[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/balanced-ternary-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/balanced-ternary-wikipedia\/","headline":"Balanced ternary – Wikipedia","name":"Balanced ternary – Wikipedia","description":"Numeral system that uses the digits -1, 0 and, 1 Balanced ternary is a ternary numeral system (i.e. base 3","datePublished":"2016-09-07","dateModified":"2016-09-07","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed525b93bf294538e4252b31cc93182d3444609b","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed525b93bf294538e4252b31cc93182d3444609b","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/balanced-ternary-wikipedia\/","about":["Wiki"],"wordCount":18169,"articleBody":"Numeral system that uses the digits -1, 0 and, 1Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values \u22121, 0, and 1. This stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2.The balanced ternary system 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. The balanced ternary system is an example of a non-standard positional numeral system. It was used in some early computers[1] and also in some solutions of balance puzzles.[2]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 \u22121, while 0 and 1 represent themselves. Other conventions include using ‘\u2212’ and ‘+’ to represent \u22121 and 1 respectively, or using Greek letter theta (\u0398), which resembles a minus sign in a circle, to represent \u22121. In publications about the Setun computer, \u22121 is represented as overturned 1: “1“.[1]Balanced ternary makes an early appearance in Michael Stifel’s book Arithmetica Integra (1544).[3] It also occurs in the works of Johannes Kepler and L\u00e9on 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.[2]Table of ContentsDefinition[edit]Ternary integer evaluation[edit]Conversion to decimal[edit]Conversion to and from a fraction[edit]Irrational numbers[edit]Conversion from ternary[edit]Conversion to balanced ternary from any integer base[edit]Addition, subtraction and multiplication and division[edit]Multi-trit addition and subtraction[edit]Multi-trit multiplication[edit]Multi-trit division[edit]Square roots and cube roots[edit]Applications[edit]In computer design[edit]Other applications[edit]See also[edit]References[edit]External links[edit]Definition[edit]Let D3{displaystyle {mathcal {D}}_{3}} denote the set of symbols (also called glyphs or characters) D3={T,0,1}{displaystyle {mathcal {D}}_{3}=lbrace operatorname {T} ,0,1rbrace }, where the symbol 1\u00af{displaystyle {bar {1}}} is sometimes used in place of T.{displaystyle operatorname {T} .}Define an integer-valued function f=fD3:D3\u2192Z{displaystyle f=f_{{mathcal {D}}_{3}}:{mathcal {D}}_{3}to mathbb {Z} } byf(T)=\u22121,{displaystyle f_{}(operatorname {T} )=-1,}f(0)=0,{displaystyle f_{}(0)=0,}[note 1] andf(1)=1{displaystyle f_{}(1)=1}where the right hand sides are integers with their usual (decimal) values. This function, f,{displaystyle f_{},} is what rigorously and formally establishes how integer values are assigned to the symbols\/glyphs in D3.{displaystyle {mathcal {D}}_{3}.