Random? Yes. But this is a blog, not an accumulation of scholarly and well-ordered writings. Each individual post is, I think, well-contained enough, but one can’t expect uniformity and order among posts on a personal blog (though one might, and should, expect it from a blog devoted to a certain topic; some excellent such blogs are listed in the links section on the sidebar). So, I’ve moved to another exciting and unbelievably obscure topic: non-decimal mathematics.
See, prior to placeholder notation (what we use to write numbers now; if you don’t get this, don’t worry, I’ll be getting there shortly), most civilizations wrote their numbers in an almost insufferably clumsy way. Take Roman numerals, for instance, which have been rightly relegated only to the most formal locations in our day (and sadly sometimes, as on tombstones and book publication dates and the like, not even there). The Romans had no concept of placeholder notation; as a result, they simply had a different symbol for each number they considered important enough to deserve one. Therefore, there was “I” for one, “V” for five, “X” for ten, “L” for fifty, “C” for a hundred, “M” for a thousand, and sometimes “D” for five hundred, as well. They shuffled these symbols as appropriate to make up more complex numbers. Originally, they would simply accumulate lower numbers until they reached a higher one, like so:
I, II, III, IIII, V, VI, VII, VIII, VIIII, X…
Later, however, they came up with a clever way of keeping their numbers shorter: if a smaller number was placed to the left of a larger one, that meant “subtract the smaller from the larger.” Thus, “IV” replaced “IIII,” “IX” replaced “VIIII,” and so on. This made things shorter, but not much easier; indeed, one has to examine a Roman numeral carefully to ensure proper parsing.
Imagine arithmetic with a number system such as this. Take simple addition, for example: one hundred and forty-seven plus two hundred and twelve. First, write them out as Roman numerals:
Now, just jumble all the symbols together, organizing them in kind. But be careful to keep, say, the “X” that is being subtracted from something away from the “X” that’s being added to something, lest you confuse them and ruin your result.
Now, reduce it to some sensible (sensible for Roman numerals, anyway) form:
Three hundred and fifty-nine. Now, imagine something painfully simple in our current numeric notation: multiplication by a single-digit multiplier; say, three. One hundred and forty-seven multiplied by three. This can’t be hard, can it? Well, first let’s write our numbers. It should be easy, as the large one has already been written above:
To do this, we must take CXLVII, write it three times, then reduce it in the same way that we did before after addition.
CXLVII + CXLVII + CXLVII –> CCCXXXLLLVVVIIIIII
Now, we have to reduce that monstrous number to something reasonably sensible. Remember that our “XLs” are equal to “LLL – XXX,” yielding a new total of “CXX,” and our “Vs” are equal to “XV,” and our “IIIIII” is equal to “VI.”
CCCCXXVVVIIIIII –> CDXXXVVI –> CDXXXXI –> CDXLI
So, there you have it. Four hundred and forty-one. Is this doable? Certainly. Now imagine doing it with one hundred and forty-seven and, say, seven hundred and twenty-two. Extremely cumbersome. Not only that, but the location of the digits within their numbers gives us no assistance in calculating the result. Since the numerical symbols have to be arranged together somehow, it would be nice if that arrangement itself could help us in our calculations.
Enter placeholder notation. That funny little digit that isn’t even really a number, “0,” saves the day. Zero is nothing more than a placeholder, a filler; it means nothing by itself. But with it, placeholder notation was made possible, and mathematical calculations became, while not easy, at least much easier.
In placeholder notation, the location of an individual digit is vital in determining what the number as a whole is. Let’s use a “base-ten,” or decimal, base to begin with, since we’re all familiar with it. Because we are using a base of ten, we have ten different symbols: one through nine, plus zero. So, while we are counting our units, it’s easy to go from one to ten:
1, 2, 3, 4, 5, 6, 7, 8, 9…
But wait! What to do? We’re out of symbols, it seems. How will we represent in writing the next number, which is spelled “ten?” The answer, of course, is easy, because we’re not out of symbols. We have one more: “0.” We’re using a base of ten. So, after we’ve counted to nine, in order to move on to the next unit we just move to another place. We write the number “10,” in which the location of the one and the zero make all the different (if we wrote “01,” for example, the digits would have a clearly different meaning). In the number written “10,” in a decimal base, the zero means “zero units of one,” while the one means “one unit of ten.” Thus, we are easily able to keep counting even to ninety-nine, using numbers like “74,” which, in a decimal base, means “seven units of ten and four units of one.”
