Shack's Base Six Dialectic

There must be a better way

This 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 bold font when I am referring to base ten and the number base may otherwise be ambiguous. So 100 (heximal) equals 36 (decimal), and 10000 (heximal) equals 1296 (decimal).

A given number does require, on average, about 29% more digits to express in heximal, but this is a small price to pay given that number pads require 40% fewer digits, and thus can be made smaller to allow room for the larger displays. Also, since smaller numbers tend to be used more frequently than larger numbers, the 29% figure (derived from the ratio of the logarithms of the number bases) overestimates the penalty.

Learning the system

Most 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.

0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 10 12 14
3 0 3 10 13 20 23
4 0 4 12 20 24 32
5 0 5 14 23 32 41

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 100 entries. Leaving out the trivial cases of 0-times and 1-times leaves 64 entries. Using the commutative property leaves still leaves 36 entries. Many of these are easy to learn, but they still do have to be learned.

In base six, however the table has only 36 entries. Leaving out the trivial cases of 0-times and 1-times leaves 16 entries. Using the commutative property leaves just 10 entries. And of those, 4 are 2-times, 3 are 3-times, and 2 are 5-times, leaving only 4x4.

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 7x8 = 56. With base six, that hurdle is much, much smaller.

But we have ten fingers

Yes, 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.

Rule 3. Sec. 5. Art. 12.a “The following numbers are legal: 0, 1, 2, 3, 4, 5, 00, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 55. Team rosters can include 0 or 00 but not both.”

An aside for the Geeks

Some computer Geeks enjoy counting in binary on each hand. Using this method, a person can easily count to 31 on one hand, and to 1023 on two hands. It is a matter of assigning a binary place to each finger and then swinging the fingers up or down for the digits 0 and 1. In a base six world, however, it is convenient to follow 31 (which is 51 in base six), with 52, 53, 54, and 55 while touching the thumb to the other four fingers in turn. Thus the Geek can count to 35 on one hand (55 base six), or 1295 on two hands (5555 in base six).

Naming the numbers

It 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.

Formatting

Whereas in base ten, it is traditional to separate the digits of large numbers into groups of three, as in 1,000,000 (or 1.000.000 outside the US), in base six it is more convenient to use groups of two or four. It just works out that way. I propose eliminating the English/European ambiguity with respect to the comma/period (the meaning of 12.345 is different by a factor of 1000 depending on where it is being read). This could be done by separating digits with either spaces or colons, and then reserving the comma or period for the "heximal" point. Being American, in this paper I will use the period ("full stop") for the heximal point.

Divisibility

In base ten, one can tell at a glance if a number is divisible by 2, 5, or 10. With a second glance, divisibility by 4. There are rules to detect divisibility by 3 or by 9 (a number is divisible by 3 or by 9 if and only if the sum of its digits is divisible by 3 or by 9). In base six, divisibility by 2, 3, or six can be seen at a glance. With a second glance, divisibility by 4. The sum of the digits rule applies to check for divisibility by 5.

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.

 Decimal fraction Heximal fraction Heximal expansion Decimal expansion 1/2 1/2 0.3 0.5 1/3 1/3 0.2 0.333... 1/4 1/4 0.13 0.25 1/5 1/5 0.111... 0.2 1/6 1/10 0.1 0.1666... 1/7 1/11 0.050505... 0.142857... 1/8 1/12 0.043 0.125 1/9 1/13 0.04 0.111... 1/10 1/14 0.0333... 0.1 1/12 1/20 0.03 0.08333...

Measuring distance

Base 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 (1296 in decimal) and "mega" to mean 1 0000 0000 (1,679,616 in decimal). I will follow my convention for numerals and bold-face those terms when they have their decimal meaning. So the metric inch may also be called a heximal centimeter.

So a heximal kilometer would be 1 0000 meters, which works out to 1.296 (decimal) kilometers or about 0.8 miles. So a heximal mile/kilometer is a unit which may usefully be used in circumstances where the decimal mile/kilometer is used today. The heximal system adds a convenient unit, the heximal chain or hectometer, of 100 meters.

Measuring time

The 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 40 decimal minutes), divide the heximal hour into 100 heximal minutes (each equal to about 1.11 decimal minutes), and divide the heximal minute into 100 heximal seconds (each equal to about 1.85 decimal seconds).

Time is currently measured in base 60 (minutes and seconds) and base 12/24 (hours). Base 60 means that the hour can be divided into a whole number of minutes and seconds by halves, thirds, quarters, fifths, sixths, eights, ninths, tenths, twelfths, fifteenths, twentieths, etc. A very useful system, except that it does not match the decimal number base most commonly used, so most time and calculations are difficult. With heximal time, the hour may be evenly divided into halves, thirds, quarters, sixths, eights, ninths, twelfths, etc, so it is still very convenient.

The conversion between heximal and decimal hours is fairly easy, every three heximal hours is two decimal hours. So 10:00:00 heximal is 4:00 AM decimal. Here's a table.

 heximal decimal 03:00 2:00 AM 10:00 4:00 AM 13:00 6:00 AM 20:00 8:00 AM 23:00 10:00 AM 30:00 noon 33:00 2:00 PM 40:00 4:00 PM 43:00 6:00 PM 50:00 8:00 PM 53:00 10:00 PM 00:00 midnight

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 conversions

One 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 volume

In 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 liter--1000 cubic centimeters. There are 1000 liters per cubic meter. In the heximal system, we use a heximal liter which is 1/1 0000 of a heximal cubic meter, and this volume is 100 heximal volumetric (fluid) ounces, each of which is one cubic heximal centimeter. So the heximal liter is about 0.77 decimal liters (about 0.82 decimal quarts), and the heximal fluid ounce is about 0.72 US decimal fluid ounces. The heximal milliliter is about 0.6 of a decimal milliliter.

Measuring mass

The 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).

Currency

Useful 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 \$10 bills because we are decimalized, and \$5 bills because it is inconvenient to have to count up ten \$1 bills before grouping them. But \$5 bills are an awkward denomination because there is no reason to carry more than one of them. In base six we group six \$1 bills into a \$10 bill, six \$10 bills to a \$100 bill, and six \$100 bills to a \$1000 bill. We have no need for the uneven gaps that occur in the decimal system.

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 29% more digits is a problem. This problem can be circumvented, however, by using a base 100 number system in these cases. In this system, the digits are 0-9 and A-Z, where Z represents 55. In the table, the entries represent the encoding of a two-digit heximal number, the rows are indexed by the first digit of that number and the columns are indexed by the second digit.

0 1 2 3 4 5
0 0 1 2 3 4 5
1 6 7 8 9 A B
2 C D E F G H
3 I J K L M N
4 O P Q R S T
5 U V W X Y Z

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.)