## Shack's Base Six Dialectic## There must be a better wayThis arose out of my familiarity, typical of US citizens, with both the metric system and the English system of measurement. Naturally, there are some areas where the metric system is the obvious choice, most notably scientific measurements. On the other hand, the English system works out very nicely in some other circumstances, most notably when constructing quantities that must be divided evenly. Ten divides well into halves or fifths, but not so well into fourths and badly into thirds. The metric system excels because it uses the number base most often used for computation, the English system excels because it uses a number base that is more convenient when the computation is very simple, such as a division into three parts. The obvious solution is to switch to a more convenient number base for computation, such as base six. With a computational system that allows easy division into common fractions, such as thirds, a measurement system that utilizes the computation system will have all the advantages of both the metric and the English system. ## Does it really work?Actually, it works out surprisingly well. It turns out that base six has advantages over base ten. Many of these derive from the fact that an order of magnitude in base six is not too small and not too large. This means that a good compromise unit is nearly always available (the lack of a good compromise unit is one of the reasons that there has been so much reluctance to break the day into a decimalized unit). Most of the advantages derive from human factors, human psychology is better adapted to grouping into sixes than into tens. ## What is base six?Briefly, base six (or "heximal") is a number system that uses the number symbols 0 through 5
in each digit rather than 0 through 9. So the number that would be
expressed as 6 in base ten is expressed as 10 in base six. We count
1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20, and so on. In what
follows, I will use a A given number does require, on average, about ## Learning the systemMost parents know the difficulty with which children learn their base ten multiplication tables, so it seems foolish to replace this with something that seems even more complicated (owing to its unfamiliarity). But the base six multiplication table is trivial to learn. Let's have a look.
The 0-times row is trivial, as in base ten. So is the 1-times row. The 2-times row is the same idea as in base ten, very easy. The 3-times row in base six works like the 5-times row in base ten--this rarely gives anyone trouble. For the 5-times row, there is a trick that is similar to the 9-times trick in base ten--subtract one from the multiplier to get the first digit and the sum of the two digits is always five. Since multiplication is commutative, the columns work the same way as the rows. This leaves only 4x4 = 24 that must be learned by rote, and even that is pretty easy since 2x4 = 12 and twice that is clearly 24. In base ten, the table has In base six, however the table has only I believe that many children get turned off to mathematics largely because
their first exposure is the rote memorization that is involved in the mastery of
elementary arithmetic. Many children may have
"mathematical" minds in the sense of being comfortable with
abstraction, but believe themselves to be "no good at math" because in
third grade they cannot quickly recall that ## But we have ten fingersYes, five fingers on each hand. With those five fingers it is very natural for a small child to count 1, 2, 3, 4, 5 on one hand. And then, the child closes those fingers while raising one finger on the other hand to count 10, and then counts on the first hand 11, 12, 13, 14, 15. Then the child again closes those fingers while counting to two on the other hand for 20, and so on. Thus the child learns "place value" almost from the beginning, and the association of the digits 1-5 with the fingers of each hand seems very natural. Base six has already been adopted in the numbering of basketball jerseys, for precisely the reason that it is easy for referees to express with their fingers.
## An aside for the GeeksSome computer Geeks enjoy counting in binary on each hand. Using
this method, a person can easily count to ## Naming the numbersIt is very tempting to simply call 15 "fifteen", simply understanding it to be base six. As an alternative, the word "hex" could be used for six, and the t's in the names for numbers changed to h's. Where the h's are hard to pronounce, we just drop them. So, for example, "twenty" becomes "twenny". I think that words like "dozen" and "gross" should continue their base-independence, meaning 20 and 400, respectively. ## FormattingWhereas in base ten, it is traditional to separate the digits of large
numbers into groups of three, as in ## DivisibilityIn base ten, one can tell at a glance if a number is divisible by 2, 5, or In base ten, the common fractions 1/3 and 1/6 result in repeating decimals. In base six, one-half is 0.3, one third is 0.2, one fourth is 0.13, one sixth is 0.1. One fifth is a repeating heximal (0.111...), but that is a much less commonly used fraction. Most common expansions are easier in base six.
## Measuring distanceBase six works very nicely with the English system of yards and inches, 100
inches to the yard. The system can also just as easily be based on the
meter, however, with the metric inch equaling 1/100 of a meter. I will
usurp a practice from computer science, of fudging the meaning of terms like
"kilo" and "mega" to refer to the closest corresponding
round value in the number base of choice. So I will use, for
example, "kilo" to mean 1 0000 ( So a heximal kilometer would be 1 0000 meters, which works out to ## Measuring timeThe public has not accepted metric time (division of the day into parts based
on a decimal system) largely because ten hours per day, with ten minutes
per hour is too coarse a measurement, whereas one hundred hours per day with one
hundred minutes per hour is too fine. The heximal system, however,
works very nicely with time. Divide the day into 100 heximal
hours (each equal to Time is currently measured in base The conversion between heximal and decimal hours is fairly easy, every
three heximal hours is two decimal hours. So 10:00:00 heximal is
Note that when it is 33:00:00, it is 33 hours past midnight, or 3300 minutes past midnight, or 33 0000 seconds past midnight. So all of the current, complicated, time conversions are eliminated. If you have JavaScript enabled, you should be able to see the current time of day here: 00:00:00.0 . Note also that heximal time retains the notion of "30" being "half-past", since 30 is one-half of 100. Also "20" has its time-like notion of one-third, and "10" its time-like notion of one-sixth. That these notions carry over from time measurement may be one of the reasons that base six seems natural so quickly. ## Speed conversionsOne of the advantages of eliminating the complex time conversions is that the speed conversions become simpler as well. If you are traveling at 200 heximal miles per hour, that is also 200 heximal meters per second (or 200 heximal chains per minute). ## Measuring volumeIn the decimal metric system, the cubic meter is sometimes used as a measure
for moving earth, but the more common unit of measure is the ## Measuring massThe metric kilogram is approximately the mass of one liter of water. I propose a heximal gram which is about 1/1 0000 the mass of a heximal liter of water. Then 100 heximal grams would make a heximal ounce (or hectogram), and 100 heximal ounces would make a heximal pound (or kilogram). ## CurrencyUseful denominations would include the $1, $10, $100, and $1000 bills, and
coins for $0.01 and $0.1 (ok, we can keep the $1 coin). We now (in
the decimal system) have $ ## Telephone numbers, etc.One of the disadvantages of heximal comes when numbers of many digits must be
remembered. In this case, the fact that a typical number has
So a telephone number like 44:25:14:20:32 would become SHACK. (Of course, the usual programmer's care would have to be exercised to distinguish between O/I and 0/1, which can easily be done by decorating the digits appropriately--a slash or tick on the zero, and writing 1 and 7 as the Europeans do.) ## SadlyThe world is not going to change to base six any time soon, despite the advantages. What is needed is a good transition to base six. It is, however, convenient and fun to use in one's personal computations, even if the larger world remains stuck with the decimal system. |