Balanced ternary
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Balanced ternary is a non-standard positional numeral system (a balanced form), useful for comparison logic. It is a ternary system, but unlike the standard (unbalanced) ternary system, the digits have the values −1, 0, and 1. This combination is especially valuable for ordinal relationships between two values, where the three possible relationships are less-than, equals, and greater-than. Balanced ternary can represent all integers without resorting to a separate minus sign.
Balanced ternary is counted as follows. (In this example, the symbol 1 denotes the digit −1, but alternatively for easier parsing "−" may be used to denote −1 and "+" to denote +1.)
| Decimal | −6 | −5 | −4 | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Balanced ternary | 110 | 111 | 11 | 10 | 11 | 1 | 0 | 1 | 11 | 10 | 11 | 111 | 110 |
Unbalanced ternary can be converted to balanced ternary notation by adding 1111.. with carry, then subtracting 1111... without borrow (the string of 1s must be the same length, so if the result of the addition has more digits, subtract nothing from these extra digits). For example, 0213 + 1113 = 2023, 2023 − 1113 = 1113(bal) = 710.
Balanced ternary is easily represented as electronic signals, as potential can either be negative, neutral, or positive. Utilizing a third state encompasses more data per digit; linearly approximately 1.5849 (log23) bits per trit.
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[edit] Applications
Balanced ternary has other applications besides computing. For example, a classical "2-pan" balance, with one weight for each power of 3, can weigh relatively heavy objects accurately with a small number of weights, by moving weights between the two pans and the table. For example, with weights for each power of 3 through 81, a 60-gram object (6010 = 11110) will be balanced perfectly with an 81 gram weight in the other pan, the 27 gram weight in its own pan, the 9 gram weight in the other pan, the 3 gram weight in its own pan, and the 1 gram weight set aside. This is an optimal solution in terms of the number of weights needed to weigh any object.
Similarly, a currency system using ternary values would save visits to the bank — customers would be likely to have exact change, or be able to get a small number of coins in change, and sellers would just occasionally need to deposit a large coin or two. The system works by representing positive values as coins the customer gives the merchant, and negative values as coins the merchant gives the customer. For example, if a merchant sells an item for five zorkmids, the customer would give the merchant a nine-zorkmid coin, and the merchant would give the customer a three-zorkmid coin and a one-zorkmid coin.
[edit] Computation
There were a few experimental Russian computers in the early days of computing that were built with balanced ternary instead of binary. The notation has a number of computational advantages over regular ternary. Particularly, the one-digit multiplication table has no carries in balanced ternary, and the addition table has only two symmetric carries instead of three.
This notation has the property that the leading non-zero digit is the sign of the full number. To compare two numbers, simply compare digits from the most significant to the least significant. The direction of the magnitude compare of the first digits that are different is the direction of the magnitude compare of the full numbers.
A number is divisible by three if the last digit is zero. The quick test for even is the analog of the base ten divide-by-nine test: add up all the digits and repeat until you have a one-digit number; the number is even if the final sum is zero.
[edit] Fractional balanced ternary
Balanced ternary can be extended to fractional numbers similarly to how decimal numbers are written past the decimal point.[1] For instance, ⅓ is 0.1 and ⅔ is 1.1.
Donald Knuth has pointed out that truncation and rounding are the same operation in balanced ternary — they produce exactly the same result. Moreover, there is no ambiguity in rounding (a property shared with other odd bases) since the number ½ is represented by the repeating fraction 0.1 (and 1.1).
Unlike in standard base notation, where integer values and terminating fractions have multiple representations (e.g. in decimal, ¼ = 0.2510 = 0.24910), in balanced ternary such numbers have only one representation (¼ = 0.113(bal)). On the other hand, numbers half of a terminating fraction (i.e. whose denominator is 2 times a power of 3) do have multiple representations e.g. ⅙ = 0.013(bal) = 0.113(bal).
The basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary, except that there are some special considerations on the size compare. Multiply 2 by 10 in ternary and then divide the result by 2 to see the issue. The intermediate results are the same (in reverse order) in the two cases, and so you can use one as a check on the other.
Digit shifts multiply or divide by three instead of by two as in binary.
Multiplication by two can be done by adding a number to itself. Division by two can be done with the same operation count as an add by LeRoy Eide's algorithm (an algorithm that returns the result least significant digit first, instead of most significant digit first as standard division).
As in binary, there are balanced ternary equivalents of shift and add multiplication, and shift and multiply exponentiation algorithms.
A common convention for balanced ternary is to represent the digits by "+" for +1, "−" for −1, and "0" for zero. Using this convention, the multiplication and addition tables are:
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