Editing Ternary numeral system (section)

== Comparison to other bases ==

Representations of [[integer number]]s in ternary do not get uncomfortably lengthy as quickly as in [[binary numeral system|binary]]. For example, [[decimal]] [[365 (number)|365]]{{sub|(10)}} or [[senary]] {{gaps|1|405}}{{sub|(6)}} corresponds to binary {{gaps|1|0110|1101}}{{sub|(2)}} (nine [[bit]]s) and to ternary {{gaps|111|112}}{{sub|(3)}} (six digits). However, they are still far less compact than the corresponding representations in bases such as [[decimal]] – see below for a compact way to codify ternary using nonary (base&nbsp;9) and [[septemvigesimal]] (base&nbsp;27).

{| class="wikitable" style="float:right; text-align:center"
|+ A ternary [[multiplication table]]
|-
! × || '''1'''|| '''2''' || '''10''' || '''11''' || '''12''' || '''20''' || '''21''' || '''22''' || '''100''' 
|-
! '''1'''
| 1 || 2 || 10 || 11 || 12 || 20 || 21 || 22 || 100
|-
! '''2'''
| 2 || 11 || 20 || 22 || 101 || 110 || 112 || 121 || 200
|-
! '''10'''
| 10 || 20 || 100 || 110 || 120 || 200 || 210 || 220 || 1,000
|-
! '''11'''
| 11 || 22 || 110 || 121 || 202 || 220 || 1,001 || 1,012 || 1,100 
|-
! '''12'''
| 12 || 101 || 120 || 202 || 221 || 1,010
| 1,022 || 1,111 || 1,200 
|-
! '''20'''
| 20 || 110 || 200 || 220 || 1,010 || 1,100
| 1,120 || 1,210 || 2,000
|-
! '''21'''
| 21 || 112 || 210 || 1,001 || 1,022 || 1,120
| 1,211 || 2,002 || 2,100
|-
! '''22'''
| 22 || 121 || 220 || 1,012 || 1,111 || 1,210
| 2,002 || 2,101 || 2,200
|-
! '''100'''
| 100 || 200 || 1,000 || 1,100 || 1,200 || 2,000
| 2,100 || 2,200 || 10,000
|}

:{| class="wikitable"
|+ '''Numbers from 0 to 3<sup>3</sup> − 1 in standard ternary'''
|- align="center"
! Ternary
| 0 || 1 || 2 || 10 || 11 || 12 || 20 || 21 || 22
|- align="center"
! Binary
| 0 || 1 || 10 || 11 || 100 || 101 || 110 || 111 || {{gaps|1|000}}
|- align="center"
! Senary
| 0 || 1 || 2 || 3 || 4 || 5 || 10 || 11 || 12
|- align="center"
! Decimal
! 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8
|-
|colspan=10 style="background-color:white;"|
|- align="center"
! Ternary
| 100 || 101 || 102 || 110 || 111 || 112 || 120 || 121 || 122
|- align="center"
! Binary
| 1001 || 1010 || 1011 || 1100 || 1101 || 1110 || 1111
| {{gaps|1|0000}} || {{gaps|1|0001}}
|- align="center"
! Senary
| 13 || 14 || 15 || 20 || 21 || 22 || 23 || 24 || 25
|- align="center"
! Decimal
! 9 ||10 || 11 || 12|| 13 || 14 || 15 || 16 || 17
|-
|colspan=10 style="background-color:white;"|
|- align="center"
! Ternary
| 200 || 201 || 202 || 210 || 211 || 212 || 220 || 221 || 222
|- align="center"
! Binary
| {{gaps|1|0010}} || {{gaps|1|0011}} || {{gaps|1|0100}} || {{gaps|1|0101}} || {{gaps|1|0110}}
| {{gaps|1|0111}} || {{gaps|1|1000}} || {{gaps|1|1001}} || {{gaps|1|1010}}
|- align="center"
! Senary
| 30 || 31 || 32 || 33 || 34 || 35 || 40 || 41 || 42
|- align="center"
! Decimal
! 18 || 19 || 20 || 21 || 22 || 23 || 24 || 25 || 26
|}

