Balanced ternary is a non-standard positional numeral system (a balanced form), used in some early computers and useful in the solution of balance puzzles. It is a ternary (base 3) number system in which the digits have the values –1, 0, and 1, in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2. Balanced ternary can represent all integers without using a separate minus sign; the value of the leading non-zero digit of a number has the sign of the number itself. While binary numerals with digits 0 and 1 provide the simplest positional numeral system for natural numbers (or for positive integers if using 1 and 2 as the digits), balanced ternary provides the simplest self-contained[definition needed] positional numeral system for integers.
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 publications about the Setun computer, −1 is represented as overturned 1: "1".
Balanced ternary makes an early appearance in Michael Stifel's book Arithmetica Integra (1544). It also occurs in the works of Johannes Kepler and Léon Lalanne. Related signed-digit schemes in other bases have been discussed by John Colson, John Leslie, Augustin-Louis Cauchy, and possibly even the ancient Indian Vedas.
In computer design
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 traditional binary and ternary. Particularly, the plus–minus consistency cuts down the carry rate in multi-digit multiplication, and the rounding–truncation equivalence cuts down the carry rate in rounding on fractions. The one-digit multiplication table has no carries in balanced ternary, and the addition table has only two symmetric carries instead of three.
Because balanced ternary provides a uniform self-contained representation for integers, the distinction between signed and unsigned numerals no longer needs to be made; thereby eliminating the need to duplicate operator sets into signed and unsigned varieties, as most CPU architectures and many programming languages currently do.[dubious ]
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 3n. This is useful when converting between decimal and balanced ternary. In the following the strings denoting balanced ternary carry the suffix, bal3. For instance,
- 10bal3 = 1 × 31 + 0 × 30 = 310
- 10ᴛbal3 = 1 × 32 + 0 × 31 + (−1) × 30 = 810
- −910 = −1 × 32 + 0 × 31 + 0 × 30 = ᴛ00bal3
- 810 = 1 × 32 + 0 × 31 + (−1) × 30 = 10ᴛbal3
Similarly, the first place to the right of the radix point holds 3−1 = 1/, the second place holds 3−2 = 1/, and so on. For instance,
- −2/10 = −1 + 1/ = −1 × 30 + 1 × 3−1 = ᴛ.1bal3.
Dec Bal3 Expansion Dec Bal3 Expansion 0 0 0 1 1 +1 −1 ᴛ −1 2 1ᴛ +3−1 −2 ᴛ1 −3+1 3 10 +3 −3 ᴛ0 −3 4 11 +3+1 −4 ᴛᴛ −3−1 5 1ᴛᴛ +9−3−1 −5 ᴛ11 −9+3+1 6 1ᴛ0 +9−3 −6 ᴛ10 −9+3 7 1ᴛ1 +9−3+1 −7 ᴛ1ᴛ −9+3−1 8 10ᴛ +9−1 −8 ᴛ01 −9+1 9 100 +9 −9 ᴛ00 −9 10 101 +9+1 −10 ᴛ0ᴛ −9−1 11 11ᴛ +9+3−1 −11 ᴛᴛ1 −9−3+1 12 110 +9+3 −12 ᴛᴛ0 −9−3 13 111 +9+3+1 −13 ᴛᴛᴛ −9−3−1
An integer is divisible by three if and only if the digit in the units place is zero.
Decimal −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 Balanced Ternary ᴛ.010ᴛ ᴛ.1ᴛᴛ1 ᴛ.10ᴛ0 ᴛ.11ᴛᴛ 0.ᴛ or ᴛ.1 0.ᴛᴛ11 0.ᴛ010 0.ᴛ11ᴛ 0.0ᴛ01 0 Decimal 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Balanced Ternary 1.0ᴛ01 1.ᴛ11ᴛ 1.ᴛ010 1.ᴛᴛ11 0.1 or 1.ᴛ 0.11ᴛᴛ 0.10ᴛ0 0.1ᴛᴛ1 0.010ᴛ 0
In decimal or binary, integer values and terminating fractions have multiple representations. For example, 1/ = 0.1 = 0.10 = 0.09. And, 1/ = 0.12 = 0.102 = 0.012. Some balanced ternary fractions have multiple representations too. For example, 1/ = 0.1ᴛbal3 = 0.01bal3. 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/ 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.ᴛ (round to 1, and truncate to 1). With an odd radix, double rounding is also equivalent to directly rounding to the final precision, unlike with an even radix.
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 a-trit-left-shifting.
