Triadic Numbers

Introduction

Triadic numbers have digits with values -1, 0, and +1. The centrality of zero is possible with (and only with) any odd-numbered base. In bases 5, 7, 9, etc. the sides "weigh more" than the center. In base 5, for instance, there are two negative values, two positive values, and zero by itself at the center. With base 3, the "mass" of the center is equal to the "mass" of its left and right (- and +) sides. Each of the three parts (-, 0, and +) holds an equal fraction of the whole.

Bases other than ten

We are accustomed to using and thinking with base ten, though other numeric bases are part of our every-day world as well. We have a dozen and a gross (base twelve), minutes and seconds (base sixty, whether for time or angular measure). The computer's binary (base two) numbers and the derivative octal and hexadecimal have made the concept familiar to others. In any base the number of distinct digits is equal to the base. Base two uses zero and one, base ten uses zero through nine. In numbers with two or more digits, the digits to the left of the initial digit indicate multiples of the base to some power. In base two, two is expressed as 10-one times two (to the first power) plus zero times one. Four is 100 (two to the second power), eight is 1000 (two to the third power). In any base x, the value x is expressed as 10-one times x plus zero times one. The number 100 represents the base to the second power, 1000 is the base to the third power. Digits to the right of the units digit (separated by a "base point") represent fractions.

Base Two: Binary (base 2) numbers have only two digits, 0 and 1. Left from the "binary point" are the one's, two's, four's, eight's, etc. positions. Going to the right of the "binary point" are the half's, fourth's, eighth's, etc. The binary number 1101.101 represents 8 + 4 + 0 + 1 + 1/2 + 0/4 + 1/8, or 13.625 (1*10 + 3 + 6/10 + 2/100 + 5/1000) in decimal form.

A triadic base, centered on zero

A variation on the usual way of expressing numbers works only in odd number bases. That is to have digits with negative value, as well as digits with positive value. In base three, the digits would have values one, zero, and minus one. For lack of a better notation, I'll use the symbols [ + 0 - ], respectively. Using these digits, there is no need for a minus sign; its purpose is implicit in one of the digits. Numbers in this version of base three have the same one's, three's, and nine's places (three to the power zero, one, and two) as with the "usual" base three. Thirteen is expressed +++ (nine plus three plus one). Negative thirteen is expressed ---. Seven is expressed +-+ (nine minus three plus one).

One property of this form of number (with the range of digit values centered on zero) that differs from conventional number systems is that truncating a number is identical to rounding it. If we lop off the fractional part of a decimal number, getting, for instance, 12, the unknown digits might represent a number closer to 13 (as with 12.6 or 12.95) or to 12 (as with 12.1 or 12.495). We have no way of knowing. The same sort of uncertainty occurs at any level of precision.
That doesn't happen with the new scheme. The number +.---... (1 - 1/3 - 1/9 - 1/27 - ...) when extended to any finite number of places will never get as small as 1/2, though one can get as close to 1/2 as one wants by using enough digits. Likewise, the number +.+++... (1 + 1/3 + 1/9 + 1/27 + ...) never gets as large as 1 1/2, though, again, one can get arbitrarily close with enough digits.

A zero-centered, auto-rounding base 3 number system

In the May, 1993 issue of Dramatic University, Anthony Blake made brief mention of a triadic computer in his talk (p. 36, first paragraph). I proposed to Tony that a number system such as what I have just discussed might be right for a triadic computer. If a complex calculation were carried out using this number system, each component calculation could be carried out concurrently, feeding into the final result. Components having less effect on the overall result could be calculated to less precision than the more important components. Perhaps nature works this way.

A puzzle solved using a base 3 number system

I applied the above form of base 3 to a math puzzle discussed by Ken Pledge in the May, 1993 issue of Dramatic University (what he refers to as a 'reversed notation' on pages 23-25). The puzzle gives you a balance scale and some coins, from which you are to distinguish which coin is counterfeit and whether it is heavy or light. And, you're to use as few weighings with the scale as possible. If the scale balances, you know all the coins on the scale are of the proper weight. If it does not, then either a coin on the heavy side is too heavy, or a coin on the light side is too light. Either could cause the imbalance. As I re-phrased the problem: If you have 13 coins, of which no more than one is counterfeit (too heavy or too light), and a 14th coin, known to be authentic, how many weighings will it take to determine if a coin is bad, and if so which, and whether it is heavy or light. There are 27 possible outcomes (all coins OK, coin one to thirteen light, coin one to thirteen heavy).

Three uses of the scale produces 3*3, or 27, results - making it theoretically possible to solve the above problem in three weighings.
A hint to solving the puzzle: arrange the digits expressing one to thirteen in three columns. Each column (9's, 3's, 1's) determines what gets weighed at one time. Each row determines whether a given coin will be weighed, and if so, on which side of the scale. To see the rest of the solution, go to the answer page.