{"history":[],"article":{"id":"616ae5ed-d111-4d80-8f3e-d664648c490a","title":"Balanced ternary","slug":"balanced-ternary","content":":::summary\n:::\n\n# \n**Balanced [ternary](https://bliptext.com/articles/ternary-numeral-system)** is a [ternary](https://bliptext.com/articles/ternary-numeral-system) numeral system (i.e. base 3 with three [digits](https://bliptext.com/articles/numerical-digit)) that uses a balanced [signed-digit representation](https://bliptext.com/articles/signed-digit-representation) of the [integers](https://bliptext.com/articles/integer) in which the [digits](https://bliptext.com/articles/numerical-digit) have the values [−1](https://bliptext.com/articles/[1](https://bliptext.com/articles/1)), [0](https://bliptext.com/articles/0), and [1](https://bliptext.com/articles/1).\nThis stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2.\nThe balanced ternary system can represent all integers without using a separate [minus sign](https://bliptext.com/articles/minus-sign); the value of the leading non-zero digit of a number has the sign of the number itself.\nThe balanced ternary system is an example of a [non-standard positional numeral system](https://bliptext.com/articles/non-standard-positional-numeral-systems).\nIt was used in some early computers and has also been used to solve [balance puzzles](https://bliptext.com/articles/balance-puzzle).\nDifferent sources use different glyphs to represent the three digits in balanced ternary.\nIn this article, T (which resembles a [ligature](https://bliptext.com/articles/typographical-ligature) of the minus sign and [1](https://bliptext.com/articles/1)) represents [−1](https://bliptext.com/articles/[1](https://bliptext.com/articles/1)), while [0](https://bliptext.com/articles/0) and [1](https://bliptext.com/articles/1) represent themselves.\nOther conventions include using '−' and '+' to represent −1 and 1 respectively, or using [Greek letter](https://bliptext.com/articles/greek-alphabet) [theta](https://bliptext.com/articles/theta) (Θ), which resembles a minus sign in a circle, to represent −1.\nIn publications about the [Setun](https://bliptext.com/articles/setun) computer, −1 is represented as overturned 1: \" 1 \".\nBalanced ternary makes an early appearance in [Michael Stifel](https://bliptext.com/articles/michael-stifel)'s book Arithmetica Integra (1544).\nIt also occurs in the works of [Johannes Kepler](https://bliptext.com/articles/johannes-kepler) and [Léon Lalanne](https://bliptext.com/articles/l-on-lalanne).\nRelated signed-digit schemes in other bases have been discussed by [John Colson](https://bliptext.com/articles/john-colson), [John Leslie](https://bliptext.com/articles/john-leslie-physicist), [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy), and possibly even the ancient Indian [Vedas](https://bliptext.com/articles/vedas).\n\n# Definition\nLet denote the set of [symbols](https://bliptext.com/articles/symbol-mathematics) (also called glyphs or characters), where the symbol \\bar{1} is sometimes used in place of Define an [integer](https://bliptext.com/articles/integer)-valued function by\nwhere the right hand sides are integers with their usual values.\nThis function, f_{}, is what rigorously and formally establishes how integer values are assigned to the symbols/glyphs in One benefit of this formalism is that the definition of \"the integers\" (however they may be defined) is not conflated with any particular system for writing/representing them; in this way, these two distinct (albeit closely related) concepts are kept separate.\nThe set together with the function f_{} forms a balanced [signed-digit representation](https://bliptext.com/articles/signed-digit-representation) called the balanced ternary system.\nIt can be used to represent integers and real numbers.