:

:{| class="wikitable"
|+ '''Powers of 3 in ternary'''
|- align="center"
! Ternary
| 1 || 10 || 100 || {{gaps|1|000}} || {{gaps|10|000}}
|- align="center"
! Binary
| 1 || 11 || 1001 || {{gaps|1|1011}} || {{gaps|101|0001}}
|- align="center"
! Senary
| 1 || 3 || 13 || 43 || 213
|- align="center"
! Decimal
| 1 || 3 || 9 || 27 || 81
|- align="center"
! Power
! {{big|3}}{{sup|0}} || {{big|3}}{{sup|1}} || {{big|3}}{{sup|2}}
! {{big|3}}{{sup|3}} || {{big|3}}{{sup|4}}
|-
|colspan=10 style="background-color:white;"|
|- align="center"
! Ternary
| {{gaps|100|000}} || {{gaps|1|000|000}} || {{gaps|10|000|000}}
| {{gaps|100|000|000}} || {{gaps|1|000|000|000}}
|- align="center"
! Binary
| {{gaps|1111|0011}} || {{gaps|10|1101|1001}} || {{gaps|1000|1000|1011}}
| {{gaps|1|1001|1010|0001}} || {{gaps|100|1100|1110|0011}}
|- align="center"
! Senary
| {{gaps|1|043}} || {{gaps|3|213}} || {{gaps|14|043}} || {{gaps|50|213}} || {{gaps|231|043}}
|- align="center"
! Decimal
| 243 || 729 || {{gaps|2|187}} || {{gaps|6|561}} || {{gaps|19|683}}
|- align="center"
! Power
! {{big|3}}{{sup|5}} || {{big|3}}{{sup|6}} || {{big|3}}{{sup|7}}
! {{big|3}}{{sup|8}} || {{big|3}}{{sup|9}}
|}

As for [[rational number]]s, ternary offers a convenient way to represent {{sfrac|1|3}} as same as senary (as opposed to its cumbersome representation as an infinite string of [[recurring decimal|recurring digits]] in decimal); but a major drawback is that, in turn, ternary does not offer a finite representation for {{sfrac|1|2}} (nor for {{sfrac|1|4}}, {{sfrac|1|8}}, etc.), because [[2 (number)|2]] is not a [[Prime number|prime]] [[factorization|factor]] of the base; as with base two, one-tenth (decimal{{sfrac|1|10}}, senary {{sfrac|1|14}}) is not representable exactly (that would need e.g. decimal); nor is one-sixth (senary {{sfrac|1|10}}, decimal {{sfrac|1|6}}).

:{| class="wikitable"
|+ '''Fractions in ternary'''
|- align="center"
! Fraction
| '''{{sfrac|1|2}}''' || '''{{sfrac|1|3}}''' || '''{{sfrac|1|4}}''' || '''{{sfrac|1|5}}''' || '''{{sfrac|1|6}}''' || '''{{sfrac|1|7}}''' || '''{{sfrac|1|8}}''' || '''{{sfrac|1|9}}''' || '''{{sfrac|1|10}}''' || '''{{sfrac|1|11}}''' || '''{{sfrac|1|12}}''' || '''{{sfrac|1|13}}'''
|- align="center"
! Ternary
| 0.{{overline|1}} || 0.1 || 0.{{overline|02}} || 0.{{overline|0121}} || 0.0{{overline|1}} || 0.{{overline|010212}} || 0.{{overline|01}} || 0.01 || 0.{{overline|0022}} || 0.{{overline|00211}} || 0.0{{overline|02}} || 0.{{overline|002}}
|- align="center"
! Binary
| 0.1 || 0.{{overline|01}} || 0.01 || 0.{{overline|0011}} || 0.0{{overline|01}} || 0.{{overline|001}} || 0.001 || 0.{{overline|000111}} || 0.0{{overline|0011}} || 0.{{overline|0001011101}} || 0.00{{overline|01}} || 0.{{overline|000100111011}}
|- align="center"
! Senary
| 0.3 || 0.2 || 0.13 || 0.{{overline|1}} || 0.1 || 0.{{overline|05}} || 0.043 || 0.04 || 0.0{{overline|3}} || 0.{{overline|0313452421}} || 0.03 || 0.{{overline|024340531215}}
|- align="center"
! Decimal
! 0.5 || 0.{{overline|3}} || 0.25 || 0.2 || 0.1{{overline|6}} || 0.{{overline|142857}} || 0.125
! 0.{{overline|1}} || 0.1 || 0.{{overline|09}} || 0.08{{overline|3}} || 0.{{overline|076923}}
|}

