The Bases of Counting

Rough Draft (rev. 10)


§1

It may suprise you to learn that the way in which you count is only one of an infinite number of methods. When counting, in all likelihood you utilize ten different symbols, 0-9, to represent the quanitites you intend. This method is known as decimal or base-ten. But any number of symbols can be used. In fact the most evident number system, and the one first used by children and our ancient ancestors, is unary or base-one. In this system the number of symbols directly represents the quantity. Five, for instance, would simply be represented with five marks, |||||. Unary is still used today when we count by ones with our fingers, and Roman numerals are an progression from unary toward base-five. But eventually it is realized that a purely postional system of counting, a system with a base greater than one, is much more efficient and useful.

Number historians generally agree that two systems in particular ultimately locked-horns in the battle for minds of men: decimal and dozenal, base-ten and base-twelve. Base-ten is of course the obvious contendor since we have ten fingers on which we first learn to count. This handedly eases the transition to a decimal postional system. (Interestingly the Mayans, who developed independent of the western heritage created a base-twenty system, apparently they included their toes.) Appearently, as is the case with most traditions, politics may have ultimately lead to the demise of the dozenal system and the firm entrenchment of the decimal system on which most of the world depends today. In ..., as an outgrowth of the French Revolution and the Age of Elightenment, leading scientists of the day created the metric system, a comprehensive system of measures, all cast in the unquestioned counting system of the day, decimal. A common system of measures, is of course, of great use in the interchange of ideas, and the metric system being so complete and the triumph of Wlightenment principles, it is understandable that it has, over the years, become the world's dominant standard. Nonetheless base-twelve is often more useful because, unlike ten, twelve it is evenly divisibile by 3 and 4. These divisors are esspecially applicable to number intensive fields such as music, astronomy geometry and even trade. This addtional utility of base-twelve has never allowed it to completely fade away, despite the metric push. Today we still have twelve hours on our clocks and sixty minutes to an hour (five times twelve). And as everyone knows, eggs are by the dozen.

And so, the always curious events of history have led us to the number system we use today. But was ten really the best choice? The fact that we have five digits on each hand is a facet of our development far predating the dawn of arithmetic. It has everything to do with the physical task of survival, and little, if anything, to do with counting as an art in and of itself. As with nearly every innovation of Mother Nature, what was first set out to serve one task later become the mold of another. The count of our fingers choose our number system for us. We did not choose the count of our fingers. Yet this is not the whole story. Counting must serve a purpose, otherwise there would be no point in having it. In the course of serving this purpose, one system of counting may in fact hold advantages over another. For instance, of great importance is the convience to divide by two. To readily divide a set of items into two shares is undoubtedly a fundemental requirement of any number system. It is reasonable then to assume that even if man were rather adorned with an odd number of fingers, we would not be counting with that odd base. Dividing by two is simply too important. There is then good reason to sepculate and to wonder. Is it possible to set out proper cirteria and this determine on their basis the best system possible system for human counting?


§2

In the present digital era, another base of counting has come come strongly to the fore. Binary, or Base-two, is the simpilist positional number system possible, consisting of only two symbols: 0 and 1. The reason for the rise of binary is, of course, that it is the system upon which computers are built. Digital computers use simple on and off switches to represent numbers, which is in essence the very meaning of the word digital. To exmplify the simplicty of binary have a look at the addition and multiplcation tables:

+ 0 1
0 0 1
1 110
 
× 0 1
0 0 0
1 0 1

While binary may be the simplist number simple, it is far from the most useful for human beings. Binary numbers take nearly five times as many digits to represent the equivelant decimal number. For example, 237decimal is equivalent to 11101101binary. Obviously 237 is much less cumbersome. Here then is one criteria we may use to to judge a number system: how many digits does it require to represent a a given quantity. In this regard binary does not fair well.