\n\n## Ternary integer evaluation\nLet be the [Kleene plus](https://bliptext.com/articles/kleene-plus) of, which is the set of all finite length [concatenated](https://bliptext.com/articles/concatenation) [strings](https://bliptext.com/articles/string-computer-science) of one or more symbols (called its digits) where n is a non-negative integer and all n + 1 digits are taken from The start of is the symbol d_0 (at the right), its end is d_n (at the left), and its length is n + 1.\nThe ternary evaluation is the function defined by assigning to every string the integer\nThe string represents (with respect to v) the integer The value may alternatively be denoted by The map is [surjective](https://bliptext.com/articles/surjective-map) but not injective since, for example, However, every integer has exactly one representation under v that does not end (on the left) with the symbol 0, i.e. d_n = 0.\nIf and n > 0 then v satisfies:\nwhich shows that v satisfies a sort of [recurrence relation](https://bliptext.com/articles/recurrence-relation).\nThis recurrence relation has the initial condition\nwhere \\varepsilon is the empty string.\nThis implies that for every string\nwhich in words says that leading 0 symbols (to the left in a string with 2 or more symbols) do not affect the resulting value.\nThe following examples illustrate how some values of v can be computed, where (as before) all integer are written in decimal (base 10) and all elements of are just symbols.\nand using the above recurrence relation\n\n# Conversion to decimal\nIn the balanced ternary system the value of a digit n places left of the [radix point](https://bliptext.com/articles/radix-point) is the product of the digit and 3n.\nThis is useful when converting between decimal and balanced ternary.\nIn the following the strings denoting balanced ternary carry the suffix, bal3.\nFor instance,\nSimilarly, the first place to the right of the radix point holds 3−1 = 1⁄3, the second place holds 3−2 = 1⁄9, and so on.\nFor instance,\nAn integer is divisible by three if and only if the digit in the units place is zero.\nWe may check the [parity](https://bliptext.com/articles/parity-mathematics) of a balanced ternary integer by checking the [parity](https://bliptext.com/articles/parity-mathematics) of the sum of all trits.\nThis sum has the same parity as the integer itself.\nBalanced ternary can also be extended to fractional numbers similar to how decimal numbers are written to the right of the [radix point](https://bliptext.com/articles/radix-point).\n! Decimal\n! style=\"text-align: right\" | −0.9\n! style=\"text-align: right\" | −0.8\n! style=\"text-align: right\" | −0.7\n! style=\"text-align: right\" | −0.6\n! style=\"text-align: right\" | −0.5\n! style=\"text-align: right\" | −0.4\n! style=\"text-align: right\" | −0.3\n! style=\"text-align: right\" | −0.2\n! style=\"text-align: right\" | −0.1\n! style=\"text-align: right\" | 0\n! Balanced Ternary\n! Decimal\n! style=\"text-align: right\" | 0.9\n! style=\"text-align: right\" | 0.8\n! style=\"text-align: right\" | 0.7\n! style=\"text-align: right\" | 0.6\n! style=\"text-align: right\" | 0.5\n! style=\"text-align: right\" | 0.4\n! style=\"text-align: right\" | 0.3\n! style=\"text-align: right\" | 0.2\n! style=\"text-align: right\" | 0.1\n! style=\"text-align: right\" | 0\n! Balanced Ternary\nIn decimal or binary, integer values and terminating fractions have multiple representations.\nFor example, 1⁄10 = 0.1 = 0.10 = 0.09.\nAnd, 1⁄2 = 0.12 = 0.102 = 0.012.\nSome balanced ternary fractions have multiple representations too.\nFor example, 1⁄6 = 0.1𝖳bal3 = 0.01bal3.\nCertainly, 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.\nBut, 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.\n[Donald Knuth](https://bliptext.com/articles/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).\nThe 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).\nWith an odd [radix](https://bliptext.com/articles/radix), [double rounding](https://bliptext.com/articles/rounding) is also equivalent to directly rounding to the final precision, unlike with an even [radix](https://bliptext.com/articles/radix).