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Balanced ternary

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Numeral_system" title="Numeral system">Numeral systems</a></th></tr><tr><td class="sidebar-content-with-subgroup">

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<dl><dt>Contemporary</dt></dl>

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<ul><li><a href="History_of_ancient_numeral_systems" title="History of ancient numeral systems">History</a></li></ul>

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<dl><dt><a href="Ancient_history" title="Ancient history">Ancient</a></dt></dl>

<ul><li><a href="Babylonian_cuneiform_numerals" title="Babylonian cuneiform numerals">Babylonian</a></li></ul>

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<dl><dt><a href="Post-classical_history" title="Post-classical history">Post-classical</a></dt></dl>

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<ul><li><a href="Aksharapalli" title="Aksharapalli">Akṣarapallī</a></li>

<li><a href="%C4%80ryabha%E1%B9%ADa_numeration" title="Āryabhaṭa numeration">Āryabhaṭa</a></li>

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Balanced ternary is a <a href="Ternary_numeral_system" title="Ternary numeral system">ternary numeral system</a> (i.e. base 3 with three <a href="Numerical_digit" title="Numerical digit">digits</a>) that uses a balanced <a href="Signed-digit_representation" title="Signed-digit representation">signed-digit representation</a> of the <a href="Integer" title="Integer">integers</a> in which the digits have the values <a href="%E2%88%921" title="−1">−1</a>, <a href="0" title="0">0</a>, and <a href="1" title="1">1</a>. 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 <a href="Minus_sign" class="mw-redirect" title="Minus sign">minus sign</a>; 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 <a href="Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">non-standard positional numeral system</a>. It was used in some early computers<a href="#cite_note-setun-1">[1]</a> and has also been used to solve <a href="Balance_puzzle" title="Balance puzzle">balance puzzles</a>.<a href="#cite_note-hayes-2">[2]</a>

Different sources use different glyphs to represent the three digits in balanced ternary. In this article, T (which resembles a <a href="Typographical_ligature" class="mw-redirect" title="Typographical ligature">ligature</a> of the minus sign and 1) represents <a href="%E2%88%921" title="−1">−1</a>, while <a href="0" title="0">0</a> and <a href="1" title="1">1</a> represent themselves. Other conventions include using '−' and '+' to represent −1 and 1 respectively, or using <a href="Greek_alphabet" title="Greek alphabet">Greek letter</a> <a href="Theta" title="Theta">theta</a> (Θ), which resembles a minus sign in a circle, to represent −1. In publications about the <a href="Setun" title="Setun">Setun</a> computer, −1 is represented as overturned 1: "1".<a href="#cite_note-setun-1">[1]</a>

Balanced ternary makes an early appearance in <a href="Michael_Stifel" title="Michael Stifel">Michael Stifel</a>'s book Arithmetica Integra (1544).<a href="#cite_note-3">[3]</a> It also occurs in the works of <a href="Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> and <a href="L%C3%A9on_Lalanne" title="Léon Lalanne">Léon Lalanne</a>. Related signed-digit schemes in other bases have been discussed by <a href="John_Colson" title="John Colson">John Colson</a>, <a href="John_Leslie_(physicist)" title="John Leslie (physicist)">John Leslie</a>, <a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a>, and possibly even the ancient Indian <a href="Vedas" title="Vedas">Vedas</a>.<a href="#cite_note-hayes-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Signed-digit_representation" title="Signed-digit representation">Signed-digit representation</a></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{3}:=\lbrace \operatorname {T} ,0,1\rbrace }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{3}:=\lbrace \operatorname {T} ,0,1\rbrace }</annotation>

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</math><img src="./50768e0c94862cfbf8a4702debd777b6b257a4cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.987ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}_{3}:=\lbrace \operatorname {T} ,0,1\rbrace }" loading="lazy"> denote the set of <a href="Symbol_(mathematics)" class="mw-redirect" title="Symbol (mathematics)">symbols</a> (also called glyphs or characters), where the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {1}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {1}}}</annotation>

</semantics>

</math><img src="./33787267e125dbadb71bf531837ecb9858be9838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle {\bar {1}}}" loading="lazy"> is sometimes used in place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {T} .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {T} .}</annotation>

</semantics>

</math><img src="./9d8c4291d8f11ea7c60fe7eff0f4b18c621314a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \operatorname {T} .}" loading="lazy">

Define an <a href="Integer" title="Integer">integer</a>-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=f_{{\mathcal {D}}_{3}}:{\mathcal {D}}_{3}\to \mathbb {Z} }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f=f_{{\mathcal {D}}_{3}}:{\mathcal {D}}_{3}\to \mathbb {Z} }</annotation>

</semantics>

</math><img src="./ad167ebb812484eb938450e8f58be1f8ef67ed8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.795ex; height:2.843ex;" alt="{\displaystyle f=f_{{\mathcal {D}}_{3}}:{\mathcal {D}}_{3}\to \mathbb {Z} }" loading="lazy"> by

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<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{}(\operatorname {T} )&=-1,\\f_{}(0)&=0,\\f_{}(1)&=1,\end{aligned}}}</annotation>

</semantics>

</math><img src="./85dfad073ee59b9bf0dd5f40490deb3bfe70bb2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.233ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}f_{}(\operatorname {T} )&=-1,\\f_{}(0)&=0,\\f_{}(1)&=1,\end{aligned}}}" loading="lazy"><a href="#cite_note-4">[4]</a></dd></dl>

where the right hand sides are integers with their usual values. This function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{},}</annotation>

</semantics>

</math><img src="./cf9664765be195d689b4164b887ca210c62d856b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f_{},}" loading="lazy"> is what rigorously and formally establishes how integer values are assigned to the symbols/glyphs in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{3}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{3}.}</annotation>

</semantics>

</math><img src="./c601269213d6d205741f5fd436099d3f8171169c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.493ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}_{3}.}" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{3}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{3}}</annotation>

</semantics>

</math><img src="./ed525b93bf294538e4252b31cc93182d3444609b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.846ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}_{3}}" loading="lazy"> together with the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{}}</annotation>

</semantics>

</math><img src="./27d9bb421086ca85e928410a414136d78f1df850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f_{}}" loading="lazy"> forms a balanced <a href="Signed-digit_representation" title="Signed-digit representation">signed-digit representation</a> called the balanced ternary system.

