Balanced ternary
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Balanced ternary is a nonstandard positional numeral system (a balanced form), useful for comparison logic. While it is a ternary (base 3) number system, in the standard (unbalanced) ternary system, digits have values 0, 1 and 2. The digits in the balanced ternary system have values −1, 0, and 1.
Different sources use different glyphs used to represent the three digits in balanced ternary. In this article, T (which resembles a ligature of the minus sign and 1) represents −1, while 0 and 1 represent themselves. Other conventions include using '−' and '+' to represent −1 and 1 respectively, or using Greek letter theta (Θ), which resembles a minus sign in a circle, to represent −1.
In Setun printings, −1 is represented as overturned 1: "1".^{[1]}
Computational properties
In the early days of computing, a few experimental Soviet computers were built with balanced ternary instead of binary, the most famous being the Setun, built by Nikolay Brusentsov and Sergei Sobolev. The notation has a number of computational advantages over regular binary. Particularly, the plusminus consistency cuts down the carry rate in multidigit multiplication, and the roundingtruncation equivalence cuts down the carry rate in rounding on fractions.
Balanced ternary also has a number of computational advantages over traditional ternary. Particularly, the onedigit multiplication table has no carries in balanced ternary, and the addition table has only two symmetric carries instead of three.
A possible use of balanced ternary is to represent if a list of values in a list is less than, equal to or greater than the corresponding value in a second list. Balanced ternary can also represent all integers without using a separate minus sign; the value of the leading nonzero digit of a number has the sign of the number itself.
Conversion to decimal
In the balanced ternary system the value of a digit n places left of the radix point is the product of the digit and 3^{n}. This is useful when converting between decimal and balanced ternary. In the following the strings denoting balanced ternary carry the suffix, bal3. For instance,
 10_{bal3} = 1×3^{1} + 0×3^{0} = 3_{dec}
 10T_{bal3} = 1×3^{2} + 0×3^{1} + −1×3^{0} = 8_{dec}
 −9_{dec} = −1×3^{2} + 0×3^{1} + 0×3^{0} = T00_{bal3}
 8_{dec} = 1×3^{2} + 0×3^{1} + −1×3^{0} = 10T_{bal3}
Similarly, the first place to the right of the radix point holds 3^{−1} = 1/3, the second place to the right of the decimal place holds 3^{−2} = 1/9, and so on. For instance,
 −2/3_{dec} = −1 + 1/3 = −1×3^{0} + 1×3^{−1} = T.1_{bal3}.
Dec  Bal3  Expansion  Dec  Bal3  Expansion  Dec  Bal3  Expansion  Dec  Bal3  Expansion  Dec  Bal3  Expansion 

−12  TT0  −9−3  −7  T1T  −9+3−1  −2  T1  −3+1  3  10  +3  8  10T  +9−1 
−11  TT1  −9−3+1  −6  T10  −9+3  −1  T  −1  4  11  +3+1  9  100  +9 
−10  T0T  −9−1  −5  T11  −9+3+1  0  0  0  5  1TT  +9−3−1  10  101  +9+1 
−9  T00  −9  −4  TT  −3−1  1  1  +1  6  1T0  +9−3  11  11T  +9+3−1 
−8  T01  −9+1  −3  T0  −3  2  1T  +3−1  7  1T1  +9−3+1  12  110  +9+3 
An integer is divisible by three if and only if the digit in the units place is zero.
We may check the parity of a balanced ternary integer by checking the parity of the sum of all trits. This sum has the same parity as the integer itself.
Balanced ternary can also be extended to fractional numbers similar to how decimal numbers are written to the right of the radix point.^{[2]}
Decimal  −0.9  −0.8  −0.7  −0.6  −0.5  −0.4  −0.3  −0.2  −0.1  0 

Balanced Ternary  T.010T  T.1TT1  T.10T0  T.11TT  0.T or T.1  0.TT11  0.T010  0.T11T  0.0T01  0 
Decimal  0.9  0.8  0.7  0.6  0.5  0.4  0.3  0.2  0.1  0 
Balanced Ternary  1.0T01  1.T11T  1.T010  1.TT11  0.1 or 1.T  0.11TT  0.10T0  0.1TT1  0.010T  0 
In decimal or binary, integer values and terminating fractions have multiple representations. For example, = 0.1 = 0.10 = 0.09. And, = 0.1_{bin} = 0.10_{bin} = 0.01_{bin}. Some balanced ternary fractions have multiple representations too. For example, = 0.1T_{bal3} = 0.01_{bal3}. Certainly, in the decimal and binary, we may omit the rightmost trailing infinite 0s after the radix point and gain a representations of integer or terminating fraction. But, in balanced ternary, we can't omit the rightmost trailing infinite 1s after the radix point in order to gain a representations of integer or terminating fraction.
Donald Knuth has pointed out that truncation and rounding are the same operation in balanced ternary — they produce exactly the same result (a property shared with other balanced numeral systems). The number 1/2 is not exceptional; it has two equally valid representations, and two equally valid truncations: 0.1 (round to 0, and truncate to 0) and 1.T (round to 1, and truncate to 1).
The basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary. Multiplication by two can be done by adding a number to itself, or subtracting itself after atritleftshifting.
An arithmetic shift left of a balanced ternary number is the equivalent of multiplication by a (positive, integral) power of 3; and an arithmetic shift right of a balanced ternary number is the equivalent of division by a (positive, integral) power of 3.
Conversion to and from a fraction
Fraction  Balanced ternary  Fraction  Balanced ternary  