=== Sum of the digits in ternary as opposed to binary ===
The value of a binary number with ''n'' bits that are all 1 is {{math|2<sup>''n''</sup>&nbsp;−&nbsp;1}}.

Similarly, for a number ''N''(''b'', ''d'') with base ''b'' and ''d'' digits, all of which are the maximal digit value {{math|''b''&nbsp;−&nbsp;1}}, we can write:

: {{math|1=''N''(''b'', ''d'') = (''b''&nbsp;−&nbsp;1)''b''<sup>''d''−1</sup> + (''b''&nbsp;−&nbsp;1)''b''<sup>''d''−2</sup> + … + (''b''&nbsp;−&nbsp;1)''b''<sup>1</sup> + (''b''&nbsp;−&nbsp;1)''b''<sup>0</sup>,}}
: {{math|1={{white|''N''(''b'', ''d'')}} = (''b''&nbsp;−&nbsp;1)(''b''<sup>''d''−1</sup> + ''b''<sup>''d''−2</sup> + … + ''b''<sup>1</sup> + 1),}}
: {{math|1={{white|''N''(''b'', ''d'')}} = (''b''&nbsp;−&nbsp;1)''M''}}.
: {{math|1=''bM'' = ''b''<sup>''d''</sup> + ''b''<sup>''d''−1</sup> + … + ''b''<sup>2</sup> + ''b''<sup>1</sup>}} and
: {{math|1=−''M'' = −''b''<sup>''d''−1</sup>&nbsp;−&nbsp;''b''<sup>''d''−2</sup>&nbsp;−&nbsp;...&nbsp;−&nbsp;b<sup>1</sup>&nbsp;−&nbsp;1}}, so
: {{math|1=''bM''&nbsp;−&nbsp;''M'' = ''b''<sup>''d''</sup>&nbsp;−&nbsp;1}}, or
: {{math|1=''M'' = {{sfrac|''b''<sup>''d''</sup>&nbsp;−&nbsp;1|''b''&nbsp;−&nbsp;1}}.}}

Then

: {{math|1=''N''(''b'', ''d'') = (''b''&nbsp;−&nbsp;1)''M'',}}
: {{math|1={{white|''N''(''b'', ''d'')}} = {{sfrac|(''b''&nbsp;−&nbsp;1)(''b''<sup>''d''</sup>&nbsp;−&nbsp;1)|''b''&nbsp;−&nbsp;1}},}}
: {{math|1={{white|''N''(''b'', ''d'')}} = ''b''<sup>''d''</sup>&nbsp;−&nbsp;1.}}

For a three-digit ternary number, {{math|1=''N''(3, 3) = 3<sup>3</sup>&nbsp;−&nbsp;1 = 26 = 2 × 3<sup>2</sup> + 2 × 3<sup>1</sup> + 2 × 3<sup>0</sup> = 18 + 6 + 2}}.

=== Compact ternary representation: base 9 and 27 ===
{| class="wikitable" style="float:right; text-align:center"
|+ Comparison between ternary and nonary
|-
! ternary || nonary
|-
| 00 || 0
|-
| 01 || 1
|-
| 02 || 2
|-
| 10 || 3
|-
| 11 || 4
|-
| 12 || 5
|-
| 20 || 6
|-
| 21 || 7
|-
| 22 || 8
|}

'''Nonary''' {{IPAc-en|ˈ|n|ɒ|n|ər|i}}  (base 9, each digit is two ternary digits) or [[septemvigesimal]] (base 27, each digit is three ternary digits) can be used for compact representation of ternary, similar to how [[octal]] and [[hexadecimal]] systems are used in place of [[binary numeral system|binary]].