} 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.The set D3{displaystyle {mathcal {D}}_{3}} together with the function f{displaystyle f_{}} forms a balanced signed-digit representation called the balanced ternary system.It can be used to represent integers and real numbers.Ternary integer evaluation[edit]Let D3+{displaystyle {mathcal {D}}_{3}^{+}} be the Kleene plus of D3{displaystyle {mathcal {D}}_{3}}, which is the set of all finite length concatenated strings dn\u2026d0{displaystyle d_{n}ldots d_{0}} of one or more symbols (called its digits) where n{displaystyle n} is a non-negative integer and all n+1{displaystyle n+1} digits dn,\u2026,d0{displaystyle d_{n},ldots ,d_{0}} are taken from D3={T,0,1}.{displaystyle {mathcal {D}}_{3}=lbrace operatorname {T} ,0,1rbrace .} The start of dn\u2026d0{displaystyle d_{n}ldots d_{0}} is the symbol d0{displaystyle d_{0}} (at the right), its end is dn{displaystyle d_{n}} (at the left), and its length is n+1{displaystyle n+1}. The ternary evaluation is the function v=v3\u00a0:\u00a0D3+\u2192Z{displaystyle v=v_{3}~:~{mathcal {D}}_{3}^{+}to mathbb {Z} } defined by assigning to every string dn\u2026d0\u2208D3+{displaystyle d_{n}ldots d_{0}in {mathcal {D}}_{3}^{+}} the integerv(dn\u2026d0)\u00a0=\u00a0\u2211i=0nf(di)3i.{displaystyle vleft(d_{n}ldots d_{0}right)~=~sum _{i=0}^{n}f_{}left(d_{i}right)3^{i}.}The string dn\u2026d0{displaystyle d_{n}ldots d_{0}} represents (with respect to v{displaystyle v}) the integer v(dn\u2026d0).{displaystyle vleft(d_{n}ldots d_{0}right).} The value v(dn\u2026d0){displaystyle vleft(d_{n}ldots d_{0}right)} may alternatively be denoted by dn\u2026d0bal\u20613.{displaystyle {d_{n}ldots d_{0}}_{operatorname {bal} 3}.}The map v:D3+\u2192Z{displaystyle v:{mathcal {D}}_{3}^{+}to mathbb {Z} } is surjective but not injective since, for example, 0=v(0)=v(00)=v(000)=\u22ef.{displaystyle 0=v(0)=v(00)=v(000)=cdots .} However, every integer has exactly one representation under v{displaystyle v} that does not end (on the left) with the symbol 0,{displaystyle 0,} i.e. dn=0.{displaystyle d_{n}=0.}If dn\u2026d0\u2208D3+{displaystyle d_{n}ldots d_{0}in {mathcal {D}}_{3}^{+}} and 0″\/> then v{displaystyle v} satisfies:v(dndn\u22121\u2026d0)\u00a0=\u00a0f(dn)3n+v(dn\u22121\u2026d0){displaystyle vleft(d_{n}d_{n-1}ldots d_{0}right)~=~f_{}left(d_{n}right)3^{n}+vleft(d_{n-1}ldots d_{0}right)}which shows that v{displaystyle v} satisfies a sort of recurrence relation. This recurrence relation has the initial conditionv(\u03b5)=0{displaystyle vleft(varepsilon right)=0}where \u03b5{displaystyle varepsilon } is the empty string.This implies that for every string dn\u2026d0\u2208D3+,{displaystyle d_{n}ldots d_{0}in {mathcal {D}}_{3}^{+},}v(0dn\u2026d0)=v(dn\u2026d0){displaystyle vleft(0d_{n}ldots d_{0}right)=vleft(d_{n}ldots d_{0}right)}which in words says that leading 0{displaystyle 0} symbols (to the left in a string with 2 or more symbols) do not affect the resulting value.The following examples illustrate how some values of v{displaystyle v} can be computed, where (as before) all integer are written in decimal (base 10) and all elements of D3+{displaystyle {mathcal {D}}_{3}^{+}} are