Counting Systems Compared: Decimal, Dozenal (Duodecimal), Hexadecimal, and Octal

Originated: 07 September 2004

Revised: 07 February 2005

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Introduction:

Ten fingers, ten toes – that’s the proper number for humans. Among the pre-English cultures of America, the traditional greeting between strangers was to hold out the hand so that each could count the other’s fingers – those with more than five fingers were viewed with terror. Many of the things we use are designed with five-fingered humans in mind; our fingers have a variety of uses and what we use is often shaped by our number of fingers. In mathematics, we use our fingers to count – and the most straightforward way is to use each digit as a whole; thus, we tend to keep track of things in tens. And this has been the great downfall of human math (though, it wasn’t a huge problem until compulsory metrication and decimalization appeared on the world scene). You see, since most people do what seems obvious, many cultures end up with base-10 (decimal) counting systems (except, of course, the Sumerians with their base-60 (sexagesimal) system, the Mayans with their base-20 (vigesimal) system, and the Maldivians with their base-12 (duodecimal) system). And this is a shame since other counting bases are much more utilitarian.

We English-speakers are fortunate in that we come from a North-European culture that, while counting in tens, used other number bases where convenient. Hence, we have terms in our language such as dozen, gross, and great gross. In modern times, we have moved away from our older holistic approach to number bases and more toward a more uniform, rigid, decimal orthodoxy. This has certain advantages that look very good on paper. Counting seems easier – it’s harder to figure out one gross multiplied by four than it is to figure out one hundred multiplied by four. And it’s really easy to count by fives as opposed to the difficulty faced in counting by sixes or by eights. But the trouble with the ‘advantageous’ nature of decimal counting is that it is all an illusion.

Decimal Unmasked:

You have to remember that all numbers are numbers. They are just numbers. They don’t naturally occur in any set repeating series – that is a human construct to make the handling of numbers possible. Ten is an easy number to work with because it is the base – the demarcator of the set. When you multiply a number by ten, you are essentially determining how many sets the equation will yield. No extras to deal with when multiplying complete sets, you see. Four times ten? No problem, that’s forty. The answer is obvious – four sets. But what about four times thirteen? Oh dear, that is troubling. You have four sets that you know of; but each set has three extra. What’s three times four? Twelve – that’s one more set and two extra. So we have five sets and two, or fifty-two. Now that you’ve seen how it works, you can accept the fact that this process works the same for any number-base. The only purpose of the demarcator (the base number) is to define the value of the set. In decimal, the value of the set is ten.

The reason the number five is similarly easy to count with is because of it’s relationship to the number base. Five is half of ten. This is a binary relationship. Binary patterns are always easy to work with. The trouble is that some numbers are more binary than others. Five’s ease of use only works in one direction. Two times five is ten, two times ten is twenty, and so forth. But what happens when you divide by two? five divided by two is two and one half, two and one half divided by two is one and a quarter, one and a quarter divided by two is 0.625, and it gets messier from there.

As you can see, ten’s special powers come from its being the demarcator of the set. Five’s special powers, in turn, come from its binary relationship to ten – but those powers break down in binary division (it is interesting to note that numbers like eight can yield manageable numbers ad infinitum from binary division). The idea that the decimal system is superior to other systems is an illusion – an product of a biology that gave us 5 digits on each hand instead of four or six.

The Utility of a Number Base:

Having debunked decimal’s superiority, we are now free to analyze the utility of each number base in a more rational manner. Now, think for a minute – what makes a number useful? Or a better question – do you like working certain numbers more than others? When dealing with whole numbers, addition, subtraction, and multiplication don’t present any problems. Division is a different story (I think we’re all familiar with the dangers of division: the chance of coming up with pesky remainders or repeating decimals, depending on the calculation medium). Imagine, for a moment, that you are being presented with a division problem (you don’t need to know the denominator); which number would you rather have as the numerator? Thirteen, or sixteen? Twelve, or nineteen? Of those two sets, which numbers would you choose? Why?

Well, unless you’re a glutton for punishment, you would most likely choose sixteen and twelve, respectively. The reason is simple – you know that there is a greater chance of the denominator being something that divides evenly into the numerator, of it being a factor. In a random division problem, if you could choose your numerator, not knowing what the denominator might be, you would most likely want to choose a number that is divisible by as many factors as possible. The factors a given number has, the more utilitarian it is. We like utilitarian numerators for division problems. Similarly, when looking at what number might be a useful base for a counting system, we look for just this sort of utility.