To overcome the length disadvantage of binary, computer scientists use number systems with bases which are simple increaments on the power of base-two. Specifically, 23 which is base-eight, aka octal, and 24 which is base-sixteen, aka hexidecimal. Because they are simple powers of two, a group of digits can be directly converted to-and-from binary. Every three binary digits can be direclty represented by an octal digit, and every four binary digits by a hexidecimal digit. Octal and hexidecimal are in this fashion quite elegant systems, easy to understand and convert between one another. Yet, though they may be conceptually elegant, this is of limitesd utility. What is of greater importance is that they are both easily divisible by 2 and 4. This is very useful, as any english chef or carpenter will tell you. Yet despite being more ergonomic in this manner, base-two systems also suffer form a related disadvantage. Their base is not evenly divisable by 3. Which is a common enough division as to work against them.

Here are the basic arithmetic tables for octal.

+ 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 710
2 2 3 4 5 6 71011
3 3 4 5 6 7101112
4 4 5 6 710111213
5 5 6 71011121314
6 6 7101112131415
7 710111213141516
 
× 0 1 2 3 4 5 6 7
0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7
2 0 2 4 610121416
3 0 3 61114172225
4 0 4101420243034
5 0 5121724313643
6 0 6142230364452
7 0 7162534435261

This then presents another criteria: the number and relative utility of a base's divisors. Precidence, of course, goes to utility over number of divisors, and unquestionably utility follows the simplicity of the number: 2 is more useful than 3, which is more useful than 4, which is in turn more useful than 5, and so down the line. It is impossible to say just how much more useful one divisor is over another, but this ranking serves as useful enough ordinal guide.

A final note on the base-two systems: the elegance and digital utility of these number systems is strongly appealing and is indeed almost enough to mute the division by three failing. But not quite. Nonetheless leaning binary and the related octal and hexideciaml systems is very rewarding; well worth the small effort. Consider, for instance, this very simple and fun way to count in binary: Let each finger represent a binary digit. Use your thumb to hold down the fingers that are zeros and extend the other fingers which represent ones. In this manner you have a total of eight binary digits (four in each hand) at your disposal. This allows you to count from 0 to 255! See the references section below for links to turorials on binary, octal and hexidecimal.


§3

If major shortcoming of base-two and derivative systems was the lack of convienent trisection, what then of a base-three system? Unfortunately, base-three lacks convienent bisection --division by 2, which is a mcuh worse disadvantage. So much so, in fact, that we can immediately exclude any system that does not allow it.

Yet we must take a moment to consider base-three more carefully for one simple reason. It has been shown to be the most efficient of all number systems. We will not get into the scientific details of this discovery, suffice to say that it isn't terribly difficult to figure out that an abacus based on ternary will need the least number of beads to represent any given number. This fact is directly due to its closeness to the natural logorithmic number e (2.718...). It is interesting also to consider this fact in relation to the findings of psychologists regarding the inate abilities of the humn mind. The typical human can only inately count to four, a base nearly as efficent as ternary. You can test this on yourself by trying to instantly recognize a quantity of items --instantly mind you is the key word here. You will likely discover that you can only instantly recognize quantities up to four. Quantities greater than four are only recognized if they fall into particular patterns that you have come to associate with those numbers, such as a cross for five, or a three-by-two grid for six. Beyond this, the quanity must be broken into subgroups and systematically counted and multiplied. Interestingly, because of ternary's greater efficiency we may eventually see the existence of ternary based computers. We should point out though, that despite ternary's efficency, like binary, it is not highly compact in written form. For example, the current year of 2004 decimal is 2202020.

For completeness here are the addition and mutiplication tables for ternary.

+ 0 1 2
0 0 1 2
1 1 210
2 21011
 
x 0 1 2
0 0 0 0
1 0 1 2
2 0 211

§4

Now we address base-ten, the number system with which we are all familiar. Decimal's obvious advantage is, of course, that we are born with ten fingers. This simplifies the transistion we make as pupils from the unary system with which we learned to count, to the postional method we will subsequently use for the rest of our lives.

This indicates another important criteria: ease-of-learning. The consideration of important in this regard is a number system's correlation to our everyday natural abilities. This works to decimal's advantage with rgrad to the number of fingers we have, but this may be considered rather minor when compared to the complexity of related tables and figures. If the base of the system is too great then the arthmetic tables large and more difficult to memorize. If the base is too small then numbers themselves, as we have already seen will be too large for convienient use. So we will need to strike a reasonable balance between the two.

