Every once in a while, an idea comes along that is so perfect that it can only be held onto by the biggest ivory tower eggheads in the world. Occasionally, an economist, game theorist, or mathematician will reveal to his other cloistered colleagues an idea that actually holds some practical benefit for the world at large, but mostly these ideas are meant to entertain people who want to think that it matters to speculate on the implications of time travel or cold fusion or the like. One such idea is base 12, which is simultaneously one of the most sensible and most ridiculous ideas in the world.
Last summer I was interviewed for a book on people's quirks, And I Thought I Was Crazy!. The author, Judy Reiser, was trying to get at why one of mine had to do with the number 12. (To find out what the quirk is, you'll have to buy the book.) I told her that 12 has always been my favorite number and that a lot of it had to do with the special place that 12 has in our world. Twelve describes an awful lot of things considering that we live in a base 10 world, which I've never understood but always thought was noteworthy. I thought this was pretty idiosyncratic, but I recently discovered that I have a few kindred spirits.
When I say we live in a base 10 world, I simply mean that we use 10 digits to count and perform arithmetic. The Romans based their number system on 5s and 10s, and Arabic numerals (which we use today) have standardized base 10. Thanks to this system, we can multiply by 10 just by adding a zero, which is much easier than resorting to Roman numerals.
Though we live in a base 10 world, it seems as though we really want to be in a base 12 world. Whenever possible, we express important measurements in terms of 12 rather than 10. We use 12 inches in a foot, 12 months in a year, 12 hours on a clock, and we group items for sale in dozens more often than we do tens. We do this because thinking in terms of twelve is natural, while thinking in terms of ten results from our awkward attempt to make something from the fact that we have five fingers on each hand.
In 1934, F. Emerson Andrews proposed simplifying our numbers by moving to base 12. Base 12 conforms to more of our units of measure, plus it is easier to multiply and divide in. Because 2, 3, 4, and 6 divide evenly into 12 (as opposed to only 2 and 5 for base 10), multiplication tables are much simpler, and fractions like 1/3 and 1/4 could be expressed simply as 0.4 and 0.3, respectively. Andrews suggested that we take what we now know as the numeral 12 and begin writing it as “10,” while inventing two new numerals (he called them dek and el) to replace the old 10 and 11. (For more information on Andrews and his ideas, please visit the Dozenal Society of America.) This little change has innumerable effects on the way we view the world, which Andrews detailed in his book New Numbers: How Acceptance of a Duo-Decimal System Would Simplify Mathematics.
Americans have often been ridiculed for our stubborn refusal to accept the metric system, but it’s possible that we are holding on to a superior form of measurement that is waiting only for a superior form of counting to come along. Most nations have no trouble reconciling their counting and their measuring because they have adopted the metric system, which means that they have no problem saying that a football player is 185 centimeters (1.85 meters) tall. In the United States, 72 inches does not come close to meaning 7.2 feet because it is based on twelves. If, however, we went to base 12, it would instantly become much easier to make these kinds of comparisons.
Basic arithmetic would become amazingly simple, too. I won't reproduce a multiplication table here, but suffice it to say that if you saw it you would wish we were using it. Because 2, 3, 4, and 6 all divide evenly into 12, they would all essentially be as easy as multiplying by 2 or 5 in base 10.
This is an excellent exercise in new math, but what does it all matter? Could we ever actually make the change from 10 to 12? I enjoy reading about how much better base 12 is, but it bothers me that I have yet to find anyone who has set out to determine whether we could get there.
Transitioning to base 12 may in fact be the most challenging net present value problem ever devised. The costs are unknown, the benefits are difficult (if not impossible) to quantify, the time frame is unknown, and the discount rate--well, good luck.
The costs would occur mostly in the first few years. With two new digits, you would have to remake everything that displays arabic numerals (clocks, calendars, phones, calculators, computer keyboards, etc.). Computers would need to be reprogrammed. Everyone would have to be retaught, either through a sink-or-swim instantaneous transition or by learning to use base 10 and base 12 side-by-side for a few years, as some have advocated. Either way, the training costs would be phenomenal. Money would have to change, too. Given all the changes we would need to make (and their ripple effects) it wouldn't be difficult to imagine the costs being close to a year's worth of global GDP.
What of books that contain base 10? Would they be reprinted? What of history? Did Columbus sail the ocean blue in X44 (pronounced dek four four, or ten gross four dozen and four)? Was the Declaration of Independence signed in 1040? Do we change FDR's proclamation to read, "Yesterday, December 7, 1159, a date which will live in infamy..."? Will we all remember the tragic events of 9E (nine el), 11X9? These are ridiculous questions, but they would have to be dealt with if we were to attempt a transition.
How about the benefits of duodecimal? Forget that it would be easier to count in a duodecimal system--what does that mean? Well, if you and two of your friends went in together on something costing $100, you could pay for it with $40 each, rather than having to scrounge for 33 dollars and 33.333333333 cents. Area codes could contain 20% more phone numbers, which is important in this age of cell phones. And as far as all those little calculations that would become so much easier in base 12, for the sake of argument we can represent the value of that as $0.0001 (the incremental value of each measurement or calculation compared to base 10) times 50 (the number of calculations per day) times 365 (the number of days in a year) times 6.5 billion (the population of the earth), carried out every year until the end of time.
There are other benefits. We would probably be a much more musical society. Scales and chords can be tough to learn, and there are many people who have started to learn piano, or some other instrument, only to give up. But since an octave has twelve semitones, learning music in a base-12 world would be much more intuitive, and more people would be likely to stick with it. The fact that there are untold Lennons and McCartneys out there waiting for a number base that accomodates their lack of patience with music instruction has some elusive dollar value associated with it as well.
If you did a back-of-the-envelope calculation using these costs and benefits (even asssuming the most rosy scenarios), I can't see any way you would end up saying that base 12 makes economic sense. Which is why all the mathemeticians out there, when describing their base 12 utopia, should probably keep saying, "If man had been born with 6 fingers and 6 toes..." rather than "If we were to make the switch to base 12..."
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