\nThe basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary.\nMultiplication by two can be done by adding a number to itself, or subtracting itself after a-trit-left-shifting.\nAn 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.\n\n# Conversion to and from a fraction\nThe conversion of a repeating balanced ternary number to a fraction is analogous to [converting a repeating decimal](https://bliptext.com/articles/repeating-decimal).\nFor example (because of 111111bal3 = (36 − 1⁄3 − 1)dec):\n\n# Irrational numbers\nAs in any other integer base, algebraic irrationals and transcendental numbers do not terminate or repeat.\nFor example:\n!Decimal !!\nBalanced ternary\nThe balanced ternary expansions of \\pi is given in [OEIS](https://bliptext.com/articles/oeis) as, that of e in.\n\n# Conversion from ternary\nUnbalanced ternary can be converted to balanced ternary notation in two ways:\nIf the three values of [ternary logic](https://bliptext.com/articles/three-valued-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](https://bliptext.com/articles/offset-binary) system.\nIf the ternary number has n trits, then the bias b is which is represented as all ones in either conventional or biased form.\nAs 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.\nFurthermore, if x and y are balanced numbers, their balanced sum is x + y − b when computed using conventional unsigned ternary arithmetic.\nSimilarly, if x and y are conventional unsigned ternary numbers, their sum is x + y + b when computed using balanced ternary arithmetic.\n\n# Conversion to balanced ternary from any integer base\nWe may convert to balanced ternary with the following formula:\nwhere,\nFor instance,\n−25.4dec = −(1T×1011 + 1TT×1010 + 11×101−1)\n= −(1T×101 + 1TT + 11÷101)\n= −10T1.11TT\n= T01T.TT11\n1010.12 = 1T10 + 1T1 + 1T−1\n= 10T + 1T + 0.1\n= 101.1\n\n# Addition, subtraction and multiplication and division\nThe single-trit addition, subtraction, multiplication and division tables are shown below.\nFor subtraction and division, which are not [commutative](https://bliptext.com/articles/commutative-property), the first operand is given to the left of the table, while the second is given at the top.\nFor instance, the answer to 1 − T = 1T is found in the bottom left corner of the subtraction table.\n:{| class=\"wikitable\" style=\"width: 8em; text-align: center;\"\n|+ Addition\n|- align=\"right\"\n! + !! T !!\n0 !! 1 |- |- ! T | T1 || T || 0\n|- ! 0 | T || 0 || 1\n|- ! 1 | 0 || 1 || 1T\n|} :{| class=\"wikitable\" style=\"width: 8em; text-align: center;\"\n|+ Subtraction\n|- align=\"right\"\n! − !! T !!\n0 !! 1 |- ! T | 0 || T || T1 |-\n! 0 | 1 || 0 || T |-\n! 1 | 1T || 1 || 0\n|} :{| class=\"wikitable\" style=\"width: 8em; text-align: center;\"\n|+ Multiplication\n|- align=\"right\"\n! × !! T !!\n0 !! 1 |- |- ! T | 1 || 0 || T |-\n! 0 | 0 || 0 || 0 |- ! 1 | T || 0 || 1\n|} :{| class=\"wikitable\" style=\"text-align: center;\"\n|+ Division\n|- align=\"right\"\n! ÷ !! T !!\n1 |- |- ! T | 1 || T |-\n! 0 | 0 || 0 |- ! 1 | T || 1\n|}\n\n## Multi-trit addition and subtraction\nMulti-trit addition and subtraction is analogous to that of binary and decimal.\nAdd and subtract trit by trit, and add the carry appropriately.\nFor example:\n1TT1TT.1TT1 1TT1TT.1TT1 1TT1TT.1TT1 1TT1TT.1TT1\n+ 11T1.T − 11T1.T − 11T1.T → + TT1T.1\n______________ ______________ _______________ 1T0T10.0TT1 1T1001.TTT1 1T1001.TTT1\n+ 1T + T T1 + T T ______________ ________________ ________________\n1T1110.0TT1 1110TT.TTT1 1110TT.TTT1\n+ T + T 1 + T 1\n______________ ________________ ________________ 1T0110.0TT1 1100T.TTT1 1100T.TTT1\n\n## Multi-trit multiplication\nMulti-trit multiplication is analogous to that of binary and decimal.\n1TT1.TT × T11T.1\n_____________ 1TT.1TT multiply 1\nT11T.11 multiply T 1TT1T.T multiply 1\n1TT1TT multiply 1\nT11T11 multiply T _____________\n0T0000T.10T\n\n## Multi-trit division\nBalanced ternary division is analogous to that of binary and decimal.