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/ 0.01ᴛ11 1/ 0.1 1.ᴛ 1/ 0.01ᴛ 1/ 0.1 1/ 0.01ᴛ 1/ 0.1ᴛ 1/ 0.01ᴛ0ᴛ1 1/ 0.1ᴛᴛ1 1/ 0.01ᴛᴛ1 1/ 0.01 0.1ᴛ 1/ 0.01ᴛᴛ 1/ 0.0110ᴛᴛ 1/ 0.01ᴛᴛᴛ10ᴛ0ᴛ111ᴛ01 1/ 0.01 1/ 0.001 0.01ᴛ 1/ 0.01 1/ 0.00111ᴛ10100ᴛᴛᴛ1ᴛ0ᴛ 1/ 0.010ᴛ 1/ 0.0011
The conversion of a repeating balanced ternary number to a fraction is analogous to converting a repeating decimal. For example (because of 111111bal3 = (36 − 1/)10):
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 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 = 1T1bal3 = 710.
- If a 2 is present in ternary, turn it into 1T. For example,
- 02123 = 0010bal3 + 1T00bal3 + 001Tbal3 = 10TTbal3 = 2310
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 n trits, then the bias b is
which is represented as all ones in either conventional or biased form.
As a result, if these two representations are used for balanced and unsigned ternary numbers, an unsigned n-trit positive ternary value can be converted to balanced form by adding the bias b and a positive balanced number can be converted to unsigned form by subtracting the bias b. Furthermore, if x and y are balanced numbers, their balanced sum is x + y − b when computed using conventional unsigned ternary arithmetic. Similarly, if x and y are conventional unsigned ternary numbers, their sum is x + y + b when computed using balanced ternary arithmetic.
Conversion to balanced ternary from any integer base
We may convert to balanced ternary with the following formula:
- anan−1...a1a0.c1c2c3... is the original representation in the original numeral system.
- b is the original radix. b is 10 if converting from decimal.
- ak and ck are the digits k places to the left and right of the radix point respectively.
−25.410 = −(1T×1011 + 1TT×1010 + 11×101−1) = −(1T×101 + 1TT + 11÷101) = −10T1.11TT = T01T.TT11
1010.12 = 1T10 + 1T1 + 1T−1 = 10T + 1T + 0.1 = 101.1
Addition, subtraction and multiplication and division
The single-trit 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 1 − T = 1T is found in the bottom left corner of the subtraction table.
Addition + T 0 1 T T1 T 0 0 T 0 1 1 0 1 1T Subtraction − T 0 1 T 0 T T1 0 1 0 T 1 1T 1 0 Multiplication × T 0 1 T 1 0 T 0 0 0 0 1 T 0 1 Division ÷ T 1 T 1 T 0 0 0 1 T 1
Multi-trit addition and subtraction
Multi-trit 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 T1 + T T ______________ ________________ ________________ 1T1110.0TT1 1110TT.TTT1 1110TT.TTT1 + T + T 1 + T 1 ______________ ________________ ________________ 1T0110.0TT1 1100T.TTT1 1100T.TTT1
Multi-trit multiplication is analogous to that of binary and decimal.
1TT1.TT × T11T.1 _____________ 1TT.1TT multiply 1 T11T.11 multiply T 1TT1T.T multiply 1 1TT1TT multiply 1 T11T11 multiply T _____________ 0T0000T.10T
Balanced ternary division is analogous to that of binary and decimal.
However, 0.510 = 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 111T > 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
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
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√ = 1.25992110 = 1.1T1 000 111 001 T01 00T 1T1 T10 111bal3.
Balanced ternary has other applications besides computing. For example, a classical two-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 = 1T1T0bal3) 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¤ (710 = 1T1bal3), the buyer pays 1¤ + 9¤ and receives 3¤ in change.
They may also provide a more natural representation for the Qutrit and systems that use it.
- Setun, a ternary computer
- Ternary logic
- Numeral system
- Methods of computing square roots
- Salamis Tablet
- N.A.Krinitsky; G.A.Mironov; G.D.Frolov (1963). "Chapter 10. Program-controlled machine Setun". In M.R.Shura-Bura (ed.). Programming (in Russian). Moscow.
- Hayes, Brian (2001), "Third base" (PDF), American Scientist, 89 (6): 490–494, doi:10.1511/2001.40.3268. Reprinted in Hayes, Brian (2008), Group Theory in the Bedroom, and Other Mathematical Diversions, Farrar, Straus and Giroux, pp. 179–200, ISBN 9781429938570
- Stifel, Michael (1544), Arithmetica integra (in Latin), p. 38.
- Bhattacharjee, Abhijit (24 July 2006). "Balanced ternary". Archived from the original on 2009-09-19.
- Knuth, Donald (1997). The art of Computer Programming. 2. Addison-Wesley. pp. 195–213. ISBN 0-201-89684-2.
- Douglas W. Jones, Ternary Number Systems, October 15, 2013.
- Andrews, George E. (2007). "Euler's "De Partitio numerorum"". Bulletin of the American Mathematical Society. New Series. 44 (4): 561–573. doi:10.1090/S0273-0979-07-01180-9. MR 2338365.
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