When we get to ninety-nine, we might seem to have a problem again. However, this answer is once again easy: use the zero. We write the next number after ninety-nine “100,” which once again uses placeholder notation to make its meaning immediately clear. The one means “one unit of ten tens,” or in more colloquial language, “one unit of one hundred.” The first zero means “zero units of ten,” and the second means, as before, “zero units of one.” So we can now count all the way up to nine hundred and ninety-nine, using numbers like “487,” which means “four units of one hundred, eight units of ten, and seven units of one.” After 999, we just add another digit, which will mean “one unit of ten hundreds,” or “one unit of one thousand.” And so on, as long as we are blessed enough to need to continue counting.
This seems so simple to us in the modern day that even discussing it is absurd. It is taught in our schools as a given, never really explained in its workings because no one can imagine any other way of writing figures.
However, this system became such second nature that, when the Revolution came, for some reason the number ten was invested with a mystical importance. Because it happened to be the base of the number system that had arisen in India and the Arab world, the Revolutionaries decided that it was the high holy number which would brook no opposition. A seven day week? Absurd! We need a ten day week. While we’re at it, we need a ten hour day, too, and a ten month year. All our measurements should be based on ten, as well. We’ll call it the “metric system,” as though no other system of measuring can be called “metric” (which, after all, just means “measuring”). The great number-god, ten, will be the victor. One number to rule them all, one number to find them; one number to bring them all, and in the darkness bind them.
Indeed, the worship of the number-god has become so powerful that almost no one realizes that it’s really not very remarkable at all. The great convenience of placeholder notation, for example: metricists like to praise the metric system because it’s so easy to convert units. 718 kilometers; how many meters? Just move the decimal point to the right; 7180 meters. Easy. Much easier than multiplying by 5,280 to get feet from miles.
And it surely is easier, should you find yourself needing to know how many feet there are between you and Hoboken. However, why does it have to be ten? These conversions would work with any base system, as long as your measuring system had the same base. Indeed, another base might be considerably more convenient than ten. Suppose you wanted to measure out precisely one third of a meter of wood and saw it off of a larger piece. This will be rather difficult, as one third of a meter is 33.333333333333333333333, repeating to infinity, centimeters. The best you can do is eyeball a third, say 333 millimeters, and call it a day. But if there were a base in which three divided evenly, you could get exactly a third, and need no inaccurate estimation. With ten, of course, one can get even tenths and even fifths, but not even fourths, thirds, or sixths. Given the frequency with which most people need to deal in thirds and quarters, and the comparative rarity of dealing with fifths, this would seem a major defect of the decimal system.
Let us assume, then, that we were intent upon inventing a number system for our society. We would want a number that was not too large; we don’t want to have to have too many independent symbols (one needs the same number of symbols as one’s base divides into one; so, for example, we needed ten different symbols in base-ten). We would also not want a base that is too small; otherwise our numbers would quickly become unwieldy. Take the number 10000, for example; in base-two, or “binary,” that number equals only 16 in decimal. To get ten thousand in binary, one would require no less than fourteen digits. Clearly, we want to avoid having to juggle so many digits for such small numbers.
We would also want a number with as many even fractions as possible. Seven or eleven, for example, which divide evenly only by themselves and by one (these are called “prime” numbers), while being of convenient size, are inconvenient because of their lack of whole factors (“whole factors” is mathematics talk for “even fractions”).
