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# Encyclopedia

(Redirected to 9 (number) article)

9
Cardinal 9
nine
Ordinal 9th
ninth
Numeral system nonary
Factorization 32
Divisors 1, 3, 9
Amharic
Roman numeral IX
Roman numeral (Unicode) Ⅸ, ⅸ
prefixes ennea- (from Greek)

nona- (from Latin)

Binary 1001
Octal 11
Duodecimal 9
Arabic-Indic numeral ٩
Armenian numeral Թ
Bengali
Chinese/Japanese numeral

Devanāgarī
Greek numeral θ´
Hebrew numeral ט (Tet)
Tamil numeral
Khmer
Thai numeral

9 (nine) is the natural number following 8 and preceding 10. The ordinal adjective is ninth.

## Mathematics

Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. 9 is a Motzkin number. It is the first composite lucky number.

Nine is the highest single-digit number in the decimal system. It is the second non-unitary square prime of the form (p2) and the first that is odd. All subsequent squares of this form are odd. It has a unique aliquot sum 4 which is itself a square prime. 9 is; and can be, the only square prime with an aliquot sum of the same form. The aliquot sequence of 9 has 5 members (9,4,3,1,0) this number being the second composite member of the 3-aliquot tree. It is the aliquot sum of only one number the discrete semiprime 15.

There are nine Heegner numbers.

Since $9 = 3^{2^1}$, 9 is an exponential factorial.

8 and 9 form a Ruth-Aaron pair under the second definition that counts repeated prime factors as often as they occur.

A polygon with nine sides is called a nonagon or enneagon. A group of nine of anything is called an ennead.

In base 10 a number is evenly divisible by nine if and only if its digital root is 9. That is, if you multiply nine by any natural number, and repeatedly add the digits of the answer until it is just one digit, you will end up with nine:

• 2 × 9 = 18 (1 + 8 = 9)
• 3 × 9 = 27 (2 + 7 = 9)
• 9 × 9 = 81 (8 + 1 = 9)
• 121 × 9 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)
• 234 × 9 = 2106 (2 + 1 + 0 + 6 = 9)
• 578329 × 9 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27 (2 + 7 = 9))
• 482729235601 × 9 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45 (4 + 5 = 9))

The only other number with this property is three. In base N, the divisors of N − 1 have this property. Another consequence of 9 being 10 − 1, is that it is also a Kaprekar number.

The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples:

• The sum of the digits of 41 is 5, and 41-5 = 36. The digital root of 36 is 3+6 = 9, which, as explained above, demonstrates that it is evenly divisible by nine.
• The sum of the digits of 35967930 is 3+5+9+6+7+9+3+0 = 42, and 35967930-42 = 35967888. The digital root of 35967888 is 3+5+9+6+7+8+8+8 = 54, 5+4 = 9.

Subtracting two base-10 positive integers that are transpositions of each other yields a number that is a whole multiple of nine. Some examples:

• 41-14 = 27. The digital root of 27 is 2+7 = 9.
• 36957930-35967930 = 990000, which is obviously a multiple of nine.

This works regardless of the number of digits that are transposed. For example, the largest transposition of 35967930 is 99765330 (all digits in descending order) and its smallest transposition is 03356799 (all digits in ascending order); subtracting pairs of these numbers produces:

• 99765330-35967930 = 63797400; 6+3+7+9+7+4+0+0 = 36, 3+6 = 9.
• 99765330-03356799 = 96408531; 9+6+4+0+8+5+3+1 = 36, 3+6 = 9.
• 35967930-03356799 = 32611131; 3+2+6+1+1+1+3+1 = 18, 1+8 = 9.

Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers, known as long ago as the 12th Century.

Every prime in a Cunningham chain of the first kind with a length of 4 or greater is congruent to 9 mod 10 (the only exception being the chain 2, 5, 11, 23, 47).

Six recurring nines appear in the decimal places 762 through 767 of pi. This is known as the Feynman point.

If an odd perfect number is of the form 36k + 9, it has at least nine distinct prime factors.

Nine is the binary complement of number six:

9 = 1001
6 = 0110


### Probability

In probability, the nine is a logarithmic measure of probability of an event, defined as the negative of the base-10 logarithm of the probability of the event's complement. For example, an event that is 99% likely to occur has an unlikelihood of 1% or 0.01, which amounts to −log10 0.01 = 2 nines of probability. Zero probability gives zero nines (−log10 1 = 0). The effectivity of processes and the availability of systems can be expressed in nines. For example, "five nines" (99.999%) availability implies a total downtime of no more than five minutes per year.