Base-ten also has the advantage of being evenly divisibile by 5, a divison used in statistical percentile comparisions, for example. But convenient divisiblity by 5, while sometimes useful, does not outweigh the greater disadvantage of not being evenly divisible by 3 and 4. As we mentioned easlier, this readily noticed in everyday life. Applications in which division by 3 or 4 is especially useful tend to have a strong following of alternate measurement. We need only consider that there are 16 ounces per cup, 12 inches per foot and 12 hours per half a day.

The only other advantage base-ten has, is its entrenched user base. This may not seem a fair comparison, but we must give it at least a small credit considering the long and difficult process that would be invloved in transitioning to another system. We certainly should ask ourselves, is it really worth the effort?


§5

Senary, or base-six, is the first system that overcomes both the 2 and 3 divisor issue. But this is not the only great advantage that senary has over the other systems we have thus far considered. Sernary also has the advantage of being much easier to learn. All of its bae numbers are within the upper bound (5) of man's innate comprension. Thus its arthmetic tables are very straightforward and extremely easy to memorize.

+ 0 1 2 3 4 5
0 0 1 2 3 4 5
1 1 2 3 4 510
2 2 3 4 51011
3 3 4 5101112
4 4 510111213
5 51011121314
 
x 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4101214
3 0 310132023
4 0 412202432
5 0 514233241

Counting in base-six is also simple and strightfoward. While base-ten may naturally correlate to our ten unary fingers, base-six naturally correlates to our hands-as-abacus --a one-for-one representation of postional counting. To count in base-six simply use one hand to represent the ones place and the other hand to represent the sixes place. With this technique anyone can easily count to thirty-five (556).

But senary is not perfect. It does have two noticable disadvantages. The first is that senary numbers tend to be a bit larger than their decimal equivalents, although argueably mitigated by the fact that senary numbers require four less symbols to denotate. The more signifficant disadvantage is that senary is not readily divisibile by 4, a division that is perhaps just as common as 3.


§6

Dozenal, or base-twelve, addresses the disadvantages of senary. It is more compact than decimal and is evenly divisible by 2, 3, 4 and 6. But for what it gains in compactness and divisibility it also looses in learnability. The arthemetic tables for base-twelve contain 44 more entries than thier decimal counterparts. This too may well help account for dozenal's poorer fairing against decimal over the course of history.

+ 0 1 2 3 4 5 6 7 8 9 X J
0 0 1 2 3 4 5 6 7 8 9 X J
1 1 2 3 4 5 6 7 8 9 X J10
2 2 3 4 5 6 7 8 9 X J1011
3 3 4 5 6 7 8 9 X J101112
4 4 5 6 7 8 9 X J10111213
5 5 6 7 8 9 X J1011121314
6 6 7 8 9 X J101112131415
7 7 8 9 X J10111213141516
8 8 9 X J1011121314151617
9 9 X J101112131415161718
X X J10111213141516171819
J J1011121314151617181920
 
x 0 1 2 3 4 5 6 7 8 9 X J
0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 X J
2 0 2 4 6 8 X10121416181X
3 0 3 6 91013161920232629
4 0 4 8101418202428303438
5 0 5 X131821262J34394247
6 0 610162026303640465056
7 0 71219242J364148535X65
8 0 814202834404854606874
9 0 916233039465360697683
X 0 X18263442505X68768492
J 0 J1X2938475665748392X1

As for counting in base-twelve, the verdict is rather neutral. Like base-six, counting in base-twelve is also a direct representation of the postional system, but it is less intuitive. Again, one hand represents the ones place and the other hand represents the twelves place. But instead of using each finger as a single digit, each finger is divided into three parts: a root (just above the palm), a center and a tip. Since there are four fingers per hand (excluding the thumb), this gives twelve positions. The thumb is then used to point to each of these spaces on the fingers in order, and applying this method to both hands, one is able to count up to 156d (110dz). Although unintuitive, this high number is at least an advantage.

Dozenal does have one other distadvantage which should be mentioned. Its decimal-point representation of fractions (not to be confused, with the decimal number system) can be rather complex, most notably 1/5 turns out to be 0.2497... (repeating). Numerous other fractions require at least four or more repeating digits as well. Although it is perhaps not the most improtant of criteria, it is something to consider. Fortunatly none of the other systems we have examined really suffer too greatly from this problem. In fact decimal would probably be the next runner-up in this area and its decimal-point representations are fairly better than dozenals.