It can be used to represent integers and real numbers.

<div class="mw-heading mw-heading3"><h3 id="Ternary_integer_evaluation">Ternary integer evaluation</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{3}^{+}}">

</mrow>

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</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{3}^{+}}</annotation>

</semantics>

</math><img src="./8a699f577a3031dcb819ac6cdfb23bad3d2d2153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.303ex; height:3.176ex;" alt="{\displaystyle {\mathcal {D}}_{3}^{+}}" loading="lazy"> be the <a href="Kleene_plus" class="mw-redirect" title="Kleene plus">Kleene plus</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{3}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{3}}</annotation>

</semantics>

</math><img src="./ed525b93bf294538e4252b31cc93182d3444609b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.846ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}_{3}}" loading="lazy">, which is the set of all finite length <a href="Concatenation" title="Concatenation">concatenated</a> <a href="String_(computer_science)" title="String (computer science)">strings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}\ldots d_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n}\ldots d_{0}}</annotation>

</semantics>

</math><img src="./c7fd6676c95e1327e06a57e8a02330e386e44fea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.188ex; height:2.509ex;" alt="{\displaystyle d_{n}\ldots d_{0}}" loading="lazy"> of one or more symbols (called its digits) where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is a non-negative integer and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

</semantics>

</math><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"> digits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n},\ldots ,d_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n},\ldots ,d_{0}}</annotation>

</semantics>

</math><img src="./135fd9f425e91a1acd53b8e01bcc5f1ca4867561.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.869ex; height:2.509ex;" alt="{\displaystyle d_{n},\ldots ,d_{0}}" loading="lazy"> are taken from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{3}=\lbrace \operatorname {T} ,0,1\rbrace .}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{3}=\lbrace \operatorname {T} ,0,1\rbrace .}</annotation>

</semantics>

</math><img src="./246f45ddd8fd84a389672acd8b41a59f6559f21a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.987ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}_{3}=\lbrace \operatorname {T} ,0,1\rbrace .}" loading="lazy"> The start of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}\ldots d_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n}\ldots d_{0}}</annotation>

</semantics>

</math><img src="./c7fd6676c95e1327e06a57e8a02330e386e44fea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.188ex; height:2.509ex;" alt="{\displaystyle d_{n}\ldots d_{0}}" loading="lazy"> is the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{0}}</annotation>

</semantics>

</math><img src="./4740381c16ea98c4132510daa642e93c1e42c049.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{0}}" loading="lazy"> (at the right), its end is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n}}</annotation>

</semantics>

</math><img src="./858aa7b9662c90d0e01c9615aac9b1fdd84a02dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.427ex; height:2.509ex;" alt="{\displaystyle d_{n}}" loading="lazy"> (at the left), and its length is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

</semantics>

</math><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy">. The ternary evaluation is the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{3}~:~{\mathcal {D}}_{3}^{+}\to \mathbb {Z} }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v=v_{3}~:~{\mathcal {D}}_{3}^{+}\to \mathbb {Z} }</annotation>

</semantics>

</math><img src="./fbb183b7e25dba742bb0322b60c8fd3b40c28620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.973ex; height:3.176ex;" alt="{\displaystyle v=v_{3}~:~{\mathcal {D}}_{3}^{+}\to \mathbb {Z} }" loading="lazy"> defined by assigning to every string <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+}}</annotation>

</semantics>

</math><img src="./232a84cf75c219ec6c83978ad1965ecc75daee9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.331ex; height:3.176ex;" alt="{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+}}" loading="lazy"> the integer

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\left(d_{n}\ldots d_{0}\right)~=~\sum _{i=0}^{n}f_{}\left(d_{i}\right)3^{i}.}</annotation>

</semantics>

</math><img src="./49812183cb6a0c501c3550f4d28d6cc1f66571eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.768ex; height:6.843ex;" alt="{\displaystyle v\left(d_{n}\ldots d_{0}\right)~=~\sum _{i=0}^{n}f_{}\left(d_{i}\right)3^{i}.}" loading="lazy"></dd></dl>

The string <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}\ldots d_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n}\ldots d_{0}}</annotation>

</semantics>

</math><img src="./c7fd6676c95e1327e06a57e8a02330e386e44fea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.188ex; height:2.509ex;" alt="{\displaystyle d_{n}\ldots d_{0}}" loading="lazy"> represents (with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">) the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\left(d_{n}\ldots d_{0}\right).}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\left(d_{n}\ldots d_{0}\right).}</annotation>

</semantics>

</math><img src="./6438c7e57ebe477b4f64cc71b651c22a5979965d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.546ex; height:2.843ex;" alt="{\displaystyle v\left(d_{n}\ldots d_{0}\right).}" loading="lazy"> The value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\left(d_{n}\ldots d_{0}\right)}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\left(d_{n}\ldots d_{0}\right)}</annotation>

</semantics>

</math><img src="./a168ea9bd5f706d3c64dd21d782fb5ab9326eb23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.512ex; height:2.843ex;" alt="{\displaystyle v\left(d_{n}\ldots d_{0}\right)}" loading="lazy"> may alternatively be denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d_{n}\ldots d_{0}}_{\operatorname {bal} 3}.}">

<msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {d_{n}\ldots d_{0}}_{\operatorname {bal} 3}.}</annotation>

</semantics>

</math><img src="./24cf66cf7e005435ee760c81e505303667ac06e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.469ex; height:2.676ex;" alt="{\displaystyle {d_{n}\ldots d_{0}}_{\operatorname {bal} 3}.}" loading="lazy">

The map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v:{\mathcal {D}}_{3}^{+}\to \mathbb {Z} }">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v:{\mathcal {D}}_{3}^{+}\to \mathbb {Z} }</annotation>

