View source for Balanced ternary

{{Short description|Numeral system using the values -1, 0 and 1}}
{{numeral systems}}
'''Balanced ternary''' is a [[Ternary numeral system|ternary]] numeral system (i.e. base 3 with three [[Numerical digit|digits]]) that uses a balanced [[signed-digit representation]] of the [[integer]]s in which the digits have the values [[−1]], [[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 systems|non-standard positional numeral system]]. It was used in some early computers<ref name=setun/> and has also been used to solve [[balance puzzle]]s.<ref name=hayes/>

Different sources use different glyphs to represent the three digits in balanced ternary. In this article, T (which resembles a [[typographical ligature|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 alphabet|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: "<span class="plainlinks"></span><span style="display:inline-block;vertical-align:-0.05em;transform:matrix(-1, 0, 0, -1, 0, 0);">1</span>".<ref name=setun>{{cite book|title=Programming|year=1963|location=Moscow|author=N. A. Krinitsky |author2=G. A. Mironov |author3=G. D. Frolov|editor=M. R. Shura-Bura|language=ru|chapter=Chapter 10. Program-controlled machine Setun}}</ref>

Balanced ternary makes an early appearance in [[Michael Stifel]]'s book ''Arithmetica Integra'' (1544).<ref>{{citation
 | last = Stifel | first = Michael | author-link = Michael Stifel
 | language = Latin
 | page = 38
 | title = Arithmetica integra
 | url = https://archive.org/stream/bub_gb_ywkW9hDd7IIC#page/n85/mode/2up
 | year = 1544| publisher = apud Iohan Petreium }}.</ref> 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 (physicist)|John Leslie]], [[Augustin-Louis Cauchy]], and possibly even the ancient Indian [[Vedas]].<ref name=hayes>{{citation | first=Brian|last=Hayes|authorlink=Brian Hayes (scientist)|title=Third base|journal=American Scientist|url=http://bit-player.org/bph-publications/AmSci-2001-11-Hayes-ternary.pdf|year=2001|volume=89|issue=6|pages=490&ndash;494|doi=10.1511/2001.40.3268}}. Reprinted in {{citation|title=Group Theory in the Bedroom, and Other Mathematical Diversions|first=Brian|last=Hayes|authorlink=Brian Hayes (scientist)|publisher=Farrar, Straus and Giroux|year=2008|isbn=9781429938570|pages=179–200|url=https://books.google.com/books?id=1ZkYEFi3DMMC&pg=PA179}}</ref>

== Definition ==
{{See also|Signed-digit representation}}

Let <math>\mathcal{D}_{3} := \lbrace \operatorname{T}, 0, 1 \rbrace</math> denote the set of [[Symbol (mathematics)|symbols]] (also called ''glyphs'' or ''characters''), where the symbol <math>\bar{1}</math> is sometimes used in place of <math>\operatorname{T}.</math> 
Define an [[integer]]-valued function <math>f = f_{\mathcal{D}_{3}} : \mathcal{D}_{3} \to \mathbb{Z}</math> by

:<math>\begin{align}
f_{}(\operatorname{T}) &= -1, \\
f_{}(0) &= 0, \\
f_{}(1) &= 1,
\end{align}</math><ref>The symbols <math>0</math> and <math>1</math> appear twice in the equalities <math>f_{}(0) = 0</math> and <math>f_{}(1) = 1,</math> but these instances do not represent the same thing. The right hand side <math>0</math> and <math>1</math> mean the integers <math>\in\Z,</math> but the instances inside <math>f</math>'s parentheses (which belong to <math>\mathcal{D}_{3}</math>) should be thought of as being nothing more than symbols.</ref>
where the right hand sides are integers with their usual values. This function, <math>f_{},</math> is what rigorously and formally establishes how integer values are assigned to the symbols/glyphs in <math>\mathcal{D}_{3}.</math> 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 <math>\mathcal{D}_{3}</math> together with the function <math>f_{}</math> forms a balanced [[signed-digit representation]] called the ''balanced ternary'' system. 
It can be used to represent integers and real numbers.

