9

This article is about the number. For the years, see 9 BC and AD 9. For other uses, see 9 (disambiguation) and Number nine (disambiguation).

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Cardinal	nine
Ordinal	9th (ninth)
Numeral system	nonary
Factorization	32
Divisors	1,3,9
Greek numeral	Θ´
Roman numeral	IX, ix
Greek prefix	ennea-
Latin prefix	nona-
Binary	10012
Ternary	1003
Senary	136
Octal	118
Duodecimal	912
Hexadecimal	916
Amharic	፱
Arabic, Kurdish, Persian, Sindhi, Urdu	٩
Armenian numeral	Թ
Bengali	৯
Chinese numeral	九, 玖
Devanāgarī	९
Greek numeral	θ´
Hebrew numeral	ט
Tamil numerals	௯
Khmer	៩
Telugu script	౯
Thai numeral	๙
Malayalam	൯
Babylonian numeral	𒐝
Egyptian hieroglyph	𓐂
Morse code	____.
NATO phonetic alphabet	NINER

9 (nine) is the natural number following 8 and preceding 10.

Evolution of the Hindu–Arabic digit

See also: Hindu–Arabic numeral system

Circa 300 BC, as part of the Brahmi numerals, various Indians wrote a digit 9 similar in shape to the modern closing question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. ^[1] How the numbers got to their Gupta form is open to considerable debate. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the sign @ 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 glyph for the digit 9 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

The form of the number nine (9) could possibly derived from the Arabic letter waw , in which its isolated form (و) resembles the number 9.

The modern digit resembles an inverted 6. To disambiguate the two on objects and labels that can be inverted, they are often underlined. It is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letter q, which distinguishes it from the 6. Similarly, in seven-segment display, the number 9 can be constructed either with a hook at the end of its stem or without one. Most LCD calculators use the former, but some VFD models use the latter.

Mathematics

9 is the fourth composite number, and the first odd composite number. 9 is also a refactorable number. ^[2]

Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers in decimal, a method known as long ago as the 12th century. ^[3]

If an odd perfect number exists, it will have at least nine distinct prime factors. ^[4]

Non-intersecting chords between four points on a circle

9 is the sum of the cubes of the first two non-zero positive integers ${\displaystyle 1^{3}+2^{3}}$ which makes it the first cube-sum number greater than one. ^[5] A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes. ^[6]

Four concentric magic circles with 9 in the center (by Yang Hui), where numbers on each circle and diameter around the center generate a magic sum of 138.

There are nine Heegner numbers, or square-free positive integers ${\displaystyle n}$ that yield an imaginary quadratic field ${\displaystyle \mathbb {Q} \left[{\sqrt {-n}}\right]}$ whose ring of integers has a unique factorization, or class number of 1. ^[7]

Geometry

A polygon with nine sides is called a nonagon. ^[8] A regular nonagon can be constructed with a regular compass, straightedge, and angle trisector. ^[9]

The lowest number of squares needed for a perfect tiling of a rectangle is 9. ^[10]

9 is the largest single-digit number in the decimal system.

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	20	25	50	100	1000
9 × x	9	18	27	36	45	54	63	72	81	90	99	108	117	126	135	144	180	225	450	900	9000

Division	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
9 ÷ x	9	4.5	3	2.25	1.8	1.5	1.285714	1.125	1	0.9	0.81	0.75	0.692307	0.6428571	0.6
x ÷ 9	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	1	1.1	1.2	1.3	1.4	1.5	1.6

Exponentiation	1	2	3	4	5	6	7	8	9	10
9x	9	81	729	6561	59049	531441	4782969	43046721	387420489	3486784401
x9	1	512	19683	262144	1953125	10077696	40353607	134217728	387420489	1000000000

Culture and mythology

Indian culture

Nine is a number that appears often in Indian culture and mythology. ^[11] For example, there are nine influencers attested to in Indian astrology. In the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements: Earth, Water, Air, Fire, Ether, Time, Space, Soul, and Mind. ^[12] And Navaratri is a nine-day festival dedicated to the nine forms of Durga. ^{[13] [14]}