</semantics>

</math><img src="./755a524c9b7870b636ce6218241e64e330f25eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.532ex; height:3.176ex;" alt="{\displaystyle v:{\mathcal {D}}_{3}^{+}\to \mathbb {Z} }" loading="lazy"> is <a href="Surjective_map" class="mw-redirect" title="Surjective map">surjective</a> but not injective since, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=v(0)=v(00)=v(000)=\cdots .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0=v(0)=v(00)=v(000)=\cdots .}</annotation>

</semantics>

</math><img src="./0da75b0801b7246d094703095864bedd9960a365.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.099ex; height:2.843ex;" alt="{\displaystyle 0=v(0)=v(00)=v(000)=\cdots .}" loading="lazy"> However, every nonzero integer has exactly one representation under <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> that does not end (on the left) with the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0,}</annotation>

</semantics>

</math><img src="./95547343453ea34a314dd174f8458012f5a39ca3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 0,}" loading="lazy"> i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}=0.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n}=0.}</annotation>

</semantics>

</math><img src="./bd03b8f73837901fa3cc54462599e8f7762e982e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.335ex; height:2.509ex;" alt="{\displaystyle d_{n}=0.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+}}</annotation>

</semantics>

</math><img src="./232a84cf75c219ec6c83978ad1965ecc75daee9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.331ex; height:3.176ex;" alt="{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>

</semantics>

</math><img src="./27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> satisfies:

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 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)}</annotation>

</semantics>

</math><img src="./6ab06a84f4ffab2e2aae24e33af99ded16ceb591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.81ex; height:2.843ex;" alt="{\displaystyle 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)}" loading="lazy"></dd></dl>

which shows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> satisfies a sort of <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>. This recurrence relation has the initial condition

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\left(\varepsilon \right)=0}</annotation>

</semantics>

</math><img src="./6a277b7e14bec0b34bf874dbac11e55c9752a9c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.668ex; height:2.843ex;" alt="{\displaystyle v\left(\varepsilon \right)=0}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> is the empty string.

This implies that for every string <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+},}</annotation>

</semantics>

</math><img src="./eb7fee18eae739f65f76b39663396fe4fadfb9eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.978ex; height:3.176ex;" alt="{\displaystyle d_{n}\ldots d_{0}\in {\mathcal {D}}_{3}^{+},}" loading="lazy">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\left(0d_{n}\ldots d_{0}\right)=v\left(d_{n}\ldots d_{0}\right)}</annotation>

</semantics>

</math><img src="./29ff031187ede3e6e697ce7c90d3f6209f89c969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.285ex; height:2.843ex;" alt="{\displaystyle v\left(0d_{n}\ldots d_{0}\right)=v\left(d_{n}\ldots d_{0}\right)}" loading="lazy"></dd></dl>

which in words says that leading <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> can be computed, where (as before) all integer are written in decimal (base 10) and all elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{3}^{+}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{3}^{+}}</annotation>

</semantics>

</math><img src="./8a699f577a3031dcb819ac6cdfb23bad3d2d2153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.303ex; height:3.176ex;" alt="{\displaystyle {\mathcal {D}}_{3}^{+}}" loading="lazy"> are just symbols.

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{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(11\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(10\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{alignedat}}}</annotation>

</semantics>

</math><img src="./14ba4b982fdea5a325077a5a2cf9f0b75e0dcceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.073ex; margin-bottom: -0.265ex; width:69.706ex; height:19.843ex;" alt="{\displaystyle {\begin{alignedat}{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(11\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(10\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{alignedat}}}" loading="lazy"></dd></dl>

and using the above recurrence relation

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\left(101\operatorname {T} \right)=f_{}\left(1\right)3^{3}+v\left(01\operatorname {T} \right)=(1)27+v\left(1\operatorname {T} \right)=27+2=29.}</annotation>

</semantics>

</math><img src="./a4404186d05bb49bc943bf6e1250f853f8c7ca65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.054ex; height:3.176ex;" alt="{\displaystyle v\left(101\operatorname {T} \right)=f_{}\left(1\right)3^{3}+v\left(01\operatorname {T} \right)=(1)27+v\left(1\operatorname {T} \right)=27+2=29.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Conversions_to/from_other_representations">Conversions to/from other representations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Conversion_to_decimal">Conversion to decimal</h3></div>

In the balanced ternary system the value of a digit n places left of the <a href="Radix_point" class="mw-redirect" title="Radix point">radix point</a> 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,

Similarly, the first place to the right of the radix point holds 3−1 = <style data-mw-deduplicate="TemplateStyles:r1214402035">

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style>⁠1/3⁠, the second place holds 3−2 = ⁠1/9⁠, and so on. For instance,

<th>Expansion

</th></tr>

<tr>

<td>0

</td></tr>

<tr>

<td>+1

</td></tr>

<tr>

<td>+3−1

</td></tr>

<tr>

<td>+3

</td></tr>

<tr>

<td>+3+1

</td></tr>

<tr>

<td>+9−3−1

</td></tr>

<tr>

<td>+9−3

</td></tr>

<tr>

<td>+9−3+1

</td></tr>

<tr>

<td>+9−1

</td></tr>

<tr>

<td>+9

</td></tr>

<tr>

<td>+9+1

</td></tr>

<tr>

<td>+9+3−1

</td></tr>

<tr>

<td>+9+3

</td></tr>

<tr>

<td>+9+3+1

</td></tr></tbody></table>

<th>Expansion

</th></tr>

<tr>

<td>0

</td></tr>

<tr>

<td>−1

</td></tr>

<tr>

<td>−3+1

</td></tr>

<tr>

<td>−3

</td></tr>

<tr>

<td>−3−1

</td></tr>

<tr>

<td>−9+3+1

</td></tr>

<tr>

<td>−9+3

</td></tr>

<tr>

<td>−9+3−1

</td></tr>

<tr>

<td>−9+1

</td></tr>

<tr>

<td>−9

</td></tr>

<tr>

<td>−9−1

</td></tr>

<tr>

<td>−9−3+1

</td></tr>

<tr>

<td>−9−3

</td></tr>

<tr>

<td>−9−3−1

</td></tr></tbody></table>

</div>

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

We may check the <a href="Parity_(mathematics)" title="Parity (mathematics)">parity</a> 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 <a href="Radix_point" class="mw-redirect" title="Radix point">radix point</a>.<a href="#cite_note-5">[5]</a>

