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| {{Merge to |Ternary numeral system |discuss=Talk:Ternary_numeral_system#Merge_from_.22Nonary.22_and_.22Base_27.22 |date=July 2013}}
| | Eusebio is the name [http://Www.Twitpic.com/tag/workers workers] use to call our family and I think in which sounds quite good when you say it. Idaho is our birth install. I second-hand to be unemployed except now I am a great cashier. My friends say it's not great for me but so what on earth I love doing is also to bake but I'm just thinking on starting interesting things. I'm not good at web development but you might want to check my website: http://prometeu.net<br><br>Also visit my web-site: [http://prometeu.net clash of clans hacks] |
| {{Unreferenced|date=December 2009}}
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| {{Table Numeral Systems}}
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| '''Nonary''' (also '''novemal''') is a [[base (exponentiation)|base]]-{{Num|9}} [[numeral system]], typically using the [[numerical digit|digit]]s 0-8, but not the digit 9.
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| The first few numbers in decimal and nonary are:
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| {| class="wikitable"
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| |- align="center"
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| | Decimal || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20 || 21 || 22 || 23 || 24 || 25 || 26 || 27
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| |-
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| ! Nonary !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 20 !! 21 !! 22 !! 23 !! 24 !! 25 !! 26 !! 27 !! 28 !! 30
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| |-
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| | Ternary || 0 || 1 || 2 || 10 || 11 || 12 || 20 || 21 || 22 || 100 || 101 || 102 || 110 || 111 || 112 || 120 || 121 || 122 || 200 || 201 || 202 || 210 || 211 || 212 || 220 || 221 || 222 || 1000
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| |}
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| The multiplication table in nonary is:
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| {| class="wikitable"
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| | * || '''1''' || '''2''' || '''3''' || '''4''' || '''5''' || '''6''' || '''7''' || '''8''' || '''10''' || '''11''' || '''12'''
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| | '''1''' || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 10 || 11 || 12
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| |-
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| | '''2''' || 2 || 4 || 6 || 8 || 11 || 13 || 15 || 17 || 20 || 22 || 24
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| |-
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| | '''3''' || 3 || 6 || 10 || 13 || 16 || 20 || 23 || 26 || 30 || 33 || 36
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| | '''4''' || 4 || 8 || 13 || 17 || 22 || 26 || 31 || 35 || 40 || 44 || 48
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| | '''5''' || 5 || 11 || 16 || 22 || 27 || 33 || 38 || 44 || 50 || 55 || 61
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| |-
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| | '''6''' || 6 || 13 || 20 || 26 || 33 || 40 || 46 || 53 || 60 || 66 || 73
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| | '''7''' || 7 || 15 || 23 || 31 || 38 || 46 || 54 || 62 || 70 || 77 || 85
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| |-
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| | '''8''' || 8 || 17 || 26 || 35 || 44 || 53 || 62 || 71 || 80 || 88 || 107
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| |-
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| | '''10''' || 10 || 20 || 30 || 40 || 50 || 60 || 70 || 80 || 100 || 110 || 120
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| |-
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| | '''11''' || 11 || 22 || 33 || 44 || 55 || 66 || 77 || 88 || 110 || 121 || 132
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| | '''12''' || 12 || 24 || 36 || 48 || 61 || 73 || 85 || 107 || 120 || 132 || 144
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| |}
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| Nonary notation can be used as a concise representation of [[Ternary numeral system|ternary]] data. This is similar to using [[Quaternary numeral system|quaternary]] notation for [[binary numeral system|binary]] data, though the digit set is closer in size to [[octal]].
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| == Mathematical characteristics ==
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| Except for [[3 (number)|three]], no primes in nonary end in 0, 3 or 6, since any nonary number ending in 0, 3 or 6 is divisible by three.
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| A nonary number is divisible by [[2 (number)|two]], [[4 (number)|four]], or [[8 (number)|eight]] if the sum of its digits is also divisible by two, four, or eight respectively.
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| If x is a triangular number, so is 9x+1.<ref group=note>
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| It can be derived naturally from the definition of [[triangular number]],
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| that if we assume <math>x</math> is <math>n_{th}</math> triangular number, then:
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| <math>x = {n+1 \choose 2} = \frac{n^2 + n}{2} </math> ;
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| We then obtain:
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| <math>9x+1 =\frac{9n^2 + 9n + 2}{2} ={(3n+2) \choose 2}</math>
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| which means <math>9x+1</math> is <math>(3n+1)_{th}</math> triangular number.
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| </ref> | |
| This means that one finds 3, 31, 311, 3111, 31111... in the [[triangular number]]s. Likewise, 6, 61, 611, 6111, ....
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| Nonary is useful for determining the sum of the sum of all numbers in a sequence's digits until a single digit is obtained. For example if one was to determine the sum of all digits in the number 382, the result would be found by 3+8+2=13 however this number has more than one digit, so the process continues, 1+3=4 therefore the number 382 would solve to be 4. This answer may be found more easily with Nonary by simply converting 382 into the base 9, which gives 464, the last digit of which will always be the result found by adding each digit up until a single digit is achieved, where 0 reflects the answer of 9. <ref name=Thayer group=Watkins>{{cite web|last=Watkins|first=Thayer|title=Digit Sums Arithmetic|url=http://www.sjsu.edu/faculty/watkins/Digitsum.htm|work=Digit Sums|publisher=Thayer Watkins|accessdate=29 November 2012}}</ref>
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| ==In popular culture==
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| Although the term "Nonary" is used in describing the written form of the language used by the fictional civilization, [[The Culture]], found in [[Iain M. Banks]]' books, the description on page 119 of ''[[Excession]]'' reads more like it's based on a binary system with a 9-bit 'byte'.
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| The "Nonary Game" is the game played by the characters in the 2009 [[Nintendo DS]] video game, ''[[999: Nine Hours, Nine Persons, Nine Doors]]''. Much of the game revolves around the number nine, hence the name.
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| ==See also==
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| *[[Numeral system#Nine|Numeral system]]
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| ==Notes==
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| {{reflist|group=note}}
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| ==References==
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| {{Reflist|group=Watkins}}
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| [[Category:Positional numeral systems]]
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Eusebio is the name workers use to call our family and I think in which sounds quite good when you say it. Idaho is our birth install. I second-hand to be unemployed except now I am a great cashier. My friends say it's not great for me but so what on earth I love doing is also to bake but I'm just thinking on starting interesting things. I'm not good at web development but you might want to check my website: http://prometeu.net
Also visit my web-site: clash of clans hacks