indessed roscivs

Yesterday I talked about the importance of the number twelve, its prevalence in measurement systems, and how the metric system was created in order to align our weights and measures with our ten-based counting system.

But why is our counting system ten-based? Many people assume it has to be that way, or that it’s more logical that way, but it turns out that it’s just an accident based on our number of phalanges. Ten-based systems aren’t inherently any more convenient than any other base—only when you’re dealing with tens. Let me illustrate.

Hundreds of years ago, positional notation was invented in India. Previously, all counting had been done in systems like Roman Numerals, where a particular symbol always stood for a particular number, like X or V. Multiplication and division was a real chore in systems like these (try it for yourself!), so positional notation was a welcome change. The novel idea was that a particular symbol could stand for many different numbers depending on where it stood in the number—its position. For example, the symbol “1″ in 123 represents a hundred, whereas the same symbol in 312 stands for ten. (The invention of the zero was vital in this scheme, so that you could represent the position of “1″ in 100 and 10 without the presence of any other numbers.)

Although using “100″ to represent a hundred seems obvious to us, having grown up with the decimal system, it’s not the only way of doing things. When learning simple arithmetic as a child, we were told that the one is in the “hundreds-place” or the “tens-place”—each column represents some value. A 3 in the “hundreds-place” column represents three of that value, or three “hundreds”. But the value of each column is arbitrary. What if to the left of the “ones-place” you had a “threes-place” and then a “nines-place”? Then “100″ would mean nine, “10″ would mean three, “200″ would mean eighteen, and “20″ would mean six! Note that in this system, we only need three digits—0, 1, and 2—to represent any value; counting looks like this: 1, 2, 10 (three), 11 (one in the threes-place, one in the ones-place = 4), 12, 20, and so on.

This is called “base three”, because we only need three digits, and each column is a power of three (threes-place, nines-place, and next would be the twenty-sevens-place). And an interesting property of any base is that, when you multiply any number written in that base times the base itself, you can simply shift digits to the left. For example, if we multiply base-three “20″ (meaning six) times three, we get eighteen—which in base three is written “200″. This happens in every base, which is why in our decimal counting system, multiplying “20″ (meaning twenty) times ten you get “200″.

You could even have “base twelve”, where you have a twelves-place and a gross-place, so multiplying “20″ (meaning twenty-four) by twelve would equal “200″ (two gross). Then you could have all the benefits of the number twelve (including its divisibility by two, three, four, and six) as well as all the benefits of the metric system (easy multiplication and division when dealing with a standard factor, which in this case would be twelve).

In fact, some argue that the French made the wrong decision when deciding to switch our useful-twelve-based measurement systems to align with our arbitrarily-ten-based counting system, and that they should have gone the other way instead. You can find their writings at the Dozenal Society. I tend to agree with them in theory, but the fact of the matter is that it was difficult enough getting everyone (well, almost everyone) to switch something as easily changed as units of measurement. Attempting to change the base of our entire counting system would be madness.

But it certainly is interesting to think about.

This entry was posted on Thursday, March 22nd, 2007 at 7:10 am and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.