# Nonary

Nonary (also novemal) is a base-9 numeral system, typically using the digits 0-8, but not the digit 9.

The first few numbers in decimal and nonary are:

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 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 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

The multiplication table in nonary is:

× 1 2 3 4 5 6 7 8 10
1 1 2 3 4 5 6 7 8 10
2 2 4 6 8 11 13 15 17 20
3 3 6 10 13 16 20 23 26 30
4 4 8 13 17 22 26 31 35 40
5 5 11 16 22 27 33 38 44 50
6 6 13 20 26 33 40 46 53 60
7 7 15 23 31 38 46 54 62 70
8 8 17 26 35 44 53 62 71 80
10 10 20 30 40 50 60 70 80 100

Nonary notation can be used as a concise representation of ternary data. This is similar to using quaternary notation for binary data, though the digit set is closer in size to octal.

## Mathematical characteristics

Except for 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.

A nonary number is divisible by two, four, or eight if the sum of its digits is also divisible by two, four, or eight respectively.

If x is a triangular number, so is 9x+1.[note 1] This means that one finds 3, 31, 311, 3111, 31111... in the triangular numbers. Likewise, 6, 61, 611, 6111, ....

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. [Watkins 1]

## In popular culture

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'.

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.

1. ^ It can be derived naturally from the definition of triangular number, that if we assume $x$ is $n_{th}$ triangular number, then: $x = {n+1 \choose 2} = \frac{n^2 + n}{2}$ ; We then obtain: $9x+1 =\frac{9n^2 + 9n + 2}{2} ={(3n+2) \choose 2}$ which means $9x+1$ is $(3n+1)_{th}$ triangular number.