just symbols.v(T\u2061T)=f(T)31+f(T)30=(\u22121)3+(\u22121)1=\u22124v(T\u20611)=f(T)31+f(1)30=(\u22121)3+(1)1=\u22122v(1T)=f(1)31+f(T)30=(1)3+(\u22121)1=2v(11)=f(1)31+f(1)30=(1)3+(1)1=4v(1T\u20610)=f(1)32+f(T)31+f(0)30=(1)9+(\u22121)3+(0)1=6v(10T)=f(1)32+f(0)31+f(T)30=(1)9+(0)3+(\u22121)1=8{displaystyle {begin{alignedat}{10}vleft(operatorname {T} operatorname {T} right)&=&&f_{}left(operatorname {T} right)3^{1}+&&f_{}left(operatorname {T} right)3^{0}&&=&&(-1)&&3&&,+,&&(-1)&&1&&=-4\\vleft(operatorname {T} 1right)&=&&f_{}left(operatorname {T} right)3^{1}+&&f_{}left(1right)3^{0}&&=&&(-1)&&3&&,+,&&(1)&&1&&=-2\\vleft(1operatorname {T} right)&=&&f_{}left(1right)3^{1}+&&f_{}left(operatorname {T} right)3^{0}&&=&&(1)&&3&&,+,&&(-1)&&1&&=2\\vleft(11right)&=&&f_{}left(1right)3^{1}+&&f_{}left(1right)3^{0}&&=&&(1)&&3&&,+,&&(1)&&1&&=4\\vleft(1operatorname {T} 0right)&=f_{}left(1right)3^{2}+&&f_{}left(operatorname {T} right)3^{1}+&&f_{}left(0right)3^{0}&&=(1)9,+,&&(-1)&&3&&,+,&&(0)&&1&&=6\\vleft(10operatorname {T} right)&=f_{}left(1right)3^{2}+&&f_{}left(0right)3^{1}+&&f_{}left(operatorname {T} right)3^{0}&&=(1)9,+,&&(0)&&3&&,+,&&(-1)&&1&&=8\\end{alignedat}}}and using the above recurrence relationv(101T)=f(1)33+v(01T)=(1)27+v(1T)=27+2=29.{displaystyle vleft(101operatorname {T} right)=f_{}left(1right)3^{3}+vleft(01operatorname {T} right)=(1)27+vleft(1operatorname {T} right)=27+2=29.}Conversion to decimal[edit]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 \u00d7 31 + 0 \u00d7 30 = 31010\ud835\uddb3bal3 = 1 \u00d7 32 + 0 \u00d7 31 + (\u22121) \u00d7 30 = 810\u2212910 = \u22121 \u00d7 32 + 0 \u00d7 31 + 0 \u00d7 30 = \ud835\uddb300bal3810 = 1 \u00d7 32 + 0 \u00d7 31 + (\u22121) \u00d7 30 = 10\ud835\uddb3bal3Similarly, the first place to the right of the radix point holds 3\u22121 = 1\/3, the second place holds 3\u22122 = 1\/9, and so on. For instance,\u22122\/310 = \u22121 + 1\/3 = \u22121 \u00d7 30 + 1 \u00d7 3\u22121 = \ud835\uddb3.1bal3.DecBal3Expansion00011+121\ud835\uddb3+3\u22121310+3411+3+151\ud835\uddb3\ud835\uddb3+9\u22123\u2212161\ud835\uddb30+9\u2212371\ud835\uddb31+9\u22123+1810\ud835\uddb3+9\u221219100+910101+9+11111\ud835\uddb3+9+3\u2212112110+9+313111+9+3+1DecBal3Expansion000\u22121\ud835\uddb3\u22121\u22122\ud835\uddb31\u22123+1\u22123\ud835\uddb30\u22123\u22124\ud835\uddb3\ud835\uddb3\u22123\u22121\u22125\ud835\uddb311\u22129+3+1\u22126\ud835\uddb310\u22129+3\u22127\ud835\uddb31\ud835\uddb3\u22129+3\u22121\u22128\ud835\uddb301\u22129+1\u22129\ud835\uddb300\u22129\u221210\ud835\uddb30\ud835\uddb3\u22129\u22121\u221211\ud835\uddb3\ud835\uddb31\u22129\u22123+1\u221212\ud835\uddb3\ud835\uddb30\u22129\u22123\u221213\ud835\uddb3\ud835\uddb3\ud835\uddb3\u22129\u22123\u22121An 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.[4]Decimal\u22120.9\u22120.8\u22120.7\u22120.6\u22120.5\u22120.4\u22120.3\u22120.2\u22120.10Balanced Ternary\ud835\uddb3.010\ud835\uddb3\ud835\uddb3.1\ud835\uddb3\ud835\uddb31\ud835\uddb3.10\ud835\uddb30\ud835\uddb3.11\ud835\uddb3\ud835\uddb30.\ud835\uddb3 or \ud835\uddb3.10.\ud835\uddb3\ud835\uddb3110.\ud835\uddb30100.\ud835\uddb311\ud835\uddb30.0\ud835\uddb3010Decimal0.90.80.70.60.50.40.30.20.10Balanced Ternary1.0\ud835\uddb3011.