Balanced Ternary

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.)

Balanced ternary
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:

  1. 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, 0213 + 113 = 1023, 1023 − 113 = 1T1Balt = 710.
  2. If a two is present in ternary, simply turn it into 1T.For example, 02123 = 0010+1T00+001T=10TTBalt=2310
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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
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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

Other articles related to "balanced ternary, balanced, ternary":

Balanced Ternary - Other Applications
... Balanced ternary has other applications besides computing ... for each power of 3 through 81, a 60-gram object (6010 = 1T1T0) 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 ...
Signed-digit Representation - Balanced Form
... In balanced form, the digits are drawn from a range to, where typically ... For balanced forms, odd base numbers are advantageous ... A notable example is balanced ternary, where the base is, and the numerals have the values −1, 0 and +1 (rather than 0, 1, and 2 as in the standard ternary numeral system) ...
Ternary Computer - Balanced Ternary
... 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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