1/1  1  1/11  0.01T11  
1/2  0.1  1.T  1/12  0.01T  
1/3  0.1  1/13  0.01T  
1/4  0.1T  1/14  0.01T0T1  
1/5  0.1TT1  1/15  0.01TT1  
1/6  0.01  0.1T  1/16  0.01TT  
1/7  0.0110TT  1/17  0.01TTT10T0T111T01  
1/8  0.01  1/18  0.001  0.01T  
1/9  0.01  1/19  0.00111T10100TTT1T0T  
1/10  0.010T  1/20  0.0011 
The conversion of a repeating balanced ternary number to a fraction is analogous to converting a repeating decimal. For example (because of ):
Irrational numbers
As in any other integer base, algebraic irrationals and transcendental numbers do not terminate or repeat. For example:
Conversion from ternary
Unbalanced ternary can be converted to balanced ternary notation in two ways:
 Add 1 tritbytrit from the first nonzero trit with carry, and then subtract 1 tritbytrit from the same trit without borrow. For example,
 021_{3} + 11_{3} = 102_{3}, 102_{3} − 11_{3} = 1T1_{bal3} = 7_{dec}.
 If a 2 is present in ternary, turn it into 1T. For example,
 0212_{3} = 0010_{bal3} + 1T00_{bal3} + 001T_{bal3} = 10TT_{bal3} = 23_{dec}
Balanced  Logic  Unsigned 

1  True  2 
0  Unknown  1 
T  False  0 
If the three values of ternary logic are false, unknown and true, and these are mapped to balanced ternary as T, 0 and 1 and to conventional unsigned ternary values as 0, 1 and 2, then balanced ternary can be viewed as a biased number system analogous to the offset binary system. If the ternary number has trits, then the bias is which is represented in as all ones in either conventional or biased form.^{[3]}
As a result, if these two representations are used for balanced and unsigned ternary numbers, an unsigned trit positive ternary value can be converted to balanced form by adding the bias and a positive balanced number can be converted to unsigned form by subtracting the bias . Furthermore, if and are balanced numbers, their balanced sum is when computed using conventional unsigned ternary arithmetic. Similarly, if and are conventional unsigned ternary numbers, their sum is when computed using balanced ternary arithmetic.
Conversion to balanced ternary from any integer base
We may convert to balanced ternary with the following formula:
where,
 is the original representation in the original numeral system.
 b is the original radix. b is 10 if converting from decimal.
 and are the digits k places to the left and right of the radix point respectively.
For instance,
−25.4_{dec} = −(1T×101^{1} + 1TT×101^{0} + 11×101^{−1}) = −(1T×101 + 1TT + 11÷101) = −10T1.11TT = T01T.TT11
1010.1_{bin} = 1T^{100} + 1T^{1} + 1T^{−1} = 10T + 1T + 0.1 = 101.1
Addition, subtraction and multiplication and division
The singletrit addition, subtraction, multiplication and division tables are shown below. For subtraction and division, which are not commutative, the first operand is given to the left of the table, while the second is given at the top. For instance, the answer to 1T=1T is found in the bottom left corner of the subtraction table.