\nHowever, 0.5dec = 0.1111...bal3 or 1.TTTT...bal3.\nIf 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.\nThe magnitude of the dividend must be compared with that of half the divisor before setting the quotient trit.\nFor example,\n1TT1.TT quotient\n0.5 × divisor T01.0 _____________\ndivisor 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 Another example, 1TTT\n0.5 × divisor 1T _______\nDivisor 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\n11 _____ 1 T1 < 1 < 1T, set 0\n___ 1T 1T = 1T, trits end, set 1.TTTTTTTTT... or 0.111111111...\n\n# Square roots and cube roots\nThe process of extracting the [square root](https://bliptext.com/articles/square-root) in balanced ternary is analogous to that in decimal or binary.\nAs in division, we should check the value of half the divisor first.\nFor example,\n1. 1 1 T 1 T T 0 0 ... _________________________\n√ 1T 1<1T<11, set 1\n− 1 _____ 1×10=10 1.0T 1.0T>0.10, set 1\n1T0 −1.T0 ________\n11×10=110 1T0T 1T0T>110, set 1\n10T0 −10T0\n________ 111×10=1110 T1T0T T1T0T111T0, set 1\n10T110 −10T110\n__________ 111T1×10=111T10 TT1TT0T TT1TT0T1TT, set 1\n1×1×1000+1=1001 −1.001 __________ T0T000\n11×100 − 1100 borrow 100×, do division\n_________ 10T000 TT1T00 TT1T001T1T01TT, set 1\n11T×11T×1000+1=11111001 − 11111001\n______________ 1T10T000\n11T1×100 − 11T100 borrow 100×, do division\n__________ 10T0T01TT 1T0T0T00 T01010T11<1T0T0T00<10T0T01TT, set 0\n11T1×11T1×1000+1=1TT1T11001 − TT1T00 return 100×\n_____________ 1T10T000000\n... Hence 3√2 = 1.259921dec = 1.1T1 000 111 001 T01 00T 1T1 T10 111bal3.\n\n# Applications\n\n## In computer design\nIn the early days of computing, a few experimental Soviet computers were built with balanced ternary instead of binary, the most famous being the [Setun](https://bliptext.com/articles/setun), built by [Nikolay Brusentsov](https://bliptext.com/articles/nikolay-brusentsov) and [Sergei Sobolev](https://bliptext.com/articles/sergei-sobolev).\nThe notation has a number of computational advantages over traditional binary and ternary.\nParticularly, 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.\nIn balanced ternary, the one-digit [multiplication table](https://bliptext.com/articles/multiplication-table) remains one-digit and has no carry and the [addition table](https://bliptext.com/articles/addition-table) has only two carries out of nine entries, compared to unbalanced ternary with one and three respectively.\nKnuth wrote that \"Perhaps the symmetric properties and simple arithmetic of this number system will prove to be quite important some day,\" noting that,\n\"The complexity of arithmetic circuitry for balanced ternary arithmetic is not much greater than it is for the binary system, and a given number requires only as many digit positions for its representation.\"\"\n# [Operation tables | upload.wikimedia.org/wikipedia/commons/4/4c/Balanced///ternary///operation///tables.svg]\n\n## Other applications\nThe theorem that every integer has a unique representation in balanced ternary was used by [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) to justify the identity of [formal power series](https://bliptext.com/articles/formal-power-series)\nBalanced ternary has other applications besides computing.\nFor example, a classical two-pan [balance](https://bliptext.com/articles/weighing-scale), 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.\nFor example, with weights for each power of 3 through 81, a 60-gram object (60dec = 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.\nSimilarly, consider a currency system with coins worth 1¤, 3¤, 9¤, 27¤, 81¤.\nIf the buyer and the seller each have only one of each kind of coin, any transaction up to 121¤ is possible.\nFor example, if the price is 7¤ (7dec = 1T1bal3), the buyer pays 1¤ + 9¤ and receives 3¤ in change.\nThey may also provide a more natural representation for the [qutrit](https://bliptext.com/articles/qutrit) and systems that use it.