\ud835\uddb311\ud835\uddb31.\ud835\uddb30101.\ud835\uddb3\ud835\uddb3110.1 or 1.\ud835\uddb30.11\ud835\uddb3\ud835\uddb30.10\ud835\uddb300.1\ud835\uddb3\ud835\uddb310.010\ud835\uddb30In decimal or binary, integer values and terminating fractions have multiple representations. For example, 1\/10 = 0.1 = 0.10 = 0.09. And, 1\/2 = 0.12 = 0.102 = 0.012. Some balanced ternary fractions have multiple representations too. For example, 1\/6 = 0.1\ud835\uddb3bal3 = 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 \u22121s after the radix point in order to gain a representations of integer or terminating fraction.Donald Knuth[5] has pointed out that truncation and rounding are the same operation in balanced ternary\u2014they 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.\ud835\uddb3 (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\u2014addition, subtraction, multiplication, and division\u2014are 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[edit]FractionBalanced ternary111\/20.11.\ud835\uddb31\/30.11\/40.1\ud835\uddb31\/50.1\ud835\uddb3\ud835\uddb311\/60.010.1\ud835\uddb31\/70.0110\ud835\uddb3\ud835\uddb31\/80.011\/90.011\/100.010\ud835\uddb3FractionBalanced ternary1\/110.01\ud835\uddb3111\/120.01\ud835\uddb31\/130.01\ud835\uddb31\/140.01\ud835\uddb30\ud835\uddb311\/150.01\ud835\uddb3\ud835\uddb311\/160.01\ud835\uddb3\ud835\uddb31\/170.01\ud835\uddb3\ud835\uddb3\ud835\uddb310\ud835\uddb30\ud835\uddb3111\ud835\uddb3011\/180.0010.01\ud835\uddb31\/190.00111\ud835\uddb310100\ud835\uddb3\ud835\uddb3\ud835\uddb31\ud835\uddb30\ud835\uddb31\/200.0011The conversion of a repeating balanced ternary number to a fraction is analogous to converting a repeating decimal. For example (because of 111111bal3 = (36 \u2212 1\/3 \u2212 1)10):0.1110TT0\u00af=1110TT0\u22121111111\u00d71T\u00d710=1110TTT111111\u00d71T0=111\u00d71000T111\u00d71001\u00d71T0=1111\u00d71T1001\u00d71T0=111110010=1T1T1TTT0=1011T10{displaystyle 0.1{overline {mathrm {110TT0} }}={tfrac {mathrm {1110TT0-1} }{mathrm {111111times 1Ttimes 10} }}={tfrac {mathrm {1110TTT} }{mathrm {111111times 1T0} }}={tfrac {mathrm {111times 1000T} }{mathrm {111times 1001times 1T0} }}={tfrac {mathrm {1111times 1T} }{mathrm {1001times 1T0} }}={tfrac {1111}{10010}}={tfrac {mathrm {1T1T} }{mathrm {1TTT0} }}={tfrac {101}{mathrm {1T10} }}}Irrational numbers[edit]As in any other integer base, algebraic irrationals and transcendental numbers do not terminate or repeat. For example:The balanced ternary expansions of \u03c0{displaystyle pi } is given in OEIS as A331313, that of e{displaystyle e} in A331990.Conversion from ternary[edit]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 \u2212 113 = 1T1bal3 = 710.If a 2 is present in ternary, turn it into 1T. For example,02123 = 0010bal3 + 1T00bal3 + 001Tbal3 = 10TTbal3 = 2310BalancedLogicUnsigned1True20Unknown1TFalse0If 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 isb=\u230a3n2\u230b{displaystyle b=leftlfloor {frac {3^{n}}{2}}rightrfloor }which is represented as all ones in either conventional or biased form.