Furthermore, which factors the base has will also be seen as quite important. First off, the daily and even academic activities of mankind demand certain fractions more often than others. While man often requires fractions such as a third and a quarter, one rarely requires an eleventh or a thirteenth. It will thus be seen as quite important to have commonly used fractions divide evenly into the base; that is, to be whole fractions. These commonly used fractions include a half, a third, and a quarter. If possible, it would also be helpful to have whole fractions representing halves of each of these fractions, which are also frequently used. Half of a half is a quarter, which we are already seeking; half of a third is a sixth, and half of a quarter is an eighth. A number with all of these fractions is unlikely to be of a conveniently small size; however, we should seek a base which contains as may of them as possible, and it would be good if, even if not whole, the fractions of the ones which our base does not contain are at least manageable; that is, are not repeating or irrational. An example of such an inconvenient number would be a third in base-ten; yielding, as it does, a placeholder representation of 0.3 repeating for a third, base-ten is inconvenient for this common fraction.
These fractions (factors of our base number) are important for another reason: the importance of the numbers themselves to mathematics and geometry. Two, of course, is universally divisible by all even numbers, and it is also the only even number which is also prime (that is, divisible only by one and itself). Also, geometrically the number two is extremely important; it is the first number which takes us beyond mere points and into having dimension. Three, as the first odd non-prime number, is also very important. Geometrically, three is the first number which yields two dimensions; three points makes the simplest polygon, the triangle, which is also vitally important for everything from simple surveying to trigonometry. The triangle is so necessary for nearly every craft, from carpentry to engineering to architecture to any number of other trades, that its importance as an even factor should not be underestimated. Four is likewise extremely important as the smallest non-prime number. Geometrically, four is the minimum number of points necessary for constructing a three-dimensional shape, the tetrahedron. Interestingly enough, the triangle and the square, regular polygons having three and four sides, tessellate together in two dimensions, further demonstrating their importance to geometry.[1]
So let’s test it, shall we? Assuming that anything lower than six is too small, and anything higher than sixteen is too high, let’s just go through the available fractions, excluding one and themselves (which all numbers have as factors), for each proposed base. We’ll put the numbers in base-ten, since we’re all used to that, for now. Remember that we’re specifically looking for two, three, and four at the very least. Additional helpful factors would be six and eight. Others, while advantageous, cannot be compared in importance or usefulness to these.
| Proposed Base |
Factors |
Number of Factors |
| 6 |
2, 3 |
2 |
| 7 |
|
0 |
| 8 |
2, 4 |
2 |
| 9 |
3 |
1 |
| 10 |
2, 5 |
2 |
| 11 |
|
0 |
| 12 |
2, 3, 4, 6 |
4 |
| 13 |
|
0 |
| 14 |
2, 7 |
2 |
| 15 |
3, 5 |
2 |
| 16 |
2, 4, 8 |
3 |
And then, of course, we get to 17, which again is prime, meaning that it has no factors other than one and itself.
One of these numbers (16) has three factors. Several (6, 8, 10, 14, and 15) have two factors. Only one, however, has more than three: twelve. Not only that, but the factors of twelve (2, 3, 4, and 6) are extremely commonly desired fractions; we often divide things into halves, thirds, and quarters (which are, after all, what you get when you divide by 2, 3, and 4); furthermore, since a sixth is half of a third, we not infrequently find ourselves in need of sixths, as well. All these convenient factors are present with a base of twelve. While twelve does lack five as a whole factor, fifths are much less often needed than those factors which twelve does have; furthermore, no other conveniently sized base which does have five as a factor also has three and four, which are certainly more important and more commonly needed.
So here we are. We have found a base which has four whole factors, more than any other number we examined (and, in fact, more than any other number below eighteen, which has two, three, six, and nine, but is too large to be convenient and also lacks the important factor four). This base, twelve, not only has more factors than any other number below eighteen, but it also contains the three most important factors we discussed, two, three, and four. Furthermore, it contains six, as well, which is half a third and thus very convenient. While it lacks five, which is occasionally convenient to have, this disadvantage is more than made up by the factors that it does have. Furthermore, it is only two higher than the base which we are already used to, and thus requires only two additional symbols and is thus very conveniently adopted.