=== Ternary integer evaluation ===

Let <math>\mathcal{D}_{3}^{+}</math> be the [[Kleene plus]] of <math>\mathcal{D}_{3}</math>, which is the set of all finite length [[concatenation|concatenated]] [[String (computer science)|strings]] <math>d_n \ldots d_0</math> of one or more symbols (called its ''digits'') where <math>n</math> is a non-negative integer and all <math>n + 1</math> digits <math>d_n, \ldots, d_0</math> are taken from <math>\mathcal{D}_{3} = \lbrace \operatorname{T}, 0, 1 \rbrace.</math> The ''start'' of <math>d_n \ldots d_0</math> is the symbol <math>d_0</math> (at the right), its ''end'' is <math>d_n</math> (at the left), and its ''length'' is <math>n + 1</math>. The ''ternary evaluation'' is the function <math>v = v_{3} ~:~ \mathcal{D}_{3}^{+} \to \mathbb{Z}</math> defined by assigning to every string <math>d_n \ldots d_0 \in \mathcal{D}_{3}^{+}</math> the integer

:<math>v\left( d_n \ldots d_0 \right) ~=~ \sum_{i=0}^{n} f_{} \left( d_{i} \right) 3^{i}.</math>

The string <math>d_n \ldots d_0</math> ''represents'' (with respect to <math>v</math>) the integer <math>v\left( d_n \ldots d_0 \right).</math> The value <math>v\left( d_n \ldots d_0 \right)</math> may alternatively be denoted by <math>{d_n \ldots d_0}_{\operatorname{bal}3}.</math> 
The map <math>v : \mathcal{D}_{3}^{+} \to \mathbb{Z}</math> is [[Surjective map|surjective]] but not injective since, for example, <math>0 = v(0) = v(00) = v(0 0 0) = \cdots.</math> However, every nonzero integer has exactly one representation under <math>v</math> that does not ''end'' (on the left) with the symbol <math>0,</math> i.e. <math>d_n = 0 .</math>

If <math>d_n \ldots d_0 \in \mathcal{D}_{3}^{+}</math> and <math>n > 0</math> then <math>v</math> satisfies:

:<math>v\left( d_n d_{n-1} \ldots d_0 \right) ~=~  f_{} \left( d_{n} \right) 3^{n} + v\left( d_{n-1} \ldots d_0 \right)</math>

which shows that <math>v</math> satisfies a sort of [[recurrence relation]]. This recurrence relation has the initial condition
<math>v\left( \varepsilon \right) = 0</math> 
where <math>\varepsilon</math> is the empty string.

This implies that for every string <math>d_n \ldots d_0 \in \mathcal{D}_{3}^{+},</math>

:<math>v\left( 0 d_n \ldots d_0 \right) = v\left( d_n \ldots d_0 \right)</math>

which in words says that ''leading'' <math>0</math> 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 <math>v</math> can be computed, where (as before) all integer are written in decimal (base 10) and all elements of <math>\mathcal{D}_{3}^{+}</math> are just symbols.

:<math>\begin{alignat}{10}
v\left( \operatorname{T} \operatorname{T} \right) 
&= && f_{}\left( \operatorname{T} \right) 3^{1} + && f_{}\left( \operatorname{T} \right) 3^{0} 
&&= &&(-1) &&3 &&\,+\, &&(-1) &&1 
&&= -4 \\

v\left( \operatorname{T} 1 \right) 
&= && f_{}\left( \operatorname{T} \right) 3^{1} + && f_{}\left( 1 \right) 3^{0} 
&&= &&(-1) &&3 &&\,+\, &&(1) &&1 
&&= -2 \\

v\left( 1 \operatorname{T} \right) 
&= && f_{}\left( 1 \right) 3^{1} + && f_{}\left( \operatorname{T} \right) 3^{0} 
&&= &&(1) &&3 &&\,+\, &&(-1) &&1 
&&= 2 \\