<th>Decimal

</th>

<th style="text-align: right">−0.9

</th>

<th style="text-align: right">−0.8

</th>

<th style="text-align: right">−0.7

</th>

<th style="text-align: right">−0.6

</th>

<th style="text-align: right">−0.5

</th>

<th style="text-align: right">−0.4

</th>

<th style="text-align: right">−0.3

</th>

<th style="text-align: right">−0.2

</th>

<th style="text-align: right">−0.1

</th>

<th style="text-align: right">0

</th></tr>

<tr>

<th>Balanced Ternary

</th>

<td>0

</td></tr>

<tr>

<th>Decimal

</th>

<th style="text-align: right">0.9

</th>

<th style="text-align: right">0.8

</th>

<th style="text-align: right">0.7

</th>

<th style="text-align: right">0.6

</th>

<th style="text-align: right">0.5

</th>

<th style="text-align: right">0.4

</th>

<th style="text-align: right">0.3

</th>

<th style="text-align: right">0.2

</th>

<th style="text-align: right">0.1

</th>

<th style="text-align: right">0

</th></tr>

<tr>

<th>Balanced Ternary

</th>

<td>0

</td></tr></tbody></table></dd></dl>

In 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 = 0.1𝖳bal3 = 0.01bal3. Certainly, in the decimal and binary, we may omit the rightmost trailing infinite 0s after the radix point and gain a representations of integer or terminating fraction. But, in balanced ternary, we can't omit the rightmost trailing infinite −1s after the radix point in order to gain a representations of integer or terminating fraction.

<a href="Donald_Knuth" title="Donald Knuth">Donald Knuth</a><a href="#cite_note-Knuth-6">[6]</a> has pointed out that truncation and rounding are the same operation in balanced ternary—they produce exactly the same result (a property shared with other balanced numeral systems). The number ⁠1/2⁠ is not exceptional; it has two equally valid representations, and two equally valid truncations: 0.1 (round to 0, and truncate to 0) and 1.𝖳 (round to 1, and truncate to 1). With an odd <a href="Radix" title="Radix">radix</a>, <a href="Rounding#Double_rounding" title="Rounding">double rounding</a> is also equivalent to directly rounding to the final precision, unlike with an even radix.

The basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary. Multiplication by two can be done by adding a number to itself, or subtracting itself after a-trit-left-shifting.

An arithmetic shift left of a balanced ternary number is the equivalent of multiplication by a (positive, integral) power of 3; and an arithmetic shift right of a balanced ternary number is the equivalent of division by a (positive, integral) power of 3.

<div class="mw-heading mw-heading3"><h3 id="Conversion_to_and_from_a_fraction">Conversion to and from a fraction</h3></div>

<th>Fraction</th>

<th colspan="2">Balanced ternary

</th></tr>

<tr>

<td colspan="2">1

</td></tr>

<tr>

<td>1.𝖳

</td></tr>

<tr>

<td colspan="2">0.1

</td></tr>

<tr>

<td colspan="2">0.1𝖳

</td></tr>

<tr>

<td colspan="2">0.1𝖳𝖳1

</td></tr>

<tr>

<td>0.1𝖳

</td></tr>

<tr>

<td colspan="2">0.0110𝖳𝖳

</td></tr>

<tr>

<td colspan="2">0.01

</td></tr>

<tr>

<td colspan="2">0.01

</td></tr>

<tr>

<td colspan="2">0.010𝖳

</td></tr></tbody></table>

<th>Fraction</th>

<th colspan="2">Balanced ternary

</th></tr>

<tr>

<td colspan="2">0.01𝖳11

</td></tr>

<tr>

<td colspan="2">0.01𝖳

</td></tr>

<tr>

<td colspan="2">0.01𝖳

</td></tr>

<tr>

<td colspan="2">0.01𝖳0𝖳1

</td></tr>

<tr>

<td colspan="2">0.01𝖳𝖳1

</td></tr>

<tr>

<td colspan="2">0.01𝖳𝖳

</td></tr>

<tr>

<td colspan="2">0.01𝖳𝖳𝖳10𝖳0𝖳111𝖳01

</td></tr>

<tr>

<td>0.01𝖳

</td></tr>

<tr>

<td colspan="2">0.00111𝖳10100𝖳𝖳𝖳1𝖳0𝖳

</td></tr>

<tr>

<td colspan="2">0.0011

</td></tr></tbody></table>

</div>

The conversion of a repeating balanced ternary number to a fraction is analogous to <a href="Repeating_decimal#Converting_repeating_decimals_to_fractions" title="Repeating decimal">converting a repeating decimal</a>. For example (because of 111111bal3 = (⁠36 − 1/3 − 1⁠)dec):

<mover>

</mrow>

</mover>

</mrow>

<mfrac>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 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} }}}</annotation>

</semantics>

</math><img src="./cf5e33c66091cb063900ba4af41d0b3a9eb35c33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:90.322ex; height:4.009ex;" alt="{\displaystyle 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} }}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Conversion_from_unbalanced_ternary">Conversion from unbalanced ternary</h3></div>

Unbalanced ternary can be converted to balanced ternary notation in two ways:

<ul><li>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,

<li>If a 2 is present in ternary, turn it into 1T. For example,

<dl><dd>02123 = 0010bal3 + 1T00bal3 + 001Tbal3 = 10TTbal3 = 23dec</dd></dl></li></ul>

<th>Balanced</th>

<th>Logic</th>

<th>Unsigned

</th></tr>

<tr>

<td>2

</td></tr>

<tr>

<td>Unknown</td>

<td>1

</td></tr>

<tr>

<td>False</td>

<td>0

</td></tr></tbody></table>

If the three values of <a href="Three-valued_logic#Kleene_logic" title="Three-valued logic">ternary logic</a> 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 <a href="Offset_binary" title="Offset binary">offset binary</a> system.