### Numeral systems

Base Numeral system
2 binary 1001
3 ternary 100
4 quaternary 21
5 quinary 14
6 senary 13
7 septenary 12
8 octal 11
9 novenary 10

### List of basic calculations

Multiplication 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 50 100 1000 $9 \times x$ 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 450 900 9000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 $9 \div x$ 9 4.5 3 2.25 1.6 1.5 $1.\overline{285714}$ 1.125 1 0.9 $0.\overline{81}$ 0.75 $0.\overline{692307}$ $0.6\overline{428571}$ 0.6 $x \div 9$ $0.\overline{1}$ $0.\overline{2}$ $0.\overline{3}$ $0.\overline{4}$ $0.\overline{5}$ $0.\overline{6}$ $0.\overline{7}$ $0.\overline{8}$ 1 $1.\overline{1}$ $1.\overline{2}$ $1.\overline{3}$ $1.\overline{4}$ $1.\overline{5}$ $1.\overline{6}$
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 $9 ^ x\,$ 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 31381059609 282429536481 2541865828329 $x ^ 9\,$ 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1000000000 2357947691 5159780352 10604499373
Radix 1 5 10 15 20 25 30 40 50 60 70 80 90 100
110 120 130 140 150 200 250 500 1000 10000 100000 1000000 $x_{9} \$ 1 5 $11_{9} \$ $16_{9} \$ $22_{9} \$ $27_{9} \$ $33_{9} \$ $44_{9} \$ $55_{9} \$ $66_{9} \$ $77_{9} \$ $88_{9} \$ $110_{9} \$ $121_{9} \$ $132_{9} \$ $143_{9} \$ $154_{9} \$ $165_{9} \$ $176_{9} \$ $242_{9} \$ $307_{9} \$ $615_{9} \$ $1331_{9} \$ $14641_{9} \$ $162151_{9} \$ $1783661_{9} \$

## Evolution of the glyph

According to Georges Ifrah, the origin of the 9 integers can be attributed to the ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0. In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot. The Kshtrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the @ character encircles a lowercase a. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic.

While the shape of the 9 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

This numeral resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, the 9 is often underlined, as is done for the 6. Another distinction from the 6 is that it is often handwritten with a straight stem. Mathematics portal

## References

1. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
2. ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
3. ^ Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
4. ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
5. ^ Eyob Delele Yirdaw, "Proving Touchard's Theorem from Euler's Form" ArXiv preprint.
6. ^ Georges Ifrah (1985). From One to Zero: A Universal History of Numbers. Viking. ISBN 0-670-37395-8.

• Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN 3791319337, ISBN 9783791319339

# Wiktionary

Up to date as of January 15, 2010

## Translingual

 Stroke order Stroke order ### Han character

(radical 5 +1, 2 strokes, cangjie input 大弓 (KN), four-corner 40017)

#### References

• KangXi: page 83, character 19
• Dai Kanwa Jiten: character 167
• Dae Jaweon: page 168, character 2
• Hanyu Da Zidian: volume 1, page 48, character 5
• Unihan data for U+4E5D

## Cantonese

### Etymology

From Old Chinese *kwyu? < Proto-Sino-Tibetan *t-kua (compare Classical Tibetan dgu)

(Yale gau2)

## Japanese

(きゅう, kyū)

### Noun

(hiragana きゅう, romaji kyū)

1. nine
2. quite many things (lead to )

### Etymology

/ko2ko2no2/: [kəkənə] > [kokono].

#### Noun

(hiragana ここの, romaji kokono)

1. nine

### Kanji

#### Compounds

• 九夏 (kyuuka):
• 九回戦 (kyuukaisen):
• 九官鳥 (kyuukanchou):
• 九牛一毛 (kyuugyuuichimou):
• 九九, 九々 (kuku): Multiplication table
• 九月 (kugatsu): September
• 九死 (kyuuji):
• 九州 (kyuushuu): Kyushu region of Japan
• 九重 (kokonoe):
• 九寸五分 (kusungobu):
• 九星 (kyuusei):
• 九折 (kyuusetsu):
• 九泉 (kyuusen):
• 九地 (kyuuchi):
• 九天 (kyuuten):
• 九拝 (kyuuhai):
• 九分 (kubu):
• 九輪 (kurin):
• 九六 (kunroku):

## Korean

### Hanja

Eumhun:

• Sound (hangeul):  (revised: gu, McCune-Reischauer: ku, Yale: kwu)
• Name (hangeul): 아홉 (revised: ahop, McCune-Reischauer: ahop, Yale: ahop)

## Mandarin

• help, file

### Hanzi

(pinyin jiǔ (jiu3), xiǎng (xiang3), xiàng (xiang4), Wade-Giles chiu3, hsiang3, hsiang4)

#### Compounds

 九一八事变 (九一八事變)

(cửu)

## Wu

### Pronunciation

IPA:

Latin Alphabet: juˇ

Zhuyin Fuhao: ㄐ一ㄡˇ