v\left( 1 1 \right) 
&= && f_{}\left( 1 \right) 3^{1} + && f_{}\left( 1 \right) 3^{0} 
&&= &&(1) &&3 &&\,+\, &&(1) &&1 
&&= 4 \\

v\left( 1 \operatorname{T} 0 \right) 
&= f_{}\left( 1 \right) 3^{2} + && f_{}\left( \operatorname{T} \right) 3^{1} + && f_{}\left( 0 \right) 3^{0} 
&&= (1) 9 \,+\, &&(-1) &&3 &&\,+\, &&(0) &&1 
&&= 6 \\

v\left( 1 0 \operatorname{T} \right) 
&= f_{}\left( 1 \right) 3^{2} + && f_{}\left( 0 \right) 3^{1} + && f_{}\left( \operatorname{T} \right) 3^{0} 
&&= (1) 9 \,+\, &&(0) &&3 &&\,+\, &&(-1) &&1 
&&= 8 \\
\end{alignat}
</math>

and using the above recurrence relation

:<math>v\left( 1 0 1 \operatorname{T} \right) = f_{}\left( 1 \right) 3^{3} + v\left( 0 1 \operatorname{T} \right) = (1) 27 + v\left( 1 \operatorname{T} \right) = 27 + 2 = 29.</math>

== Conversions to/from other representations ==
=== 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<sup>''n''</sup>. 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<sub>bal3</sub> = 1 × 3<sup>1</sup> + 0 × 3<sup>0</sup> = 3<sub>dec</sub>
: 10𝖳<sub>bal3</sub> = 1 × 3<sup>2</sup> + 0 × 3<sup>1</sup> + (−1) × 3<sup>0</sup> = 8<sub>dec</sub>
: −9<sub>dec</sub> = −1 × 3<sup>2</sup> + 0 × 3<sup>1</sup> + 0 × 3<sup>0</sup> = 𝖳00<sub>bal3</sub>
: 8<sub>dec</sub> = 1 × 3<sup>2</sup> + 0 × 3<sup>1</sup> + (−1) × 3<sup>0</sup> = 10𝖳<sub>bal3</sub>

Similarly, the first place to the right of the radix point holds 3<sup>−1</sup> = {{sfrac|1|3}}, the second place holds 3<sup>−2</sup> = {{sfrac|1|9}}, and so on. For instance,
: −{{sfrac|2|3}}<sub>dec</sub> = −1 + {{sfrac|1|3}} = −1 × 3<sup>0</sup> + 1 × 3<sup>−1</sup> = 𝖳.1<sub>bal3</sub>.

<div style="display: flex; column-gap: 1em; margin-inline-start: 1.5em;">
{| class="wikitable" style="border: none; text-align:right"
! Dec !! Bal3 !! Expansion
|-
|   0 ||   0 ||      0
|-
|   1 ||   1 ||     +1
|-
|   2 ||  1𝖳 ||   +3−1
|-
|   3 ||  10 ||   +3
|-
|   4 ||  11 ||   +3+1
|-
|   5 || 1𝖳𝖳 || +9−3−1
|-
|   6 || 1𝖳0 || +9−3
|-
|   7 || 1𝖳1 || +9−3+1
|-
|   8 || 10𝖳 || +9−1
|-
|   9 || 100 || +9
|-
|  10 || 101 || +9+1
|-
|  11 || 11𝖳 || +9+3−1
|-
|  12 || 110 || +9+3
|-
|  13 || 111 || +9+3+1
|}
{| class="wikitable" style="border: none; text-align:right"
! Dec !! Bal3 !! Expansion
|-
|   0 ||   0 ||      0
|-
|  −1 ||   𝖳 ||     −1
|-
|  −2 ||  𝖳1 ||   −3+1
|-
|  −3 ||  𝖳0 ||   −3
|-
|  −4 ||  𝖳𝖳 ||  −3−1
|-
|  −5 || 𝖳11 || −9+3+1
|-
|  −6 || 𝖳10 || −9+3
|-
|  −7 || 𝖳1𝖳 || −9+3−1
|-
|  −8 || 𝖳01 || −9+1
|-
|  −9 || 𝖳00 || −9
|-
| −10 || 𝖳0𝖳 || −9−1
|-
| −11 || 𝖳𝖳1 || −9−3+1
|-
| −12 || 𝖳𝖳0 || −9−3
|-
| −13 || 𝖳𝖳𝖳 || −9−3−1
|}
</div>

An integer is divisible by three if and only if the digit in the units place is zero.