In comparison, ten seems a weak and sad base. While it benefits from being the same as the number of fingers on the human hand, it suffers from a lack of useful factors, having only two and five, leaving out two of the most important and most frequently used factors, three and four. Writing a fourth in base-ten requires two numbers, 0.25, and writing a third accurately is impossible, as it requires an infinite number of threes. Writing a sixth, half a third, is even worse; it requires a 0.16, with the six repeating to infinity. In base twelve (“duodecimal” or “dozenal”), however, these fractions are easy, requiring only a single, even number. Furthermore, while man may have ten fingers, he has twelve segments between his knuckles on a single hand, which can be conveniently counted upon with the thumb.
Let’s examine how we’re going to write numbers as we continue our investigation into using twelve as a base. First, of course, we’ll require twelve symbols. We may as well continue using the ten that we use now; however, we will require two more, one for ten and one for eleven. Computers, when writing in hexadecimal, use A-F for the numbers ten to fifteen; we, however, are free to use whatever we want. In accordance with American practice, then, for now we’ll use “X” for ten, since it’s the Roman numeral for ten, and “E” for eleven, since eleven starts with “e.” (It’s important to note that frequently one will also see “A” and “B,” or even “*” and “#.”) We’ll pronounce them “ten” and “elv.” So now, our numbers look like this:
1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 11, 12, . . .
That would be counted, “One, two, three, four, five, six, seven, eight, nine, ten, elv, twelve.” There are a number of different proposals for how we ought to say these new numbers; for now, however, we’ll stick with the normal English “twelve,” “dozen,” and “gross” when they are needed.
Remember, when looking at a number, that you’re looking at a different type of number than you’re accustomed to. When you see the number which we write “10,” for example, you will probably read it out “ten.” This is only natural; that’s what you’ve been doing for your whole life. However, we’re using base-twelve now, not base-ten, and a one followed by a zero no longer indicates the number between nine and eleven. It’s important to remember that, when you’re using base-twelve, “10″ means “twelve,” “a dozen,” or whatever you want to call it. The number that’s said “ten” is represented by “X,” not by “10.”
It seems esoteric, but it’s really just as simple as the numbers you’ve been reading your entire life. It’s placeholder notation, pure and simple, just as we discussed long ago toward the beginning of this article. The only difference is that each place represents a multiple of twelve, rather than a multiple of ten. Take, for example, the number “1492.” In base-ten, that number represents one unit of a thousand, four units of a hundred, nine units of ten, and two units of one; added together, it equals one thousand, four hundred, and ninety-two. In base-twelve, however, that number represents one unit of a great-gross (that is, twelve gross, represented in base-ten by 1728), four units of a gross, nine units of a dozen, and two units of one. Thus, in base-twelve, “1492″ equals 2414 in base-ten. To get what “1492″ means in base-ten, one must write a different number: “X44,” which in base-twelve means “ten gross, four dozen, and four.” Thus, “1492″ in base-ten and “X44″ in base-twelve are equal.
Another example may help. To start with, however, let us form a convention to determine whether a number is written in base-ten or base-twelve. Generally, fractionals are indicated in base-ten by a “decimal point,” which looks exactly like a period, “.”. In base-twelve, the “decimal point” (which would be better referred to as a “dozenal point”) is generally a raised dot, unlike a period; however, in ASCII text such as what you are currently reading, a raised dot is difficult, so a semicolon, “;”, is used. All base-ten numbers, therefore, will end with a decimal point, such as this: “1492.”. All base-twelve numbers will end in a semicolon, such as this: “1492;”. Thus there can be no confusion as to which base a given number is relying on.