If the ternary number has n trits, then the bias b is

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<annotation encoding="application/x-tex">{\displaystyle b=\left\lfloor {\frac {3^{n}}{2}}\right\rfloor }</annotation>

</semantics>

</math><img src="./bd90a3e3408552cb19185b332958545f35c9cd2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.024ex; height:6.176ex;" alt="{\displaystyle b=\left\lfloor {\frac {3^{n}}{2}}\right\rfloor }" loading="lazy"></dd></dl>

which is represented as all ones in either conventional or biased form.<a href="#cite_note-7">[7]</a>

As a result, if these two representations are used for balanced and unsigned ternary numbers, an unsigned n-trit positive ternary value can be converted to balanced form by adding the bias b and a positive balanced number can be converted to unsigned form by subtracting the bias b. Furthermore, if x and y are balanced numbers, their balanced sum is x + y − b when computed using conventional unsigned ternary arithmetic. Similarly, if x and y are conventional unsigned ternary numbers, their sum is x + y + b when computed using balanced ternary arithmetic.

<div class="mw-heading mw-heading3"><h3 id="Conversion_from_any_integer_base_to_balanced_ternary">Conversion from any integer base to balanced ternary</h3></div>

We may convert to balanced ternary with the following formula:

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<annotation encoding="application/x-tex">{\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}.}</annotation>

</semantics>

</math><img src="./6ccdceb36308535e6a26af8e00bb06ba7f727943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.072ex; height:7.009ex;" alt="{\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}.}" loading="lazy"></dd></dl>

where,

<dl><dd>anan−1...a1a0.c1c2c3... is the original representation in the original numeral system.</dd>

<dd>b is the original radix. b is 10 if converting from decimal.</dd>

<dd>ak and ck are the digits k places to the left and right of the radix point respectively.</dd></dl>

For instance,

<pre> −25.4dec = −(1T×1011 + 1TT×1010 + 11×101−1)

= −(1T×1011 + 1TT×1010 + 11×101T)

= −(1T×101 + 1TT + 11×0.010T)

= −(1T1T + 1TT + 0.11TT)

= −10T1.11TT

= T01T.TT11

</pre>

<pre> 1010.12 = 1T10 + 1T1 + 1T−1

= 1T10 + 1T1 + 1TT

= 10T + 1T + 0.1

= 101.1

</pre>

<div class="mw-heading mw-heading2"><h2 id="Addition,_subtraction_and_multiplication_and_division">Addition, subtraction and multiplication and division</h2></div>

The single-trit addition, subtraction, multiplication and division tables are shown below. For subtraction and division, which are not <a href="Commutative_property" title="Commutative property">commutative</a>, the first operand is given to the left of the table, while the second is given at the top. For instance, the answer to 1 − T = 1T is found in the bottom left corner of the subtraction table.

<td>

<caption>Addition

</caption>

<th>1

</th></tr>

<tr>

<th>T

</th>

<td>0

</td></tr>

<tr>

<th>0

</th>

<td>1

</td></tr>

<tr>

<th>1

</th>

<td>1T

</td></tr></tbody></table></dd></dl>

</td>

<td>

<caption>Subtraction

</caption>

<th>1

</th></tr>

<tr>

<th>T

</th>

<td>T1

</td></tr>

<tr>

<th>0

</th>

<td>T

</td></tr>

<tr>

<th>1

</th>

<td>0

</td></tr></tbody></table></dd></dl>

</td>

<td>

<caption>Multiplication

</caption>

<th>1

</th></tr>

<tr>

<th>T

</th>

<td>T

</td></tr>

<tr>

<th>0

</th>

<td>0

</td></tr>

<tr>

<th>1

</th>

<td>1

</td></tr></tbody></table></dd></dl>

</td>

<td>

<caption>Division

</caption>

<th>1

</th></tr>

<tr>

<th>T

</th>

<td>T

</td></tr>

<tr>

<th>0

</th>

<td>0

</td></tr>

<tr>

<th>1

</th>

<td>1

</td></tr></tbody></table></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Multi-trit_addition_and_subtraction">Multi-trit addition and subtraction</h3></div>

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:

<pre> 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

</pre>

<div class="mw-heading mw-heading3"><h3 id="Multi-trit_multiplication">Multi-trit multiplication</h3></div>

Multi-trit multiplication is analogous to that of binary and decimal.

<pre> 1TT1.TT

× T11T.1

_____________

1TT.1TT multiply 1

T11T.11 multiply T

1TT1T.T multiply 1

1TT1TT multiply 1

T11T11 multiply T

_____________

0T0000T.10T

</pre>

<div class="mw-heading mw-heading3"><h3 id="Multi-trit_division">Multi-trit division</h3></div>

Balanced ternary division is analogous to that of binary and decimal.

However, 0.5dec = 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,

<pre> 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

</pre>

Another example,

<pre> 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

</pre>

Another example,

<pre> 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...

