Decimal vs Base-16 Arithmetic

An essay by Ken Wear, posted July 2000

We have become accustomed to the decimal system of counting and measurements and mathematics, but there is another system that is just as natural, if not actually more natural. It is also based on the number of digits on the human hand. Whether it can ever overtake in popularity the decimal system is questionable, but I present here the beginnings with the suggestion that development of symbols and constants is a necessary step in its use.

Actually, the system is already in use in computer studies, where it is called 'hexadecimal;' it has not been developed as an arithmetic or mathematical system. I prefer, I guess for reasons of personal laziness, the three-syllable term 'dioctal' as opposed to the five-syllable term 'hexadecimal.'

When I was in grade school I was told that the decimal system arose in counting because of the digits on the two hands -- 10. That there were eight fingers or that digits on hands plus feet totalled 20 escaped my notice at the time, but I have understood that the 20 was the basis for an attempt to introduce a 'vegisimal' system of counting.

Since counting is the underlying basis for arithmetic, let us examine why I contend dioctal may be more natural than decimal. If you use the fingers (only -- not thumbs) on the right hand for counting units, then use the fingers on the left hand for counting the number of repetitions of the four units. You use both hands, just as in decimal, but the use of the second hand is multiplicative instead of additive.

Now, psychologists tell us that, in counting, the sweep of the eye readily embraces two, three or four units, so there is no need for manual counting one at a time. But in counting by fives the eye -- and mind -- breaks the visual sweep into 2's and 3's; counting by 4's is both faster and more accurate.

In dealing with quantities mentally, we half and double quite readily; the decimal system is unnatural in this sense.

Whether economy in expression of quantities -- Dioctal gives a shorter string of digits than decimal just as decimal gives a shorter string than binary -- is adequate justification for switching I will not suggest. Although our computers use binary, it is not only the length of a string but also the difficulty in visually and mentally distinguishing between quantities that discourages adoption of binary arithmetic in our every-day activities. My intuition tells me that the systems based on 20 and on 25 failed to gain support because of such things as multiplication tables and the number of distinct symbols needed to express quantities.

Symbols: Retain zero since invention of the zero was crucial in the development of means of expressing quantities in counting systems. All other symbols may be combinations of horizontal dashes and vertical bars.

For decimal one use a short dash in the middle (of the height) of the line; for two use two dashes joined as in the letter 'c' but somewhat above the line; for three use three dashes joined as in two (or the Greek letter epsilon); for four replace the four joined dashes with a single vertical bar descending from above to the line because too many dashes stacked atop each other may become visually confusing..

Decimal five then becomes a dash appended to the four (similar to a capital 'L' but a suspicion of upward movement before commencing the dash); six is two dashes appended (as in the number two); seven is then three dashes appended (as in the number three). As before, we may avoid using three joined dashes by replacing them with a vertical bar; eight is then a descending bar connected to an ascending bar at a slight angle to the descending bar (similar to a Capital 'v').

Now we have used combinations of bars and dashes with each bar representing its count of decimal four. We can continue that scheme but we can anticipate that decimal twelve will require three bars and 13-15 will require appended dashes. It seems simpler to delete the first bar for digits decimal 9-12 since it is visually obvious whether the pen starts at the line or above the line. Then decimal nine is a dash appended to the ascending vertical bar with a suspicion of downward movement before commencing the dash; ten is then two dashes and 11 is three dashes appended to the vertical bar. Decimal 12 may then be a descending bar attached to the ascending bar, similar to an inverted capital 'v' or an inverted eight.

Had we retained the initial scheme the digits decimal 13-15 would have three vertical bars with attached dashes -- rather complex symbols that could be visually confusing. If we retain the simpler scheme used for decimal 9-11, then decimal 13-15 must consist of two vertical bars with attached dashes, which is still somewhat visually confusing. So, let us introduce another artifact for decimal 13-15. We could simply place the dashes between vertical bars in order to give visual clarity, but there is another, simpler, scheme.

Should dashes precede the vertical bar, there will be both visual clarity and simplicity. So decimal 13 may be a dash preceding a downward vertical bar with a slight upsweep before commencing the bar. And decimal 14 and 15 will have two and three dashes preceding the bar. Decimal 16 would, of course, be a dash for one followed by a zero since it becomes, as in hexadecimal, a two digit number.

For clarity I show here a comparison of symbols for the same quantities in five counting systems.