[6]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 \u2212 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[edit]We may convert to balanced ternary with the following formula:(anan\u22121\u22efa1a0.c1c2c3\u22ef)b=\u2211k=0nakbk+\u2211k=1\u221eckb\u2212k.{displaystyle left(a_{n}a_{n-1}cdots a_{1}a_{0}.c_{1}c_{2}c_{3}cdots right)_{b}=sum _{k=0}^{n}a_{k}b^{k}+sum _{k=1}^{infty }c_{k}b^{-k}.}where,anan\u22121…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.For instance, \u221225.410 = \u2212(1T\u00d71011 + 1TT\u00d71010 + 11\u00d7101\u22121) = \u2212(1T\u00d7101 + 1TT + 11\u00f7101) = \u221210T1.11TT = T01T.TT11 1010.12 = 1T10 + 1T1 + 1T\u22121 = 10T + 1T + 0.1 = 101.1Addition, subtraction and multiplication and division[edit]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\u00a0\u2212\u00a0T\u00a0= 1T is found in the bottom left corner of the subtraction table.Addition+T01TT1T00T011011TSubtraction\u2212T01T0TT1010T11T10Multiplication\u00d7T01T10T00001T01Division\u00f7T1T1T0001T1Multi-trit addition and subtraction[edit]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 \u2212 11T1.T \u2212 11T1.T \u2192 + 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.TTT1Multi-trit multiplication[edit]Multi-trit multiplication is analogous to that of binary and decimal. 1TT1.TT \u00d7 T11T.1 _____________ 1TT.1TT multiply 1 T11T.11 multiply T 1TT1T.T multiply 1 1TT1TT multiply 1 T11T11 multiply T _____________ 0T0000T.10TMulti-trit division[edit]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 quotient0.5 \u00d7 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 ________ 0Another example, 1TTT 0.5 \u00d7 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 ____ 0Another example, 101.TTTTTTTTT... or 100.111111111... 0.5 \u00d7 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[edit]The process of extracting the square root in balanced ternary is analogous to that in decimal or binary.(10\u22c5x+y)1T\u2212100\u22c5x1T=1T0\u22c5x\u22c5y+y1T={T10\u22c5x+1,y=T0,y=01T0\u22c5x+1,y=1{displaystyle (10cdot x+y)^{mathrm {1T} }-100cdot x^{mathrm {1T} }=mathrm {1T0} cdot xcdot y+y^{mathrm {1T} }={begin{cases}mathrm {T10} cdot x+1,&y=mathrm {T} \\0,&y=0\\mathrm {1T0} cdot x+1,&y=1end{cases}}}As in division, we should check the value of half the divisor first. For example, 1. 1 1 T 1 T T 0 0 ... _________________________ \u221a 1T 1110, set 1 10T0 \u221210T0 ________ 111\u00d710=1110 T1T0T T1T0T111T0, set 1 10T110 \u221210T110 __________ 111T1\u00d710=111T10 TT1TT0T TT1TT0T1000\u22c5x10=y10+1000\u22c5x1T\u22c5y+100\u22c5x\u22c5y1T={T+T000\u22c5x1T+100\u22c5x,y=T0,y=01+1000\u22c5x1T+100\u22c5x,y=1{displaystyle (10cdot x+y)^{10}-1000cdot x^{10}=y^{10}+1000cdot x^{mathrm {1T} }cdot y+100cdot xcdot y^{mathrm {1T} }={begin{cases}mathrm {T} +mathrm {T000} cdot x^{mathrm {1T} }+100cdot x,&y=mathrm {T} \\0,&y=0\\1+1000cdot x^{mathrm {1T} }+100cdot x,&y=1end{cases}}}Like division, we should check the value of half the divisor first too.For example: 1. 1 T 1 0 ... _____________________ \u00b3\u221a 1T \u2212 1 1"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/balanced-ternary-wikipedia\/#breadcrumbitem","name":"Balanced ternary – Wikipedia"}}]}]