We may check the [[Parity (mathematics)|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]].<ref>{{cite web |url=http://www.abhijit.info/tristate/tristate.html |title=Balanced ternary |last=Bhattacharjee |first=Abhijit |date=24 July 2006 |archiveurl=https://web.archive.org/web/20090919053547/http://www.abhijit.info/tristate/tristate.html |archivedate=2009-09-19}}</ref>

:{| class="wikitable"
|-
! Decimal
! style="text-align: right" | −0.9
! style="text-align: right" | −0.8
! style="text-align: right" | −0.7
! style="text-align: right" | −0.6
! style="text-align: right" | −0.5
! style="text-align: right" | −0.4
! style="text-align: right" | −0.3
! style="text-align: right" | −0.2
! style="text-align: right" | −0.1
! style="text-align: right" |  0
|-
! Balanced Ternary 
| 𝖳.{{overline|010𝖳}}||𝖳.{{overline|1𝖳𝖳1}}|| 𝖳.{{overline|10𝖳0}}|| 𝖳.{{overline|11𝖳𝖳}}|| 0.{{overline|𝖳}} or 𝖳.{{overline|1}} || 0.{{overline|𝖳𝖳11}} || 0.{{overline|𝖳010}} || 0.{{overline|𝖳11𝖳}} || 0.{{overline|0𝖳01}} || 0
|-
! Decimal
! style="text-align: right" | 0.9
! style="text-align: right" | 0.8
! style="text-align: right" | 0.7
! style="text-align: right" | 0.6
! style="text-align: right" | 0.5
! style="text-align: right" | 0.4
! style="text-align: right" | 0.3
! style="text-align: right" | 0.2
! style="text-align: right" | 0.1
! style="text-align: right" | 0
|-
! Balanced Ternary 
| 1.{{overline|0𝖳01}}||1.{{overline|𝖳11𝖳}}|| 1.{{overline|𝖳010}}|| 1.{{overline|𝖳𝖳11}}|| 0.{{overline|1}} or 1.{{overline|𝖳}} || 0.{{overline|11𝖳𝖳}} || 0.{{overline|10𝖳0}} || 0.{{overline|1𝖳𝖳1}} || 0.{{overline|010𝖳}} || 0
|}

In decimal or binary, integer values and terminating fractions have multiple representations. For example, {{sfrac|1|10}} = 0.1 = 0.1{{overline|0}} = 0.0{{overline|9}}. And, {{sfrac|1|2}} = 0.1<sub>2</sub> = 0.1{{overline|0}}<sub>2</sub> = 0.0{{overline|1}}<sub>2</sub>. Some balanced ternary fractions have multiple representations too. For example, {{sfrac|1|6}} = 0.1{{overline|𝖳}}<sub>bal3</sub> = 0.0{{overline|1}}<sub>bal3</sub>. 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]]<ref name="Knuth">{{Cite book
|last=Knuth
|first=Donald
|authorlink=Donald Knuth
|title=The art of Computer Programming
|volume=2
|pages=195–213
|publisher=Addison-Wesley
|year=1997
|isbn=0-201-89684-2}}</ref> has pointed out that truncation and rounding are the same operation in balanced ternary&mdash;they produce exactly the same result (a property shared with other balanced numeral systems). The number {{sfrac|1|2}} is not exceptional; it has two equally valid representations, and two equally valid truncations: 0.{{overline|1}} (round to 0, and truncate to 0) and 1.{{overline|𝖳}} (round to 1, and truncate to 1). With an odd [[radix]], [[Rounding#Double rounding|double rounding]] is also equivalent to directly rounding to the final precision, unlike with an even radix.