</pre>

In balanced ternary, there is a simpler division method that does not require comparing the remainders and operates directly according to the rules. This method originated from <a href="Donald_Knuth" title="Donald Knuth">Donald Knuth</a> trial <a rel="nofollow" class="external text" href="https://dfns.dyalog.com/n_bt.htm">quotient method</a>, but this method is not perfect. Later, it was improved by other scholars, completely abandoning the judgment of semi-closed intervals and instead using the condition of whether the highest bit of the remainder of each round of division is 0 to determine whether the round of division is over, and named it the double trial quotient method: the core logic is to split the dividend into two vectors (p and s), where the number of digits of p is at most the same as the divisor q, and s is the vector of the remaining digits; then according to the rule (double positive or double negative gives a positive quotient, one positive and one negative gives a negative quotient, and the others are 0), the quotient is subtracted at most once or twice in each round of division. If the highest bit of the remainder p is still not 0 after subtraction once, it needs to be subtracted again. If the highest bit of the remainder p is 0, the round of division is over, and the bit is the double quotient. The value is doubled and the result is given to register r; then the remainder p Shift left one bit, discard the highest bit of the previous round's divisor (already 0), pull one bit of s to supplement it, and get the new remainder p to carry on to the next round. The result r (3 times magnified) is shifted right one bit and supplemented with 0. Then, when s is exhausted, the result r and the dividend p constitute the final quotient and remainder.

<pre> +

++0- ++

_____________ _______

+0- )+0++0+ +- )+0+

-0+ -+

____________ ______

0+-+ 0++

-0+ -+

____________ ______

00-0 +-

000 -+

____________ ______

-0+ 0

+0-

____________ Add +0 and +- ,quotient get +--

0

</pre>

<div class="mw-heading mw-heading2"><h2 id="Square_roots_and_cube_roots">Square roots and cube roots</h2></div>

The process of extracting the <a href="Square_root" title="Square root">square root</a> in balanced ternary is analogous to that in decimal or binary.

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<annotation encoding="application/x-tex">{\displaystyle (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}}}</annotation>

</semantics>

</math><img src="./83c930356757c3a3a467a651aee17470c63e42c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:69.976ex; height:8.509ex;" alt="{\displaystyle (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}}}" loading="lazy"></dd></dl>

As in division, we should check the value of half the divisor first. For example,

<pre> 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

...

</pre>

Extraction of the cube root in balanced ternary is similarly analogous to extraction in decimal or binary:

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<annotation encoding="application/x-tex">{\displaystyle (10\cdot x+y)^{10}-1000\cdot x^{10}=1000\cdot x^{\mathrm {1T} }\cdot y+100\cdot x\cdot y^{\mathrm {1T} }+y^{10}={\begin{cases}\mathrm {T000} \cdot x^{\mathrm {1T} }+100\cdot x+\mathrm {T} ,&y=\mathrm {T} \\0,&y=0\\1000\cdot x^{\mathrm {1T} }+100\cdot x+1,&y=1\end{cases}}}</annotation>

</semantics>

</math><img src="./3ff80005882182a522174770e1b11cd63ab08253.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.487ex; margin-bottom: -0.184ex; width:100.604ex; height:8.509ex;" alt="{\displaystyle (10\cdot x+y)^{10}-1000\cdot x^{10}=1000\cdot x^{\mathrm {1T} }\cdot y+100\cdot x\cdot y^{\mathrm {1T} }+y^{10}={\begin{cases}\mathrm {T000} \cdot x^{\mathrm {1T} }+100\cdot x+\mathrm {T} ,&y=\mathrm {T} \\0,&y=0\\1000\cdot x^{\mathrm {1T} }+100\cdot x+1,&y=1\end{cases}}}" loading="lazy"></dd></dl>

Like division, we should check the value of half the divisor first too.

For example:

<pre> 1. 1 T 1 0 ...

_____________________

¹⁰√ 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

...

</pre>

Hence 3√2dec = 10√1Tbal3 = 1.259921dec = 1.1T1 000 111 001 T01 00T 1T1 T10 111bal3.

<div class="mw-heading mw-heading2"><h2 id="Irrational_numbers">Irrational numbers</h2></div>

As in any other integer base, algebraic irrationals and transcendental numbers do not terminate or repeat. For example:

<th>Decimal</th>

<th>Balanced ternary

</th></tr>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=1.4142135623731\ldots }</annotation>

</semantics>

</math><img src="./a599124d2f0e66f17e809c668dd70c5d40a5f843.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.228ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}=1.4142135623731\ldots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\mathrm {1T} }}=\mathrm {1.11T1TT00T00T01T0T00T00T01TT\ldots } }</annotation>

</semantics>

</math><img src="./c25c74444a63f8a0650a112bbf2f20bb602ed186.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:49.853ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\mathrm {1T} }}=\mathrm {1.11T1TT00T00T01T0T00T00T01TT\ldots } }" loading="lazy">

</td></tr>

<tr>

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}=1.7320508075689\ldots }</annotation>

</semantics>

</math><img src="./1dc19e2ae003e783cae5cabe6d79ab80cba629e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.228ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}=1.7320508075689\ldots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\mathrm {10} }}=\mathrm {1T.T1TT10T0000TT1100T0TTT011T0\ldots } }</annotation>

</semantics>

</math><img src="./e848cf41e8b8c1a07765a099ae239e57fc75e538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:51.402ex; height:2.843ex;" alt="{\displaystyle {\sqrt {\mathrm {10} }}=\mathrm {1T.T1TT10T0000TT1100T0TTT011T0\ldots } }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}=2.2360679774998\ldots }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\mathrm {1TT} }}=\mathrm {1T.1T0101010TTT1TT11010TTT01T1\ldots } }</annotation>

</semantics>

</math><img src="./a7e1d079ba9ee31810d2bcb1172303a8e31262c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:52.693ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\mathrm {1TT} }}=\mathrm {1T.1T0101010TTT1TT11010TTT01T1\ldots } }" loading="lazy">

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<annotation encoding="application/x-tex">{\textstyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887499\ldots }</annotation>

</semantics>

</math><img src="./47b62f1af1081c73913310774f45b2244af3cb68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.876ex; height:4.176ex;" alt="{\textstyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887499\ldots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\textstyle \varphi ={\frac {1+{\sqrt {\mathrm {1TT} }}}{\mathrm {1T} }}=\mathrm {1T.T0TT01TT0T10TT11T0011T10011\ldots } }</annotation>