Now, on to the example. We are presented with the decimal number 2008., which represents the current year at the time of this writing. That number is easily parsed: it represents two units of one thousand, zero units of a hundred, zero units of ten, and eight units of one. Thus, the total value of the number is achieved by adding these up:
(2 * 1000) + (0 * 100) + (0 * 10) + (8 * 1)
Once again, it’s just placeholder notation. It’s really that simple. Now, however, we are presented with the decimal number 11E4;, which also represents the current year at the time of this writing. We parse it in precisely the same way, remembering that each number represents a multiple of twelve rather than a multiple of ten:
(1 * 1000) + (1 * 100) + (E * 10) + 4
It’s really that simple. The only trick here is remember that each place represents a multiple of twelve, not a multiple of ten. But this is simply a matter of practice, of adjusting the mind, no different from remembering that, when learning Spanish, a “j” doesn’t sound like it does in “judge” anymore.
All the vaunted benefits of the number-god ten are equally present in the base-twelve system for twelve. Let’s say, for example, that we want to divide a number by one of our units. We have three gross of eggs, for example, which we want to divide into dozens. We simply take our three gross, 300;, and move the dozenal point over one digit, yielding 30;0, or more simply 30;. This is neither easier nor harder than doing something similar in base-ten; assuming that we have three hundred eggs that we want divided into tens (why, since they are sold in dozens, I’m not sure, but assuming that we do), we simply move the decimal point over one digit, getting 30.0 from 300.. It’s placeholder notation, quite possibly the most brilliant mathematical advance of all time. Division by the base is always easy, no matter what the base is. To claim this as an advantage of base-ten is simply silly.
The question is not division by the base (which, again, is always easy); it’s division by other things. With our current, decimal base, we often have a great deal of trouble dividing by common numbers. Take percentages, for example. “Percent” literally means “per one hundred,” so any fractions of a percent are going to be fractions of a hundred. But what about when we want to say that one third of our group should do this or that? We have to say that 33.333333, for as long as we’re willing to say “three,” percent should do whatever it is we’re talking about. If we’re using a dozenal base, however, and therefore are using “pergross,” the question is easy. 100; pergross is equal to 144.; a third of 100; is 40;. Four dozen pergross is one third, and they should do this particular thing; the other two thirds, or 80; pergross, should do the other thing. But we need more accuracy: we want half of that first third to do it in one way, and half to do it in another way. It’s equally easy whether we divide 40; by two, or whether we divide our whole, 100;, by six; either way we get a nice, even 20;. In the decimal system, we wind up with another repeating decimal, 16.6666666 and so on.
Are all fractions in base-twelve easy? Naturally not; no base includes every fraction evenly. However, the dozenal base does include all the most frequently used fractions, while the decimal base excludes two of them. For comparison, let us chart out the major fractions in each base, from one to twelve, listing each in their own bases.
| Number |
Dozenal |
Decimal |
| 1 |
10 |
10 |
| 2 |
0;6 |
0.5 |
| 3 |
0;4 |
0.33333 |
| 4 |
0;3 |
0.25 |
| 5 |
0;2497 |
0.2 |
| 6 |
0;2 |
0.16666 |
| 7 |
0;186X3 |
0.14285 |
| 8 |
0;16 |
0.125 |
| 9 |
0;14 |
0.11111 |
| X |
0;12497 |
0.1 |
| E |
0;11111 |
0.09090 |
| 10 |
0;1 |
0.08333 |
As we’ve noted before, all the most common fractions—halves, thirds, and fourths—come out perfectly evenly in a single digit. Half of a third (a sixth) and half of a fourth (an eighth) also come out evenly, at 0;2 and 0;16. It is also worth noting that a ninth, though not very common, comes out evenly in only two decimal places. Ten is certainly easier for fifths and tenths, but is frustratingly inadequate for thirds and sixths, and more difficult than twelve for quarters and eighths. All in all, it seems plain that twelve makes a better base than ten based on the convenience of its major fractions as well as the importance of its factors in mathematics and geometry.