The basic operations&mdash;addition, subtraction, multiplication, and division&mdash;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===
<div style="display: flex; column-gap: 1em; margin-inline-start: 1.5em;">
{| class="wikitable" style="text-align: center;"
!Fraction!!colspan="2"|Balanced ternary
|-
|1||colspan="2"|1
|-
|{{sfrac|1|2}}||0.{{overline|1}}||1.{{overline|𝖳}}
|-
|{{sfrac|1|3}}||colspan="2"|0.1
|-
|{{sfrac|1|4}}||colspan="2"|0.{{overline|1𝖳}}
|-
|{{sfrac|1|5}}||colspan="2"|0.{{overline|1𝖳𝖳1}}
|-
|{{sfrac|1|6}}||0.0{{overline|1}}||0.1{{overline|𝖳}}
|-
|{{sfrac|1|7}}||colspan="2"|0.{{overline|0110𝖳𝖳}}
|-
|{{sfrac|1|8}}||colspan="2"|0.{{overline|01}}
|-
|{{sfrac|1|9}}||colspan="2"|0.01
|-
|{{sfrac|1|10}}||colspan="2"|0.{{overline|010𝖳}}
|}
{| class="wikitable" style="text-align: center;"
!Fraction!!colspan="2"|Balanced ternary
|-
|{{sfrac|1|11}}||colspan="2"|0.{{overline|01𝖳11}}
|-
|{{sfrac|1|12}}||colspan="2"|0.0{{overline|1𝖳}}
|-
|{{sfrac|1|13}}||colspan="2"|0.{{overline|01𝖳}}
|-
|{{sfrac|1|14}}||colspan="2"|0.{{overline|01𝖳0𝖳1}}
|-
|{{sfrac|1|15}}||colspan="2"|0.0{{overline|1𝖳𝖳1}}
|-
|{{sfrac|1|16}}||colspan="2"|0.{{overline|01𝖳𝖳}}
|-
|{{sfrac|1|17}}||colspan="2"|0.{{overline|01𝖳𝖳𝖳10𝖳0𝖳111𝖳01}}
|-
|{{sfrac|1|18}}||0.00{{overline|1}}||0.01{{overline|𝖳}}
|-
|{{sfrac|1|19}}||colspan="2"|0.{{overline|00111𝖳10100𝖳𝖳𝖳1𝖳0𝖳}}
|-
|{{sfrac|1|20}}||colspan="2"|0.{{overline|0011}}
|}
</div>

The conversion of a repeating balanced ternary number to a fraction is analogous to [[Repeating decimal#Converting repeating decimals to fractions|converting a repeating decimal]]. For example (because of 111111<sub>bal3</sub> = ({{sfrac|3<sup>6</sup> − 1|3 − 1}})<sub>dec</sub>):
: <math> 0.1\overline{\mathrm{110TT0} } =\tfrac{\mathrm{1110TT0-1} }{\mathrm{111111\times 1T\times 10}}=\tfrac{\mathrm{1110TTT} } {\mathrm{111111\times 1T0}}  =\tfrac{\mathrm{111\times 1000T} } {\mathrm{111\times 1001\times 1T0}} =\tfrac{\mathrm{1111\times 1T}}{\mathrm{1001\times 1T0}} =\tfrac{1111}{10010}=\tfrac{\mathrm{1T1T}}{\mathrm{1TTT0}} =\tfrac{101}{\mathrm{1T10} } </math>

=== Conversion from unbalanced 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,
*: 021<sub>3</sub> + 11<sub>3</sub> = 102<sub>3</sub>, 102<sub>3</sub> − 11<sub>3</sub> = 1T1<sub>bal3</sub> = 7<sub>dec</sub>.
*If a 2 is present in ternary, turn it into 1T. For example,
*: 0212<sub>3</sub> = 0010<sub>bal3</sub> + 1T00<sub>bal3</sub> + 001T<sub>bal3</sub> = 10TT<sub>bal3</sub> = 23<sub>dec</sub>