Multitrit addition and subtraction
Multitrit addition and subtraction is analogous to that of binary and decimal. Add and subtract trit by trit, and add the carry appropriately. For example:
1TT1TT.1TT1 1TT1TT.1TT1 1TT1TT.1TT1 1TT1TT.1TT1 + 11T1.T  11T1.T  11T1.T > + TT1T.1    1T0T10.0TT1 1T1001.TTT1 1T1001.TTT1 + 1T + T T + T T    1T1110.0TT1 1110TT.TTT1 1110TT.TTT1 + T + T 1 + T 1    1T0110.0TT1 1100T.TTT1 1100T.TTT1
Multitrit multiplication
Multitrit multiplication is analogous to that in decimal and binary.
1TT1.TT × T11T.1  1TT.1TT multiply 1 T11T.11 multiply T 1TT1T.T multiply 1 1TT1TT multiply 1 T11T11 multiply T  0T0000T.10T
Multitrit division
Balanced ternary division is analogous to decimal or binary division.
However, 0.5_{dec} = 0.1111..._{bal3} or 1.TTTT..._{bal3}. If the dividend over the plus or minus half divisor, the trit of the quotient must be 1 or T. If the dividend is between the plus and minus of half the divisor, the trit of the quotient is 0. The magnitude of the dividend must be compared with that of half the divisor before setting the quotient trit. For example,
1TT1.TT quotient 0.5 × divisor T01.0  divisor T11T.1 ) T0000T.10T dividend T11T1 T000<T010, set 1  1T1T0 1TT1T 1T1T0>10T0, set T  111T 1TT1T 1110>10T0, set T  T00.1 T11T.1 T001<T010, set 1  1T1.00 1TT.1T 1T100>10T0, set T  1T.T1T 1T.T1T 1TT1T>10T0, set T  0
Another example,
1TTT 0.5 × divisor 1T  Divisor 11 )1T01T 1T=1T, but 1T.01>1T, set 1 11  T10 T10<T1, set T TT  T11 T11<T1, set T TT  TT TT<T1, set T TT  0
Another example,
101.TTTTTTTTT… or 100.111111111… 0.5 × divisor 1T  divisor 11 )111T 11>1T, set 1 11  1 T1<1<1T, set 0  1T 1T=1T, trits end, set 1.TTTTTTTTT… or 0.111111111…
Square roots and cube roots
The process of extracting the square root in balanced ternary is analogous to that in decimal or binary.
As in division, we should check the value of half the divisor first. For example,
1. 1 1 T 1 T T 0 0 ...  √ 1T 1<1T<11, set 1  1  1×10=10 1.0T 1.0T>0.10, set 1 1T0 1.T0  11×10=110 1T0T 1T0T>110, set 1 10T0 10T0  111×10=1110 T1T0T T1T0T<TTT0, set T 100T0 T0010  111T×10=111T0 1TTT0T 1TTT0T>111T0, set 1 10T110 10T110  111T1×10=111T10 TT1TT0T TT1TT0T<TTT1T0, set T 100TTT0 T001110  111T1T×10=111T1T0 T001TT0T T001TT0T<TTT1T10, set T 10T11110 T01TTTT0  111T1TT×10=111T1TT0 T001T0T TTT1T110<T001T0T<111T1TT0, set 0  T Return 1  111T1TT0×10=111T1TT00 T001T000T TTT1T1100<T001T000T<111T1TT00, set 0  T Return 1  111T1TT00*10=111T1TT000 T001T00000T ...
Extraction of the cube root in balanced ternary is similarly analogous to extraction in decimal or binary:
Like division, we should check the value of half the divisor first too. For example:
1. 1 T 1 0... 3 √ 1T  1 1<1T<10T,set 1  1.000 1×100=100 0.100 borrow 100×, do division  1TT 1.T00 1T00>1TT, set 1 1×1×1000+1=1001 1.001  T0T000 11×100  1100 borrow 100×, do division  10T000 TT1T00 TT1T00<T01000, set T 11×11×1000+1=1TT1001 T11T00T  1TTT01000 11T×100  11T00 borrow 100×, do division  1T1T01TT 1TTTT0100 1TTTT0100>1T1T01TT, set 1 11T×11T×1000+1=11111001  11111001  1T10T000 11T1×100  11T100 borrow 100×, do division  10T0T01TT 1T0T0T00 T01010T11<1T0T0T00<10T0T01TT, set 0 11T1×11T1×1000+1=1TT1T11001  TT1T00 return 100×  1T10T000000 ...
Hence ^{3}√2 = 1.259921_{dec} = 1.1T1 000 111 001 T01 00T 1T1 T10 111_{bal3}.
Other applications
Balanced ternary has other applications besides computing. For example, a classical twopan 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 60gram object (60_{dec} = 1T1T0_{bal3}) 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.
Similarly, consider a currency system with coins worth 1¤, 3¤, 9¤, 27¤, 81¤. If the buyer and the seller each have only one of each kind of coin, any transaction up to 121¤ is possible. For example, if the price is 7¤ (7_{dec} = 1T1_{bal3}), the buyer pays 1¤ + 9¤ and receives 3¤ in change.
See also
 Setun, a ternary computer
 Ternary logic
 Numeral system
 Methods of computing square roots
 Salamis Tablet
Notes
 ↑ N.A.Krinitsky; G.A.Mironov; G.D.Frolov (1963). "Chapter 10. Programcontrolled machine Setun". In M.R.ShuraBura. Programming (in Russian). Moscow.
 ↑ Bhattacharjee, Abhijit (24 July 2006). "Balanced ternary". Archived from the original on 20090919.
 ↑ Douglas W. Jones, Ternary Number Systems, Oct 15, 2013.
References
 Hayes, Brian (2001), "Third base", American Scientist, 89 (6): 490–494, doi:10.1511/2001.40.3268.
External links
 Balanced Ternary Calculator
 Development of ternary computers at Moscow State University
 Representation of Fractional Numbers in Balanced Ternary
 “Third base”, ternary and balanced ternary number systems
 The Balanced Ternary Number System (includes decimal integer to balanced ternary converter)
 The binomial triangle reduced to balanced ternary lists. OEIS A182929
 Balanced (Signed) Ternary Notation by Brian J. Shelburne (PDF file)
 The ternary calculating machine of Thomas Fowler by Mark Glusker