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:23:18.716Z","updated_at":"2025-01-25T19:23:18.716Z","search_vector":"'balanc':1 'ternari':2","search_content":" \n \n\n# \n**Balanced [ternary](https://bliptext.com/articles/ternary-numeral-system)** is a [ternary](https://bliptext.com/articles/ternary-numeral-system) numeral system (i.e. base 3 with three [digits](https://bliptext.com/articles/numerical-digit)) that uses a balanced [signed-digit representation](https://bliptext.com/articles/signed-digit-representation) of the [integers](https://bliptext.com/articles/integer) in which the [digits](https://bliptext.com/articles/numerical-digit) have the values [−[1](https://bliptext.com/articles/1)](https://bliptext.com/articles/[1](https://bliptext.com/articles/1)), [0](https://bliptext.com/articles/0), and [1](https://bliptext.com/articles/1).\nThis stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2.\nThe balanced ternary system can represent all integers without using a separate [minus sign](https://bliptext.com/articles/minus-sign); the value of the leading non-zero digit of a number has the sign of the number itself.\nThe balanced ternary system is an example of a [non-standard positional numeral system](https://bliptext.com/articles/non-standard-positional-numeral-systems).\nIt was used in some early computers and has also been used to solve [balance puzzles](https://bliptext.com/articles/balance-puzzle).\nDifferent sources use different glyphs to represent the three digits in balanced ternary.\nIn this article, T (which resembles a [ligature](https://bliptext.com/articles/typographical-ligature) of the minus sign and [1](https://bliptext.com/articles/1)) represents [−[1](https://bliptext.com/articles/1)](https://bliptext.com/articles/[1](https://bliptext.com/articles/1)), while [0](https://bliptext.com/articles/0) and [1](https://bliptext.com/articles/1) represent themselves.\nOther conventions include using '−' and '+' to represent −1 and 1 respectively, or using [Greek letter](https://bliptext.com/articles/greek-alphabet) [theta](https://bliptext.com/articles/theta) (Θ), which resembles a minus sign in a circle, to represent −1.\nIn publications about the [Setun](https://bliptext.com/articles/setun) computer, −1 is represented as overturned 1: \" 1 \".\nBalanced ternary makes an early appearance in [Michael Stifel](https://bliptext.com/articles/michael-stifel)'s book Arithmetica Integra (1544).\nIt also occurs in the works of [Johannes Kepler](https://bliptext.com/articles/johannes-kepler) and [Léon Lalanne](https://bliptext.com/articles/l-on-lalanne).\nRelated signed-digit schemes in other bases have been discussed by [John Colson](https://bliptext.com/articles/john-colson), [John Leslie](https://bliptext.com/articles/john-leslie-physicist), [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy), and possibly even the ancient Indian [Vedas](https://bliptext.com/articles/vedas).\n\n# Definition\nLet denote the set of [symbols](https://bliptext.com/articles/symbol-mathematics) (also called glyphs or characters), where the symbol \\bar{1} is sometimes used in place of Define an [integer](https://bliptext.com/articles/integer)-valued function by\nwhere the right hand sides are integers with their usual values.\nThis function, f_{}, is what rigorously and formally establishes how integer values are assigned to the symbols/glyphs in One benefit of this formalism is that the definition of \"the integers\" (however they may be defined) is not conflated with any particular system for writing/representing them; in this way, these two distinct (albeit closely related) concepts are kept separate.\nThe set together with the function f_{} forms a balanced [signed-digit representation](https://bliptext.com/articles/signed-digit-representation) called the balanced ternary system.\nIt can be used to represent integers and real numbers.