§7

What of larger bases, like base-twenty, base-thirty and base-sixty? Of course the advantage that these systems is their greater compactness and inclusion of many more divisors. Base-sixty, for instance, can represent a number in six digits for which decimal would require ten. And it has a plethora of divisors: 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. Which is certainly advantageous. But despite this, these larger bases suffer the disadvantage of being far too unwieldy in the number of symbols they require. Imagine memorizing a multiplication table with 60 x 60 entries --that's 3600 possibilities! Therefore these systems are best kept to very specific applications, and even then notated in a smaller base, as is now done with minutes and seconds. Perhaps if the human mind were a more capable, able to learn a table of a thousand entries as easily as it now learns a table of 100, then base-sixty would be the ultimate option. But presently it is a rather mute point.


§8

If we were now given charge to choose a number system to serve as the world's defacto standard, a universally common number system, which would it be? Unfortunately, it is still a difficult decision since there are evidently trade-offs that must be made. But we can at least remove all extreme possibilites. It seems fair to say that the base should be no less than 5 and no greater than 16, and that any base not evenly divisible by 2 should be excluded. This leaves bases 6, 8, 10, 12, 14 and 16.

At this pointt we will get bogged down in details of evaluation. If we go threw each cirteria and ascribe a point to each system that exmplifies the basis, we may get a somewhat, albiet imperfect, ... of these systems.

Statring with learnability. We certainly must ascrive a point to base-six since it is the simplist. And base-ten is also deserving a point since it is already prevelant. We will also give one to base-eight because it is relatively simple but also becasue it is easily correlated to binary.

As for compactness, hexadecimal is no doubt best, and gets a point. But not to be overly favoring, we will give a point to the two runners-up, base-12 and base-14.

If we now ascribe a point for every significant factor of divisibiltiy greater than two (3, 4 and 5), then senary, octal, decimal and hexadecimal get another point, while dozenal gets two points.

These are the major criteria. If we examine this simplistic tally, at this point we have:
    Senary         1
    Octal          1
    Decimal        1
    Dozenal        1
    Base-14
    Hexadecimal    0
  
To facilitate our determination further, lets take the criteria we have have explored and rank the top three systems in each area.

We will first notice that bo matter which system we choice we will have to sacrifice at least some potentially useful divisor(s). We again note that smaller divisors are more importatn the larger, but we make one slight exception. Despite the high utility of dividing by 4, any system that already offers division by 2 will not benefit as greatly from the addition of it as a divisor simply because one can readily divide the unit by the base again. In otherwords, six sixes and ten tens are easily divisible by four. For this reason it is argued that decimal just edges out hexidecimal. With regards to divisors then, the ranking order is: dozenal, senary, decimal.

With regards to compactness, obviously the larger the base the better. So the ranking is simply: hexidecimal, dozenal, decimal.

But we should also consider the compactness of decimal-point representation of fractions. Although not as important as the former, it should at least make a contibution. The ranking for this turns out to be the rather odd: senary, hexidecimal, decimal.

Ease of learning goes in the direction of the smallest base. With a bonus given to decimal for the fact that it is both already common and correspondent to the number of fingers on our hands. The ranking therfore follows: senary, decimal, octal.

Likewise effiency benefits from smaller bases, those being closer to the natural number e. So the ranking here is: senary, octal, decimal. But this too does not seem quite as important as the ease of learning, so it conribution should perps be a bit less.

