ruler, a very expensive widget.
Celsius’ scale only comes into its own when you start thinking about steam engines, and you want to readily remember that 100 is the magic number, not 212. So once again, if you are simply recording data and observing, the older system works better. Once you start doing sophisticated stuff, and anyway have ready access to fine tools and measuring devices, not to mention calculators, then the more modern system takes over.
