9

9
	Template:Numbers (digits)
Cardinal	9 ; nine
Ordinal	9th ; ninth
Numeral system	nonary
Factorization
Divisors	1, 3, 9
Amharic	፱
Roman numeral	IX
Roman numeral (Unicode)	Ⅸ, ⅸ
prefixes	ennea- (from Greek) nona- (from Latin)
Binary	1001
Octal	11
Duodecimal	9
Hexadecimal	9
Arabic-Indic numeral	٩
Armenian numeral	Թ
Bengali	৯
Chinese/Japanese numeral	九 ; 玖 (formal writing)
Devanāgarī	९ (Nao)
Greek numeral	θ´
Hebrew numeral	ט (Tet)
Tamil numeral	௯
Khmer	៩
Thai numeral	๙

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. Nine is a Motzkin number. It is the first composite lucky number, along with the first composite odd 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. Nine is; and can be, the only square prime with an aliquot sum of the same form. The aliquot sequence of nine 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.^[1]

Since 9 = 3^2¹, 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.

In bases 12, 18 and 24, nine is a 1-automorphic number and in base 6 a 2-automorphic number (displayed as '13').

A polygon with nine sides is called a nonagon or enneagon.^[2] 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.^[3] 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)

There are other interesting patterns involving multiples of nine:

12345679 x 9 = 111111111
12345679 x 18 = 222222222
12345679 x 81 = 999999999

This works for all the multiples of 9. n = 3 is the only other n > 1 such that a number is divisible by n if and only if its digital root is n. 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. Examples:

41 - 14 = 27 (2 + 7 = 9)
36957930 - 35967930 = 990000, 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.^[4]

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.^[5]

Nine is the binary complement of number six:

9 = 1001
6 = 0110

Coincidentally enough, Nine, when upside-down, looks just like the number six.

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). A 100% probability is considered to be impossible in most circumstances: that results in infinite improbability. The effectivity of processes and the availability of systems can be expressed (as a rule of thumb, not explictly) as a series of "nines". For example, "five nines" (99.999%) availability implies a total downtime of no more than five minutes per year - typically a very high degree of reliability; but never 100%.

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
over 9 (decimal, hexadecimal)		9

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
Radix	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}\$
$x_{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.^[6]

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 evolved from the letter "8". 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.

In sports

In baseball, 9 is the number of innings in a game.

The jersey number 9 has been retired by several North American sports teams in honor of past playing greats (or in one case, an owner):

In Major League Baseball:
- The Boston Red Sox, for Hall of Famer Ted Williams.
- The Chicago White Sox, for Minnie Miñoso.
- The New York Yankees, for Roger Maris.
- The Oakland Athletics, for Hall of Famer Reggie Jackson.
- The Pittsburgh Pirates, for Hall of Famer Bill Mazeroski.
- The St. Louis Cardinals, for Hall of Famer Enos Slaughter.
In the NBA:
- The Atlanta Hawks, for Hall of Famer Bob Pettit.
- The Phoenix Suns, for Dan Majerle.
- The Utah Jazz, for owner Larry Miller.
In the NHL:
- The Boston Bruins, for Johnny Bucyk.
- The Calgary Flames, for Lanny McDonald.
- The Chicago Blackhawks, for Bobby Hull.
- The Detroit Red Wings, for Gordie Howe.
- The Edmonton Oilers, for Glenn Anderson.
- The Montreal Canadiens, for Maurice Richard.
- The New York Islanders, for Clark Gillies.
- The New York Rangers, for Andy Bathgate and Adam Graves.
- The Winnipeg Jets, also for Hull. Although the Jets moved from Winnipeg to become the Phoenix Coyotes, the Coyotes continue to honor all numbers retired by the Jets.
- The Toronto Maple Leafs have a policy of not retiring numbers unless the player honoured either died or suffered a career-ending incident while a member of the team. Other players whose numbers would otherwise be retired instead have their numbers enshrined by the team as "Honoured Numbers", which remain in circulation for future players. The number 9 is currently honoured for Ted Kennedy and Charlie Conacher.
No NFL team has yet retired #9.

References

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

9