Chinese culture

• Nine (九; pinyin :jiǔ) is considered a good number in Chinese culture because it sounds the same as the word "long-lasting" (久; pinyin :jiǔ). ^[15]
• Nine is strongly associated with the Chinese dragon, a symbol of magic and power. There are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales – 81 yang (masculine, heavenly) and 36 yin (feminine, earthly). All three numbers are multiples of 9 (9 × 13 = 117, 9 × 9 = 81, 9 × 4 = 36). ^[16]

Anthropology

Idioms

• "To go the whole nine yards"
• "A cat has nine lives"
• "To be on cloud nine"
• The word "K-9" pronounces the same as canine and is used in many US police departments to denote the police dog unit. Despite not sounding like the translation of the word canine in other languages, many police and military units around the world use the same designation.
• Someone dressed "to the nines" is dressed up as much as they can be.
• In North American urban culture, "nine" is a slang word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide.

International maritime signal flag for 9

Religion and philosophy

A nine-pointed star

• Nine, as the largest single-digit number (in base ten), symbolizes completeness in the Baháʼí Faith. In addition, the word Baháʼ in the Abjad notation has a value of 9, and a 9-pointed star is used to symbolize the religion.
• The number 9 is revered in Hinduism and considered a complete, perfected and divine number because it represents the end of a cycle in the decimal system, which originated from the Indian subcontinent as early as 3000 BC.
• In Norse mythology, the number nine is associated with Odin, as that is how many days he hung from the world tree Yggdrasil before attaining knowledge of the runes.
• Nine is the number associated with Satan in LaVeyan Satanism. Anton LaVey wrote in The Satanic Rituals that this is because nine is the number of the ego since it "always returns to itself" even after being multiplied by any number.

Science

Chemistry

The purity of chemicals (see Nine (purity)).

Physiology

A human pregnancy normally lasts nine months, the basis of Naegele's rule.

Psychology

Common terminal digit in psychological pricing.

See also

• Mathematics portal

• 9 (disambiguation)
• 0.999...
• Cloud Nine

References

1. Lippman, David (12 July 2021). "6.0.2: The Hindu-Arabic Number System". Mathematics LibreTexts. Retrieved 31 March 2024.
2. Sloane, N. J. A. (ed.). "SequenceA033950(Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 19 June 2023.
3. Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN   0-8218-2102-4. p.91
4. Pace P., Nielsen (2007). "Odd perfect numbers have at least nine distinct prime factors". Mathematics of Computation . 76 (260). Providence, R.I.: American Mathematical Society: 2109–2126. arXiv: math/0602485 . Bibcode:2007MaCom..76.2109N. doi: 10.1090/S0025-5718-07-01990-4 . MR   2336286. S2CID   2767519. Zbl   1142.11086.
5. Sloane, N. J. A. (ed.). "SequenceA000537(Sum of first n cubes; or n-th triangular number squared.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 19 June 2023.
6. Davenport, H. (1939), "On Waring's problem for cubes", Acta Mathematica , 71, Somerville, MA: International Press of Boston: 123–143, doi: 10.1007/BF02547752 , MR   0000026, S2CID   120792546, Zbl   0021.10601
7. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
8. Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
9. Gleason, Andrew M. (1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly . 95 (3). Taylor & Francis, Ltd: 191–194. doi:10.2307/2323624. JSTOR   2323624. MR   0935432. S2CID   119831032.
10. Sloane, N. J. A. (ed.). "SequenceA219766(Number of nonsquare simple perfect squared rectangles of order n up to symmetry)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
11. DHAMIJA, ANSHUL (16 May 2018). "The Auspiciousness Of Number 9". Forbes India. Retrieved 1 April 2024.
12. "Vaisheshika | Atomism, Realism, Dualism | Britannica". www.britannica.com. Retrieved 13 April 2024.
13. "Navratri | Description, Importance, Goddess, & Facts | Britannica". www.britannica.com. 11 April 2024. Retrieved 13 April 2024.
14. Lochtefeld, James G. (2002). The illustrated encyclopedia of Hinduism. New York: the Rosen publ. group. ISBN   978-0-8239-2287-1.
15. "Lucky Number Nine, Meaning of Number 9 in Chinese Culture". www.travelchinaguide.com. Retrieved 15 January 2021.
16. Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN   1-4179-6429-4.

Further reading

• Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN   3-7913-1933-7, ISBN   978-3-7913-1933-9