Balanced ternary is a non-standard positional numeral system (a balanced form), useful for comparison logic. It is a ternary (base 3) number 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 letter "T" is uses as a ligature of "−1" in balanced ternary, but alternatively for easier parsing "−" may be used to denote −1 and "+" to denote +1.)
Decimal | −13 | −12 | −11 | −10 | −9 | −8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Balanced ternary | TTT | TT0 | TT1 | T0T | T00 | T01 | T1T | T10 | T11 | TT | T0 | T1 | T | 0 | 1 | 1T | 10 | 11 | 1TT | 1T0 | 1T1 | 10T | 100 | 101 | 11T | 110 | 111 |
Concise representation | F | G | H | J | K | L | M | N | P | Q | R | S | T | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D |
The speed of the light in vacuum is 299,792,458 metres per second. The number is 1'T10'0T0'010'001'0T1'101(1'NR2'1SA) in balanced ternary.
Unbalanced ternary can be converted to balanced ternary notation in two ways:
- add 1 trit-by-trit from the first non-zero trit with carry, and then subtract 1 trit-by-trit from the same trit without borrow. For example, 021_{3} + 11_{3} = 102_{3}, 102_{3} − 11_{3} = 1T1_{Balt} = 7_{10}.
- If a two is present in ternary, simply turn it into 1T.For example, 0212_{3} = 0010+1T00+001T=10TT_{Balt}=23_{10}
Numeral systems by culture | |
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Hindu-Arabic numerals | |
Western Arabic Eastern Arabic Indian family Tamil |
Burmese Khmer Lao Mongolian Thai |
East Asian numerals | |
Chinese Japanese Suzhou |
Korean Vietnamese Counting rods |
Alphabetic numerals | |
Abjad Armenian Āryabhaṭa Cyrillic |
Ge'ez Greek Georgian Hebrew |
other historical systems | |
Aegean Attic Babylonian Brahmi Egyptian Etruscan Inuit |
Kharosthi Mayan Quipu Roman |
Positional systems by base | |
Decimal (10) | |
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 24, 26, 27, 32, 36, 60, 64, 85 | |
Balanced ternary | |
List of numeral systems | |
Read more about Balanced Ternary: Computation, Fractional Balanced Ternary, Irrational Numbers, Transcendental Numbers, Convert A Number To Balanced Ternary, Addition, Subtraction and Multiplication and Division, Expand The Balanced Ternary To 2D, Other Applications
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... Ternary computing is commonly implemented in terms of balanced ternary, which uses the three digits −1, 0, and +1 ... The negative value of any balanced ternary digit can be obtained by replacing every + with a − and vice versa ... Balanced ternary can express negative values as easily as positive ones, without the need for a leading negative sign as with decimal numbers ...
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