</semantics>

</math><img src="./134da5439db965a4f9d70df8c95506a86366d56d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:58.745ex; height:4.176ex;" alt="{\textstyle \varphi ={\frac {1+{\sqrt {\mathrm {1TT} }}}{\mathrm {1T} }}=\mathrm {1T.T0TT01TT0T10TT11T0011T10011\ldots } }" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =6.28318530717959\ldots }</annotation>

</semantics>

</math><img src="./fc17d01d3cb58ae3bdb10ee0cd2f7a1312b6f22b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:25.494ex; height:2.176ex;" alt="{\displaystyle \tau =6.28318530717959\ldots }" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =\mathrm {1T0.10TT0T1100T110TT0T1TT000001} \ldots }</annotation>

</semantics>

</math><img src="./5df3d54b54130acbcc3d743db2090f009d18fc67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:48.087ex; height:2.176ex;" alt="{\displaystyle \tau =\mathrm {1T0.10TT0T1100T110TT0T1TT000001} \ldots }" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi =3.14159265358979\ldots }</annotation>

</semantics>

</math><img src="./6de554f9fba9eb9849cf9f5ed6d193922b0dc765.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:25.624ex; height:2.176ex;" alt="{\displaystyle \pi =3.14159265358979\ldots }" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi =\mathrm {10.011T111T000T011T1101T111111} \ldots }</annotation>

</semantics>

</math><img src="./569017f1e8f048a919c2044b697b4a1768c454bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:44.477ex; height:2.176ex;" alt="{\displaystyle \pi =\mathrm {10.011T111T000T011T1101T111111} \ldots }" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e=2.71828182845905\ldots }</annotation>

</semantics>

</math><img src="./36477e2c60a49407e38dbc4679274727973236dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:25.376ex; height:2.176ex;" alt="{\displaystyle e=2.71828182845905\ldots }" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e=\mathrm {10.T0111TT0T0T111T0111T000T11T} \ldots }</annotation>

</semantics>

</math><img src="./f052b2921df42f2fab67ecc8ebfa40823756f944.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:46.291ex; height:2.176ex;" alt="{\displaystyle e=\mathrm {10.T0111TT0T0T111T0111T000T11T} \ldots }" loading="lazy">

</td></tr></tbody></table></dd></dl>

The balanced ternary expansions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> is given in <a href="OEIS" class="mw-redirect" title="OEIS">OEIS</a> as <a href="https://oeis.org/A331313" class="extiw external" title="oeis:A331313">A331313</a>, that of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> in <a href="https://oeis.org/A331990" class="extiw external" title="oeis:A331990">A331990</a>.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="In_computer_design">In computer design</h3></div>

In the early days of computing, a few experimental Soviet computers were built with balanced ternary instead of binary, the most famous being the <a href="Setun" title="Setun">Setun</a>, built by <a href="Nikolay_Brusentsov" title="Nikolay Brusentsov">Nikolay Brusentsov</a> and <a href="Sergei_Sobolev" title="Sergei Sobolev">Sergei Sobolev</a>. 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 <a href="Multiplication_table" title="Multiplication table">multiplication table</a> remains one-digit and has no carry and the <a href="Addition_table" class="mw-redirect" title="Addition table">addition table</a> 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,"<a href="#cite_note-Knuth-6">[6]</a> noting that,

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</style><blockquote class="templatequote">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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{3}2\approx 63\%}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log _{3}2\approx 63\%}</annotation>

</semantics>

</math><img src="./46ef74db519d35cf6361c4785026ff1a3aa95257.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.935ex; height:2.843ex;" alt="{\displaystyle \log _{3}2\approx 63\%}" loading="lazy"> as many digit positions for its representation."<a href="#cite_note-Knuth-6">[6]</a></blockquote>

More recently, balanced ternary numbers have been proposed for some highly-efficient low-resolution implementations of <a href="Artificial_neural_networks" class="mw-redirect" title="Artificial neural networks">artificial neural networks</a>. In <a href="Deep_learning" title="Deep learning">deep learning</a>, 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.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a>

Balanced ternary may also provide a more natural representation for the <a href="Qutrit" title="Qutrit">qutrit</a> and quantum computing systems that use it.

<div class="mw-heading mw-heading3"><h3 id="Other_applications">Other applications</h3></div>

The theorem that every integer has a unique representation in balanced ternary was used by <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> to justify the identity of <a href="Formal_power_series" title="Formal power series">formal power series</a><a href="#cite_note-10">[10]</a>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{n=0}^{\infty }\left(x^{-3^{n}}+1+x^{3^{n}}\right)=\sum _{n=-\infty }^{\infty }x^{n}.}</annotation>

</semantics>

</math><img src="./398f6d4fab1ed573bd34741c9ace94b9a571377e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.936ex; height:6.843ex;" alt="{\displaystyle \prod _{n=0}^{\infty }\left(x^{-3^{n}}+1+x^{3^{n}}\right)=\sum _{n=-\infty }^{\infty }x^{n}.}" loading="lazy"></dd></dl>

Balanced ternary has other applications besides computing. For example, a classical two-pan <a href="Weighing_scale#Balance" title="Weighing scale">balance</a>, 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 (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.

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¤ (7dec = 1T1bal3), the buyer pays 1¤ + 9¤ and receives 3¤ in change.