{| class="wikitable floatright" style=" text-align: center"
|-
! Balanced || Logic   || Unsigned
|-
| 1        || True    || 2
|-
| 0        || Unknown || 1
|-
| T        || False   || 0
|}
If the three values of [[Three-valued logic#Kleene logic|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
:<math>b=\left\lfloor \frac{3^n}{2} \right\rfloor</math>
which is represented as all ones in either conventional or biased form.<ref>Douglas W. Jones, [http://www.cs.uiowa.edu/~jones/ternary/numbers.shtml Ternary Number Systems], October 15, 2013.</ref>

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 {{nowrap|''x'' + ''y'' − ''b''}} when computed using conventional unsigned ternary arithmetic. Similarly, if ''x'' and ''y'' are conventional unsigned ternary numbers, their sum is {{nowrap|''x'' + ''y'' + ''b''}} when computed using balanced ternary arithmetic.

===Conversion from any integer base to balanced ternary===

We may convert to balanced ternary with the following formula:
:<math>
  \left(a_na_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots\right)_b =
    \sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}.
</math>

where,
: ''a<sub>n</sub>a''<sub>''n''−1</sub>...''a''<sub>1</sub>''a''<sub>0</sub>.''c''<sub>1</sub>''c''<sub>2</sub>''c''<sub>3</sub>... is the original representation in the original numeral system. 
: ''b'' is the original radix. ''b'' is 10 if converting from decimal.
: ''a<sub>k</sub>'' and ''c<sub>k</sub>'' are the digits ''k'' places to the left and right of the radix point respectively.

For instance,
  −25.4<sub>dec</sub> = −(1T×101<sup>1</sup> + 1TT×101<sup>0</sup> + 11×101<sup>−1</sup>)
           = −(1T×101 + 1TT + 11÷101)
           = −10T1.{{overline|11TT}}
           =  T01T.{{overline|TT11}}

  1010.1<sub>2</sub> = 1T<sup>10</sup> + 1T<sup>1</sup> + 1T<sup>−1</sup>
            = 10T + 1T + 0.{{overline|1}}
            = 101.{{overline|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 property|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&nbsp;−&nbsp;T&nbsp;= 1T is found in the bottom left corner of the subtraction table.
:{|
|
 :{| class="wikitable" style="width: 8em; text-align: center;"
 |+ Addition
 |- align="right"
 ! + !! T !! 0 !! 1
 |-
 |-
 ! T 
 | T1 || T || 0
 |-
 ! 0 
 | T || 0 || 1
 |-
 ! 1 
 | 0 || 1 || 1T
 |}
|
 :{| class="wikitable" style="width: 8em; text-align: center;"
 |+ Subtraction
 |- align="right"
 ! − !! T !! 0 !! 1
 |-
 ! T
 | 0 || T || T1
 |-
 ! 0
 | 1 || 0 || T
 |-
 ! 1
 | 1T || 1 || 0
 |}
|
 :{| class="wikitable" style="width: 8em; text-align: center;"
 |+ Multiplication
 |- align="right"
 ! × !! T !! 0 !! 1
 |-
 |-
 ! T 
 | 1 || 0 || T
 |-
 ! 0 
 | 0 || 0 || 0
 |-
 ! 1 
 | T || 0 || 1
 |}
|
 :{| class="wikitable" style="text-align: center;"
 |+ Division
 |- align="right"
 ! ÷ !! 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  T1
          ______________         ________________                             ________________
            1T1110.0TT1              1110TT.TTT1                                 1110TT.TTT1
          +   T                    + T   1                                     + T   1
          ______________         ________________                             ________________
            1T0110.0TT1               1100T.TTT1                                  1100T.TTT1

===Multi-trit multiplication===

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

===Multi-trit division===
Balanced ternary division is analogous to that of binary and decimal.