\n\n## Ternary integer evaluation\nLet be the [Kleene plus](https://bliptext.com/articles/kleene-plus) of, which is the set of all finite length [concatenated](https://bliptext.com/articles/concatenation) [strings](https://bliptext.com/articles/string-computer-science) of one or more symbols (called its digits) where n is a non-negative integer and all n + 1 digits are taken from The start of is the symbol d_0 (at the right), its end is d_n (at the left), and its length is n + 1.\nThe ternary evaluation is the function defined by assigning to every string the integer\nThe string represents (with respect to v) the integer The value may alternatively be denoted by The map is [surjective](https://bliptext.com/articles/surjective-map) but not injective since, for example, However, every integer has exactly one representation under v that does not end (on the left) with the symbol 0, i.e. d_n = 0.\nIf and n > 0 then v satisfies:\nwhich shows that v satisfies a sort of [recurrence relation](https://bliptext.com/articles/recurrence-relation).\nThis recurrence relation has the initial condition\nwhere \\varepsilon is the empty string.\nThis implies that for every string\nwhich in words says that leading 0 symbols (to the left in a string with 2 or more symbols) do not affect the resulting value.\nThe following examples illustrate how some values of v can be computed, where (as before) all integer are written in decimal (base 10) and all elements of are just symbols.\nand using the above recurrence relation\n\n# Conversion to decimal\nIn the balanced ternary system the value of a digit n places left of the [radix point](https://bliptext.com/articles/radix-point) is the product of the digit and 3n.\nThis is useful when converting between decimal and balanced ternary.\nIn the following the strings denoting balanced ternary carry the suffix, bal3.\nFor instance,\nSimilarly, the first place to the right of the radix point holds 3−1 = 1⁄3, the second place holds 3−2 = 1⁄9, and so on.\nFor instance,\nAn integer is divisible by three if and only if the digit in the units place is zero.\nWe may check the [parity](https://bliptext.com/articles/parity-mathematics) of a balanced ternary integer by checking the [parity](https://bliptext.com/articles/parity-mathematics) of the sum of all trits.\nThis sum has the same parity as the integer itself.\nBalanced ternary can also be extended to fractional numbers similar to how decimal numbers are written to the right of the [radix point](https://bliptext.com/articles/radix-point).\n! Decimal\n! style=\"text-align: right\" | −0.9\n! style=\"text-align: right\" | −0.8\n! style=\"text-align: right\" | −0.7\n! style=\"text-align: right\" | −0.6\n! style=\"text-align: right\" | −0.5\n! style=\"text-align: right\" | −0.4\n! style=\"text-align: right\" | −0.3\n! style=\"text-align: right\" | −0.2\n! style=\"text-align: right\" | −0.1\n! style=\"text-align: right\" | 0\n! Balanced Ternary\n! Decimal\n! style=\"text-align: right\" | 0.9\n! style=\"text-align: right\" | 0.8\n! style=\"text-align: right\" | 0.7\n! style=\"text-align: right\" | 0.6\n! style=\"text-align: right\" | 0.5\n! style=\"text-align: right\" | 0.4\n! style=\"text-align: right\" | 0.3\n! style=\"text-align: right\" | 0.2\n! style=\"text-align: right\" | 0.1\n! style=\"text-align: right\" | 0\n! Balanced Ternary\nIn decimal or binary, integer values and terminating fractions have multiple representations.\nFor example, 1⁄10 = 0.1 = 0.10 = 0.09.\nAnd, 1⁄2 = 0.12 = 0.102 = 0.012.\nSome balanced ternary fractions have multiple representations too.\nFor example, 1⁄6 = 0.1𝖳bal3 = 0.01bal3.\nCertainly, 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.\nBut, 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.\n[Donald Knuth](https://bliptext.com/articles/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).\nThe 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).