So now we may wonder, have we convered every significant factor? Perhaps new evidence will present itself. New facts are always a welcome discovery. But if we are to presnetly come to a determination we must go on with the criteria we have available. Thereby, if we give each of the major criteria equal regard, and the two noted minor citriea half as much regard, a simple of point sytem as follows may be the best we can do in objectifying the outcome: +6 points for major 1st ranking, +4 points for major 2nd ranking, and +2 point for a major 3rd, the minors getting half so much. Applying this, the outcome is as follows:

    Senary         16
    Dozenal        10
    Decimal        10
    Hexadecimal    10
    Octal          4
  
clearly senary or dozenal would serve best, winning out over the other potential contendors: octal, hexidecimal and decimal. But two difficulites yet present themsleves. First, which of these final two do we choose? And even then, how could we ever hope to overcome the entrenchment of decimal? Addressing the later question actually gives us direction to the former. For one certain way in which a new system may be able to gain traction is for it to be simpler to learn than the its competitor. Thus senary has advantage over dozenal. Yet dozenal is slightly more capable, with its divisors of 4 and 6, so too in is own way dozenal sways better against decimal. Also, we should not fail to mention, dozenal is still commonly used in certain areas and applications. Perhaps, if we cannot settle between the two readily, there is a compromise that can be made? As luck has it, learning senary makes dozenal rather easy since dozenal is exactly a factor of two greater than senary. Consider the simple conversion table:

Twelve 0 1 2 3 4 5 6 7 8 9 X J10
Six 0 1 2 3 4 510111213141520
Decimal 0 1 2 3 4 5 6 7 8 9101112

0 1 2 3 4 5 6 7 8 9 X J
0 0 1 2 3 4 5 10 11 12 13 14 15
1 20 21 22 23 24 25 30 31 32 33 34 35
2 40 41 42 43 44 45 50 51 52 53 54 55
3100101102103104105110111112113114115
4120121122123124125130131132133134135
5140141142143144145150151152153154155
6200201202203204205210211212213214215
7220221222223224225230231232233234235
8240241242243244245250251252253254255
9300301302303304305310311312313314315
X320321322323324325330331332333334335
J340341342343344345350351352353354355

For this reason then we will ascribe that both are learned. For it makes a great deal of sense that from an early age a child will become familiar with more than a single number system, just as it is beneficial that should be familiar with more than one language.

    Decimal                Dozenal                Senary
    1/2   0.5              1/2   0.6              1/2   0.3
    1/3   0.33...          1/3   0.4              1/3   0.2
    1/4   0.25             1/4   0.3              1/4   0.13
    1/5   0.2              1/5   0.249724...      1/5   0.11...
    1/6   0.166...         1/6   0.2              1/10  0.1
    1/7   0.1428714...     1/7   0.186X3518...    1/11  0.0505...
    1/8   0.125            1/8   0.16             1/12  0.043
    1/9   0.11..           1/9   0.14             1/13  0.04
    1/10  0.1              1/X   0.12497...       1/14  0.033...
    1/11  0.0909...        1/J   0.1...           1/15
    1/12  0.0833...        1/10  0.0JJ...         1/20  0.03
    1/13  0.07692307...    1/11                   1/21
    1/14  0.07142871...    1/12                   1/22  0.02323...
    1/15  0.066...         1/13                   1/23  0.022...
    1/16  0.0635           1/14                   1/24  0.0213
  

× 0 1 2 3 4 5101112131415
0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 10 11 12 13 14 15
2 0 2 41012 14 20 22 24 30 32 34
3 0 3101320 23 30 33 40 43 50 53
4 0 4122024 32 40 44 52 60 64 72
5 0 5142332 41 50 55104113122131
10 010203040 50100110120130140150
11 011223344 55110121132143154205
12 012244052104120132144200212224
13 013304360113130143200213230243
14 014325064122140154212230244302
15 015345372131150205224243302321

2 0 1101110010111011110001001101010111100
3 0 1 210 11 12 20 21 22100101102110
4 0 1 2 3 10 11 12 13 20 21 22 23 30
5 0 1 2 3 4 10 11 12 13 14 20 21 22
6 0 1 2 3 4 5 10 11 12 13 14 15 20
7 0 1 2 3 4 5 6 10 11 12 13 14 15
8 0 1 2 3 4 5 6 7 10 11 12 13 14
9 0 1 2 3 4 5 6 7 8 10 11 12 13
10 0 1 2 3 4 5 6 7 8 9 10 11 12
11 0 1 2 3 4 5 6 7 8 9 X 10 11
12 0 1 2 3 4 5 6 7 8 9 X J 10

References

Wikipedia Entry on Numeral Systems How to Count to 1,023 on your Fingers
Third Base, Brian Hayes