However, 0.5<sub>dec</sub> = 0.1111...<sub>bal3</sub> or 1.TTTT...<sub>bal3</sub>. 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
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. 
:<math>(10\cdot x+y)^{\mathrm{1T}}-100\cdot x^{\mathrm{1T}}=\mathrm{1T0}\cdot x\cdot y+y^{\mathrm{1T}}=
\begin{cases}
\mathrm{T10}\cdot x+1, & y=\mathrm{T} \\
0, & y=0 \\
\mathrm{1T0}\cdot x+1, & y=1
\end{cases}
</math>

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: 
:<math>(10\cdot x+y)^{10}-1000\cdot x^{10}=y^{10}+1000\cdot x^{\mathrm{1T}}\cdot y+ 100\cdot x\cdot y^{\mathrm{1T}}=
\begin{cases}
\mathrm{T}+\mathrm{T000}\cdot x^{\mathrm{1T}}+100\cdot x, & y=\mathrm{T}\\
0, & y=0\\
1+1000\cdot x^{\mathrm{1T}}+100\cdot x, & y=1
\end{cases}

</math> 
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 {{radic|2|3}} = 1.259921<sub>dec</sub> = 1.1T1&nbsp;000&nbsp;111&nbsp;001&nbsp;T01&nbsp;00T&nbsp;1T1&nbsp;T10&nbsp;111<sub>bal3</sub>.

==Irrational numbers==
As in any other integer base, algebraic irrationals and transcendental numbers do not terminate or repeat. For example: 
:{| class="wikitable"
!Decimal !! Balanced ternary
|-
|<math>\sqrt{2}=1.4142135623731\ldots</math> || <math>\sqrt{\mathrm{1T}}=\mathrm{1.11T1TT00T00T01T0T00T00T01TT\ldots}</math>
|-
|<math>\sqrt{3}=1.7320508075689\ldots</math> || <math>\sqrt{\mathrm{10}}=\mathrm{1T.T1TT10T0000TT1100T0TTT011T0\ldots}</math>
|-
|<math>\sqrt{5}=2.2360679774998\ldots</math> || <math>\sqrt{\mathrm{1TT}}=\mathrm{1T.1T0101010TTT1TT11010TTT01T1\ldots}</math>
|-
|<math display="inline">\varphi=\frac{1 + \sqrt{5}}{2}=1.6180339887499\ldots</math> || <math display="inline">\varphi=\frac{1 + \sqrt{\mathrm{1TT}}}{\mathrm{1T}}=\mathrm{1T.T0TT01TT0T10TT11T0011T10011\ldots}</math>
|-
|<math>\tau=6.28318530717959\ldots</math> || <math>\tau=\mathrm{1T0.10TT0T1100T110TT0T1TT000001}\ldots</math>
|-
|<math>\pi=3.14159265358979\ldots</math> || <math>\pi=\mathrm{10.011T111T000T011T1101T111111}\ldots</math>
|-
|<math>e=2.71828182845905\ldots</math> || <math>e=\mathrm{10.T0111TT0T0T111T0111T000T11T}\ldots</math>
|}

The balanced ternary expansions of <math>\pi</math> is given in [[OEIS]] as {{OEIS link|A331313}}, that of <math>e</math> in {{OEIS link|A331990}}.


== Applications ==
=== In computer design ===
[[File:Balanced ternary operation tables.svg|thumb|Operation tables]]<!--Maybe that "al TRIT più alto" table deserves some explanation.-->
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. In balanced ternary, the one-digit [[multiplication table]] remains one-digit and has no carry and the [[addition table]] has only two carries out of nine entries, compared to unbalanced ternary with one and three respectively. Knuth wrote that "Perhaps the symmetric properties and simple arithmetic of this number system will prove to be quite important some day,"<ref name="Knuth"/> noting that,

{{quote|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 <math>\log_3 2 \approx 63 \%</math> as many digit positions for its representation."<ref name="Knuth"/>}}