\nWith an odd [radix](https://bliptext.com/articles/radix), [double rounding](https://bliptext.com/articles/rounding) is also equivalent to directly rounding to the final precision, unlike with an even [radix](https://bliptext.com/articles/radix).\nThe basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary.\nMultiplication by two can be done by adding a number to itself, or subtracting itself after a-trit-left-shifting.\nAn 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.\n\n# Conversion to and from a fraction\nThe conversion of a repeating balanced ternary number to a fraction is analogous to [converting a repeating decimal](https://bliptext.com/articles/repeating-decimal).\nFor example (because of 111111bal3 = (36 − 1⁄3 − 1)dec):\n\n# Irrational numbers\nAs in any other integer base, algebraic irrationals and transcendental numbers do not terminate or repeat.\nFor example:\n!Decimal !!\nBalanced ternary\nThe balanced ternary expansions of \\pi is given in [OEIS](https://bliptext.com/articles/oeis) as, that of e in.\n\n# Conversion from ternary\nUnbalanced ternary can be converted to balanced ternary notation in two ways:\nIf the three values of [ternary logic](https://bliptext.com/articles/three-valued-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](https://bliptext.com/articles/offset-binary) system.\nIf the ternary number has n trits, then the bias b is which is represented as all ones in either conventional or biased form.\nAs 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.\nFurthermore, if x and y are balanced numbers, their balanced sum is x + y − b when computed using conventional unsigned ternary arithmetic.\nSimilarly, if x and y are conventional unsigned ternary numbers, their sum is x + y + b when computed using balanced ternary arithmetic.\n\n# Conversion to balanced ternary from any integer base\nWe may convert to balanced ternary with the following formula:\nwhere,\nFor instance,\n−25.4dec = −(1T×1011 + 1TT×1010 + 11×101−1)\n= −(1T×101 + 1TT + 11÷101)\n= −10T1.11TT\n= T01T.TT11\n1010.12 = 1T10 + 1T1 + 1T−1\n= 10T + 1T + 0.1\n= 101.1\n\n# Addition, subtraction and multiplication and division\nThe single-trit addition, subtraction, multiplication and division tables are shown below.\nFor subtraction and division, which are not [commutative](https://bliptext.com/articles/commutative-property), the first operand is given to the left of the table, while the second is given at the top.\nFor instance, the answer to 1 − T = 1T is found in the bottom left corner of the subtraction table.\n:{| class=\"wikitable\" style=\"width: 8em; text-align: center;\"\n|+ Addition\n|- align=\"right\"\n! + !! T !!\n0 !! 1 |- |- ! T | T1 || T || 0\n|- ! 0 | T || 0 || 1\n|- ! 1 | 0 || 1 || 1T\n|} :{| class=\"wikitable\" style=\"width: 8em; text-align: center;\"\n|+ Subtraction\n|- align=\"right\"\n! − !! T !!\n0 !! 1 |- ! T | 0 || T || T1 |-\n! 0 | 1 || 0 || T |-\n! 1 | 1T || 1 || 0\n|} :{| class=\"wikitable\" style=\"width: 8em; text-align: center;\"\n|+ Multiplication\n|- align=\"right\"\n! × !! T !!\n0 !! 1 |- |- ! T | 1 || 0 || T |-\n! 0 | 0 || 0 || 0 |- ! 1 | T || 0 || 1\n|} :{| class=\"wikitable\" style=\"text-align: center;\"\n|+ Division\n|- align=\"right\"\n! ÷ !! T !!\n1 |- |- ! T | 1 || T |-\n! 0 | 0 || 0 |- ! 1 | T || 1\n|}\n\n## Multi-trit addition and subtraction\nMulti-trit addition and subtraction is analogous to that of binary and decimal.\nAdd and subtract trit by trit, and add the carry appropriately.\nFor example:\n1TT1TT.1TT1 1TT1TT.1TT1 1TT1TT.1TT1 1TT1TT.1TT1\n+ 11T1.T − 11T1.T − 11T1.T → + TT1T.1\n______________ ______________ _______________ 1T0T10.0TT1 1T1001.TTT1 1T1001.TTT1\n+ 1T + T T1 + T T ______________ ________________ ________________\n1T1110.0TT1 1110TT.TTT1 1110TT.TTT1\n+ T + T 1 + T 1\n______________ ________________ ________________ 1T0110.0TT1 1100T.TTT1 1100T.TTT1\n\n## Multi-trit multiplication\nMulti-trit multiplication is analogous to that of binary and decimal.