When choosing a base number, it is important that as many numbers as possible in the proceeding set of numbers are factors of the demarcating, or base, number. However, counting systems, unlike the ideal division problem do have extra limitations. Consider the number two hundred and forty. That number is a great choice for a numerator. Out of the numbers one through sixteen, only seven, nine, eleven, thirteen, and fourteen are not factors. Eleven numbers out of sixteen are. This number would make an excellent demarcateor for a counting set – that’s why the British divided their pound sterling into two hundred forty pennies. However, for a useful mathematical counting base – one that the average human can use – you want to look at smaller numbers for the demarcator. I know the Sumerians used a sexagesimal counting system, but sixty, though a useful numerator, demarcates a very large set of numbers. A sixty-number set conations too many symbols to be efficient for the average person to use. We probably want to stick with number sets of less than twenty (so Mayan vigesimal is out). Likewise, we don’t want out base to demarcate too small a set – that would give us more sets in a given group of sets than we already have (for example, ten sets of ten in one hundred) and that isn’t efficient. Since we already use a decimal counting base, if we change, we want to go to a larger number. Larger numbers contain more information (demarcate a set of more numbers) and are therefore more efficient symbols for the transmission of information. Luckily for me, two alternatives to the decimal system have gained popularity – and they’re between ten and twenty. Twelve and sixteen based systems have both been proposed as alternatives to the decimal system.

I’m sure that after all of this introductory material, the actual analysis will seem very anticlimactic – but this is due to the simplicity of the analysis described above. The factors are as follows:

Ten has four factors: one, two, five, and ten.

Twelve has six factors: one, two, three, four, six, and twelve.

Sixteen has five factors: one, two, four, eight, and sixteen.

Of the three, twelve is the most utilitarian demarcator. Sixteen is the second-most and ten is the least utilitarian demarcator.

Counting Systems Contrasted:

A base sixteen (hexadecimal, or hex) counting system has two major advantages over decimal in that sixteen, unlike ten, divides into whole-number quarters. Also, unlike decimal, hexadecimal is binary – this means that you can easily divide the base number into halves ad infinitum and not come up with an unwieldy decimal or fraction. Duodecimal (popularly called dozenal) has the same disadvantage relative to hex – it is not binary. However, unlike hex (and decimal), dozenal divides into thirds. Thus, where one third our current one hundred is thirty three and one third, one third of one gross is fourty-eight, a whole number. Like hex, dozenal sets also divide into quarters.

Both dozenal and hex have two advantages relative to decimal. So which one is better? Scientists would say that hex is – they like binary numbers; they make calculations easy. Dozenal, however, would be more useful to the average person – most people would probably find the ability to divide easily by three more useful than easy binary division for small numbers that they rarely even think of (most people use money and trade in goods that need to be divided up for trade purposes; remember, greater divisibility is better). And I hate to use argumentum ad populum, but since dozenal would be more useful for more people, I have to advocate dozenal as the most promising alternative to decimal.

New Symbols for Numbers:

The trouble with changing the counting base is that it will necessitate the creation of new symbols to represent the numbers within the set. If you have ever been to a hex web page, you are aware that for the numbers ten through fifteen, they use the letters A through F. That could get very confusing. A necessary characteristic of any set of symbols used in communication is the easy distinguishability between symbols. It does not do to have a number (A) that can be mistaken for a word.

The dozenists have been a little more creative than the hexadecimalists in this regard. My favorite proposal is by Don Hammond (for some years Secretary of the Dozenal Society of Great Britain). He basically rotates an altered two and three one hundred and eighty degrees clockwise to get his symbols for ten and eleven (the same way a six is rotated to get nine). I have presented my variant on this idea in my preferred dozenal characters.

A Note on Octal:

There is another viable number base that I do not discuss above. That is base eight. It has all of the advantages of hexadecimal without the hassle of coming up with six efficient new characters to use as numerals. Octal is extremely convenient and easy to learn. Each number set has only eight numbers, with 'eight' acting as the demarcator. The only difficult thing to remember is that eight would look like 10. Even though I like octal for its elegant simplicity, I am not an advocate of it because I really don't think the advantages outweigh the disadvantages of each set containing less information than even decimal sets do (in octal, 100 in octal = 64 in decimal notation, and it only gets worse – 1 000 in octal = 512 in decimal notation). My preference is that if we choose base numbers, we choose one that both contains more information than ten and is more utilitarian than ten. In my opinion, one dozen fits this bill perfectly. (And, unlike hexadecimal, my preferred dozenal characters work well with a calculator.)

For More information on Dozenal and Hexadecimal, visit the following web pages:

The Dozenal Society of Great Britain

Intuitor Hexadecimal Headquarters

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