More recently, balanced ternary numbers have been proposed for some highly-efficient low-resolution implementations of [[artificial neural networks]]. In [[deep learning]], neural nets usually use continuous (floating-point) values, but there are many works investigating quantisation and binarisation to create neural nets that can run with much lower power and/or lower memory requirements. Balanced ternary numbers are proposed to be used for the network parameters, because they are extremely compact, but can naturally represent excitatory/inhibitory/null activation patterns.<ref>{{cite arXiv
| last = Li
| first = Fengfu
| title = Ternary Weight Networks
|date = 2022
| class = cs.CV
| eprint = 1605.04711
}}</ref><ref>{{cite arxiv
| last = Ma
| first = Shuming
| title = The era of 1-bit LLMs: All large language models are in 1.58 bits
| date = 2024
| arxiv = 2402.17764
}}</ref>

Balanced ternary may also provide a more natural representation for the [[qutrit]] and quantum computing systems that use it.

=== Other applications ===
The theorem that every integer has a unique representation in balanced ternary was used by [[Leonhard Euler]] to justify the identity of [[formal power series]]<ref>{{cite journal
 | last = Andrews | first = George E.
 | doi = 10.1090/S0273-0979-07-01180-9
 | issue = 4
 | journal = Bulletin of the American Mathematical Society
 | mr = 2338365
 | pages = 561–573
 | series = New Series
 | title = Euler's "De Partitio numerorum"
 | volume = 44
 | year = 2007| doi-access = free
 }}</ref>
:<math>\prod_{n=0}^{\infty} \left(x^{-3^n}+1+x^{3^n}\right)=\sum_{n=-\infty}^{\infty}x^n.</math>

Balanced ternary has other applications besides computing. For example, a classical two-pan [[Weighing scale#Balance|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 (60<sub>dec</sub> = 1T1T0<sub>bal3</sub>) 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<sub>dec</sub> = 1T1<sub>bal3</sub>), the buyer pays 1¤ + 9¤ and receives 3¤ in change.

==See also==

* [[Signed-digit representation]]
* [[Methods of computing square roots]]
* [[Numeral system]]
* [[Qutrit]]
* [[Salamis Tablet]]
* [[Ternary computer]]
** [[Setun]], a ternary computer
* [[Ternary logic]]
* [[Generalized balanced ternary]]

==References==
{{reflist}}
{{reflist|group=note}}

==External links==
{{commons category|Balanced ternary}}
*[http://www.computer-museum.ru/english/setun.htm Development of ternary computers at Moscow State University]
*[https://web.archive.org/web/20090312094241/http://abhijit.info/tristate/tristate.html Representation of Fractional Numbers in Balanced Ternary]
*[https://www.americanscientist.org/article/third-base "Third base"], ternary and balanced ternary number systems
*[https://web.archive.org/web/20110712221950/http://www.hostsrv.com/webmaa/app1/MSP/webm1010/ternary.msp The Balanced Ternary Number System] (includes decimal integer to balanced ternary converter)
*{{OEIS el|1=A182929|2=The binomial triangle reduced to balanced ternary lists|formalname=The rows of the binomial triangle reduced to balanced ternary lists encoded as decimal numbers}} 
*[http://userpages.wittenberg.edu/bshelburne/BalancedTernaryTalkSu09.pdf Balanced (Signed) Ternary Notation] {{Webarchive|url=https://web.archive.org/web/20160303182504/http://userpages.wittenberg.edu/bshelburne/BalancedTernaryTalkSu09.pdf |date=2016-03-03 }} by Brian J. Shelburne (PDF file)
*[http://www.mortati.com/glusker/fowler/ The ternary calculating machine of Thomas Fowler] by Mark Glusker

[[Category:Computer arithmetic]]
[[Category:Non-standard positional numeral systems]]
[[Category:Ternary computers]]
[[Category:Numeral systems]]

[[de:Ternärsystem#Balanciert]]