\n1TT1.TT × T11T.1\n_____________ 1TT.1TT multiply 1\nT11T.11 multiply T 1TT1T.T multiply 1\n1TT1TT multiply 1\nT11T11 multiply T _____________\n0T0000T.10T\n\n## Multi-trit division\nBalanced ternary division is analogous to that of binary and decimal.\nHowever, 0.5dec = 0.1111...bal3 or 1.TTTT...bal3.\nIf 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.\nThe magnitude of the dividend must be compared with that of half the divisor before setting the quotient trit.\nFor example,\n1TT1.TT quotient\n0.5 × divisor T01.0 _____________\ndivisor T11T.1 ) T0000T.10T dividend T11T1 T000 10T0, set T _______ 111T 1TT1T 111T > 10T0, set T _______ T00.1 T11T.1 T001 10T0, set T ________ 1T.T1T 1T.T1T 1TT1T > 10T0, set T ________ 0 Another example, 1TTT\n0.5 × divisor 1T _______\nDivisor 11 )1T01T 1T = 1T, but 1T.01 > 1T, set 1 11 _____ T10 T10 1T, set 1\n11 _____ 1 T1 0.10, set 1\n1T0 −1.T0 ________\n11×10=110 1T0T 1T0T>110, set 1\n10T0 −10T0\n________ 111×10=1110 T1T0T T1T0T 111T0, set 1\n10T110 −10T110\n__________ 111T1×10=111T10 TT1TT0T TT1TT0T 1TT, set 1\n1×1×1000+1=1001 −1.001 __________ T0T000\n11×100 − 1100 borrow 100×, do division\n_________ 10T000 TT1T00 TT1T00 1T1T01TT, set 1\n11T×11T×1000+1=11111001 − 11111001\n______________ 1T10T000\n11T1×100 − 11T100 borrow 100×, do division\n__________ 10T0T01TT 1T0T0T00 T01010T11<1T0T0T00<10T0T01TT, set 0\n11T1×11T1×1000+1=1TT1T11001 − TT1T00 return 100×\n_____________ 1T10T000000\n... Hence 3√2 = 1.259921dec = 1.1T1 000 111 001 T01 00T 1T1 T10 111bal3.\n\n# Applications\n\n## In computer design\nIn the early days of computing, a few experimental Soviet computers were built with balanced ternary instead of binary, the most famous being the [Setun](https://bliptext.com/articles/setun), built by [Nikolay Brusentsov](https://bliptext.com/articles/nikolay-brusentsov) and [Sergei Sobolev](https://bliptext.com/articles/sergei-sobolev).\nThe notation has a number of computational advantages over traditional binary and ternary.\nParticularly, 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.\nIn balanced ternary, the one-digit [multiplication table](https://bliptext.com/articles/multiplication-table) remains one-digit and has no carry and the [addition table](https://bliptext.com/articles/addition-table) has only two carries out of nine entries, compared to unbalanced ternary with one and three respectively.\nKnuth wrote that \"Perhaps the symmetric properties and simple arithmetic of this number system will prove to be quite important some day,\" noting that,\n\"The complexity of arithmetic circuitry for balanced ternary arithmetic is not much greater than it is for the binary system, and a given number requires only as many digit positions for its representation.\"\"\n# [Operation tables | upload.wikimedia.org/wikipedia/commons/4/4c/Balanced///ternary///operation///tables.svg]\n\n## Other applications\nThe theorem that every integer has a unique representation in balanced ternary was used by [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) to justify the identity of [formal power series](https://bliptext.com/articles/formal-power-series)\nBalanced ternary has other applications besides computing.\nFor example, a classical two-pan [balance](https://bliptext.com/articles/weighing-scale), 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.\nFor example, with weights for each power of 3 through 81, a 60-gram object (60dec = 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.\nSimilarly, consider a currency system with coins worth 1¤, 3¤, 9¤, 27¤, 81¤.\nIf the buyer and the seller each have only one of each kind of coin, any transaction up to 121¤ is possible.\nFor example, if the price is 7¤ (7dec = 1T1bal3), the buyer pays 1¤ + 9¤ and receives 3¤ in change.\nThey may also provide a more natural representation for the [qutrit](https://bliptext.com/articles/qutrit) and systems that use it.","revisionCount":0},"hasMore":false,"nextCursor":null}

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