<ul><li><a href="Signed-digit_representation" title="Signed-digit representation">Signed-digit representation</a></li>

<li><a href="Methods_of_computing_square_roots" class="mw-redirect" title="Methods of computing square roots">Methods of computing square roots</a></li>

<li><a href="Numeral_system" title="Numeral system">Numeral system</a></li>

<li><a href="Qutrit" title="Qutrit">Qutrit</a></li>

<li><a href="Salamis_Tablet" title="Salamis Tablet">Salamis Tablet</a></li>

<li><a href="Ternary_computer" title="Ternary computer">Ternary computer</a>

<ul><li><a href="Setun" title="Setun">Setun</a>, a ternary computer</li></ul></li>

<li><a href="Ternary_logic" class="mw-redirect" title="Ternary logic">Ternary logic</a></li>

<li><a href="Generalized_balanced_ternary" title="Generalized balanced ternary">Generalized balanced ternary</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFN._A._KrinitskyG._A._MironovG._D._Frolov1963" class="citation book cs1 cs1-prop-foreign-lang-source">N. A. Krinitsky; G. A. Mironov; G. D. Frolov (1963). "Chapter 10. Program-controlled machine Setun". In M. R. Shura-Bura (ed.). Programming (in Russian). Moscow.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link)

</li>

<li id="cite_note-hayes-2">^ <a href="#cite_ref-hayes_2-0">a</a> <a href="#cite_ref-hayes_2-1">b</a> <cite id="CITEREFHayes2001" class="citation cs2"><a href="Brian_Hayes_(scientist)" title="Brian Hayes (scientist)">Hayes, Brian</a> (2001), <a rel="nofollow" class="external text" href="http://bit-player.org/bph-publications/AmSci-2001-11-Hayes-ternary.pdf">"Third base"</a> (PDF), American Scientist, 89 (6): 490–494, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1511%2F2001.40.3268">10.1511/2001.40.3268</a></cite>. Reprinted in <cite id="CITEREFHayes2008" class="citation cs2"><a href="Brian_Hayes_(scientist)" title="Brian Hayes (scientist)">Hayes, Brian</a> (2008), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1ZkYEFi3DMMC&pg=PA179">Group Theory in the Bedroom, and Other Mathematical Diversions</a>, Farrar, Straus and Giroux, pp. 179–200, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781429938570</bdi></cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFStifel1544" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Michael_Stifel" title="Michael Stifel">Stifel, Michael</a> (1544), <a rel="nofollow" class="external text" href="https://archive.org/stream/bub_gb_ywkW9hDd7IIC#page/n85/mode/2up">Arithmetica integra</a> (in Latin), apud Iohan Petreium, p. 38</cite>.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> The symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> appear twice in the equalities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{}(0)=0}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f_{}(0)=0}</annotation>

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</math><img src="./359cf947c64090f9d6bc2e2250a082c347c326cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f_{}(0)=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{}(1)=1,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{}(1)=1,}</annotation>

</semantics>

</math><img src="./f25386adbe6d79bce0df8e7a957f2e97a7725b63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.158ex; height:2.843ex;" alt="{\displaystyle f_{}(1)=1,}" loading="lazy"> but these instances do not represent the same thing. The right hand side <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> mean the integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \in \mathbb {Z} ,}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \in \mathbb {Z} ,}</annotation>

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</math><img src="./fc467ab47a0a52911acad5509d669c0da9805dcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.393ex; height:2.509ex;" alt="{\displaystyle \in \mathbb {Z} ,}" loading="lazy"> but the instances inside <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">'s parentheses (which belong to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{3}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{3}}</annotation>

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</math><img src="./ed525b93bf294538e4252b31cc93182d3444609b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.846ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}_{3}}" loading="lazy">) should be thought of as being nothing more than symbols.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBhattacharjee2006" class="citation web cs1">Bhattacharjee, Abhijit (24 July 2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090919053547/http://www.abhijit.info/tristate/tristate.html">"Balanced ternary"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.abhijit.info/tristate/tristate.html">the original</a> on 2009-09-19.</cite>

</li>

<li id="cite_note-Knuth-6">^ <a href="#cite_ref-Knuth_6-0">a</a> <a href="#cite_ref-Knuth_6-1">b</a> <a href="#cite_ref-Knuth_6-2">c</a> <cite id="CITEREFKnuth1997" class="citation book cs1"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1997). The art of Computer Programming. Vol. 2. Addison-Wesley. pp. 195–213. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-89684-2</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Douglas W. Jones, <a rel="nofollow" class="external text" href="http://www.cs.uiowa.edu/~jones/ternary/numbers.shtml">Ternary Number Systems</a>, October 15, 2013.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFLi2022" class="citation arxiv cs1">Li, Fengfu (2022). "Ternary Weight Networks". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1605.04711">1605.04711</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.CV">cs.CV</a>].</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFMa2024" class="citation arxiv cs1">Ma, Shuming (2024). "The era of 1-bit LLMs: All large language models are in 1.58 bits". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2402.17764">2402.17764</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.CL">cs.CL</a>].</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFAndrews2007" class="citation journal cs1">Andrews, George E. (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-07-01180-9">"Euler's "De Partitio numerorum""</a>. Bulletin of the American Mathematical Society. New Series. 44 (4): 561–573. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-07-01180-9">10.1090/S0273-0979-07-01180-9</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2338365">2338365</a>.</cite>

</li>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Balanced_ternary" class="extiw external" title="commons:Category:Balanced ternary">Balanced ternary</a>.</div></div>

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<ul><li><a rel="nofollow" class="external text" href="http://www.computer-museum.ru/english/setun.htm">Development of ternary computers at Moscow State University</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090312094241/http://abhijit.info/tristate/tristate.html">Representation of Fractional Numbers in Balanced Ternary</a></li>

<li><a rel="nofollow" class="external text" href="https://www.americanscientist.org/article/third-base">"Third base"</a>, ternary and balanced ternary number systems</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110712221950/http://www.hostsrv.com/webmaa/app1/MSP/webm1010/ternary.msp">The Balanced Ternary Number System</a> (includes decimal integer to balanced ternary converter)</li>

<li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A182929">sequence A182929 (The binomial triangle reduced to balanced ternary lists)</a></li>

<li><a rel="nofollow" class="external text" href="http://userpages.wittenberg.edu/bshelburne/BalancedTernaryTalkSu09.pdf">Balanced (Signed) Ternary Notation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303182504/http://userpages.wittenberg.edu/bshelburne/BalancedTernaryTalkSu09.pdf">Archived</a> 2016-03-03 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> by Brian J. Shelburne (PDF file)</li>

<li><a rel="nofollow" class="external text" href="http://www.mortati.com/glusker/fowler/">The ternary calculating machine of Thomas Fowler</a> by Mark Glusker</li></ul></div><div><div class="zim-footer">

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