Furthermore, those numbers that mathematicians hold dear are often more accurate and shorter when rounded in the dozenal rather than the decimal system. Take, for example, the square root of two, which in dozenal notation is 1;4E79170X07E86 and so on, while in decimal it is 1.4142135. One can round the decimal to 1.41; however, doing so is considerably less accurate than rounding to 1;5 in dozenal. Dozenal thus provides a more accurate estimation of an important mathematical number, in fewer digits, than decimal does.[2] Of course, pi and the base of the natural logarithm are still grotesquely irregular fractions; but given that dozenal provides a larger base, one does acquire greater accuracy in the same number of digits than is possible in decimal.
All in all, it seems clear that twelve makes for a better base than ten. However, our language assumes a base of ten; how are we to count by twelves without having to resort to clumsy statements like “five great-gross, three gross, five dozen, and four?” There are many suggestions as to how to handle this situation, but the easiest and most consistent is offered by Tom Pendlebury of the Dozenal Society of Great Britain. Mr. Pendlebury has not only compiled an excellent way of saying the numbers in the superior dozenal base; he has also devised a complete and coherent dozenal metrology, known as TGM. A perusal of this work is much recommended for anyone interested in dozenal mathematics.[3]
Pendlebury suggests that we call eleven “elv,” to make it easier to use in compounds. He further suggests that we shorten “dozen” to “zen,” thus allowing us to say a number such as 47; as “fourzen seven” rather than “four dozen and seven.” Finally, he advocates completely eliminating the terms “gross” and “great-gross” (a dozen dozen and a dozen gross, respectively), and instead adopts the terms “duna,” “trina,” “quedra,” and so on. Thus, the number X44; (translating to the decimal number 1492.) would be read “tenduna fourzen four,” or “tenduna fourzen and four.” As another example, the current year, 11E4;, would be read “one trina one duna elvzen and four.” For comparison, a decimal year such as 1945., when read properly, is “one thousand nine hundred and forty-five.” Abbreviations to pronunciation, such as “nineteen forty-five,” will doubtlessly arise with the use of the dozenal system, as well, and are to be encouraged as long as they fit within the dozenal milieu.
Pendlebury’s suggestions thus make reading a dozenal number just as brief and easy as reading a decimal number. Furthermore, his metrology is much more consistent, and provides much more convenient units, than the current decimal metric system, and thus bears careful reading and consideration.
Well, there you have it. Nearly four and a half thousand (or two trina seven duna and threezen) words later, you have a basic explanation of number systems, placeholder notation, and why twelve makes a better base than ten. There’s a great deal of further reading to assist you if you’re interested, including basic things like how to convert between decimal and dozenal and back again (it’s easy, don’t worry). Please go browse the Dozenal Society of America, and the more extensive resources at the Dozenal Society of Great Britain.
There are programs to help you. I’m working currently on a general text-based, reverse Polish notation calculator for Unix-based systems (it currently works in the basic four functions), and I’ve completed robust converter programs (again for Unix-based systems, though they would probably compile on DOS-based system like Windows, as well) which convert numbers of more or less arbitrary complexity from decimal to dozenal and back again, as well as one which will turn dozenal numbers into the words suggested by Pendlebury. Future projects include a robust program for converting English and metric measurements into Pendlebury’s TGM. There is already a calculator for DOS-based systems, which will run perfectly fine in Wine on Unix systems, available at DSGB. Unfortunately, it’s purely graphical, and thus doesn’t fit nicely into scripts and command lines. Nevertheless, for simple calculations, it works quite well.
Thanks for listening to my rambling on about the dozenal system. Random, yes, but also interesting. God bless you and yours.
Praise be to Christ the King!
[1] This information on the importance of these numbers in mathematics and geometry is taken largely, if not entirely, from “Polygons,” by “Troy,” available at Polygons from the The Dozenal Society of Great Britain.
[2] This fact was noted in the article “Duodecimal” on Wikipedia, at Duodecimal.
[3] Tom Pendlebury, TGM: A coherent dozenal metrology based on Time, Gravity, and Mass, published by the Dozenal Society of Great Britain, available at TGM.