9 (nine) is the natural number following 8 and preceding 10.
Nine is a composite number
, its proper divisors
. It is 3 times 3 and hence the third square number
. 9 is a Motzkin number
. It is the first composite lucky number
9 is the second non-unitary square prime of the form (p2) and the first that is odd. All subsequent squares of ths 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.
There are nine Heegner numbers.
Since , 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 an enneagon (technically) or nonagon (in common usage). 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 whole number (except zero), 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))
- (Exception) 0 x 9 = 0 (0 is not equal to 9)(though it can be considered -1 in the tens place and 10 in the ones place(10-1=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.
- 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 (see also MathWorld).
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
In numeral systems
List of basic calculations
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 has often been underlined as is done for the 6. Another distinction from the 6 is that it is often handwritten with a straight stem.
See also: Hindu-Arabic numeral system.
, 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.
probability gives zero nines (−log10
1 = 0).
The purity of chemicals (see Nine (purity)
), the effectivity of processes, the availability
of systems etc. can similarly be expressed in nines. For example, "five nines" (99.999%) availability implies a total downtime
of no more than five minutes per year. This measure can be confusing, a fact which is discussed in the myth of the nines
A human pregnancy
normally lasts nine months.
- Hindu's worship "Nava Graha" or Nine Planets and offer "Nava Dhanya" for each Graha / Planet.
- Important Buddhist rituals usually involve nine monks.
- The first nine days of the Hebrew month of Av are collectively known as "The Nine Days" (Tisha HaYamim), and are a period of semi-mourning leading up to Tisha B'Av, the ninth day of Av on which both Temples in Jerusalem were destroyed.
- Nine, as the highest single-digit number (in base ten), symbolizes completeness in the Bahá'í Faith. In addition, the word Bahá' in the Abjad notation has a value of 9, and a 9-pointed star is used to symbolize the religion.
- Nine is a significant number in Norse Mythology. Odin hung himself on an ash tree for nine days to learn the runes.
- Jesus healed ten lepers; nine did not even thank him.
- Anton LaVey applied the number to Satan.
- Ramadan, the month of fasting and prayer, is the ninth month of the Islamic calendar.
As lucky or unlucky number, Chinese culture
Nine (九 pinyin
jiǔ) is considered a good number in Chinese culture
because it sounds the same as the word "longlasting" (久 pinyin
jiǔ). The Japanese consider 9 to be unlucky, however, because it sounds similar to the Japanese word for "pain" or "distress" (苦 kunrei
ku). The fear of the number nine is enneaphobia
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 9×13 scales, 9×9 being yang (masculine, or bad influence) and 9×4 being yin (feminine, or good influence).
The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City.
The circular altar platform (Earthly Mount) of the Temple of Heaven has one circular marble plate in the center, surrounded by a ring of nine plates, then by a ring of 18 plates, and so on, for a total of nine rings, with the outermost having 81=9×9 plates.
- There is a rapper named Tech N9ne whose name stands for "Tech Nine" because 9 is the number of completion
- In music theory a ninth is the ninth note of a musical scale or the interval between the first note and the ninth. A ninth chord is a chord with a ninth.
- In classical music the curse of the ninth refers to the superstition that a composer who writes a ninth symphony will die soon. Beethoven, who left his Tenth Symphony unfinished, is regarded by the superstition as the first victim of the curse.
- Bands with the number nine in their name include Nine, Stroke 9, Nine Days, Nine Inch Nails and Alice Nine.
- The Beatles released a song called "Revolution 9" which appears on The White Album (more properly known as The Beatles); its principal vocal feature is a voice repeating, "Number nine...number nine...number nine..." John Lennon, the primary composer of "Revolution 9", also released a solo recording entitled "#9 Dream".
- "Love Potion No. 9 originally performed by The Clovers in 1955. Love Potion No. 9 was a 1992 movie starring Sandra Bullock and Tate Donovan.
- The first single of Leeds-based band ¡Forward, Russia! was called Nine and was released in April 2005. The title did not carry any symbolism, but followed the then "tradition" of the band for naming their songs in the order in which they were written.
- Composer Maury Yeston adapted Federico Fellini's 8½ for Broadway as the musical Nine, which won several Tony Awards in 1982, and was successfully revived in 2003 starring Antonio Banderas.
- 9 is also the name of various albums, including 9 by Mercyful Fate, released in 1999, and 9 by Public Image Limited. Fairport Convention also released an album with the number for a title. Irish singer/songwriter Damien Rice released an album called 9 in November 2006.
- In urban culture, 'nine' is a slang word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide.
- In hip-hop, the rapper Nine, who got his big break in late 1993 as a featured guest on First Funkmaster Flex and the Ghetto Celebs, released 2 albums (Nine Livez in 1995 and Cloud Nine in 1996) and is now making a comeback.
- Dublin pop-punk band Nine IX Lives formed in 2005 and use 'IX' in their logo.
- ISO 9 is the ISO's standard for the transliteration of Cyrillic characters into Latin characters
- In the Rich Text Format specification, 9 is the language code for the English language. All codes for regional variants of English are congruent to 9 mod 256.
- In computer system administration, uptime, a measure of the time a computer system has been up and running, is sometimes measured in "nines"—that is, the number of "9" digits in the percentage of time the computer is normally running. For example, "five nines" means 99.999% reliability, which translates to a total downtime of no longer than five minutes per year.
Nine and the circle
is one system of knowledge which shows the correspondence between the 9 integers and the circle. The 360 degrees of the circle, which can be traced back to the Rig Veda of ancient India, can also be seen to speak of the 9, via theosophical addition (3 + 6 + 0 = 9)
In other fields
- In the catholic faith there are 9 choirs of holy angels.
- Nine is the highest single-digit number in the decimal system.
- In the NATO phonetic alphabet, the digit 9 is called "Novenine".
- Channel 9, an Australian free-to-air television station
- Nine justices sit on the United States Supreme Court.
- The word "K-9" pronounces the same as canine and is used in many U.S. police departments to denote the unit working with police dog.
- The ordinal adjective is ninth.
- There are nine circles of Hell in Dante's "Divine Comedy".
- Stanines are measured on a scale of 1 to 9.
- The name of the area called Kowloon in Hong Kong literally means: nine dragons.
- Someone dressed "to the nines" is dressed up as much as they can be.
- Nine Lives Cat Food got its name from the legend that a cat has nine lives.
- Nine rank system
- The Nine Chapters on the Mathematical Art
- Nine Worthies
- 9: The Last Resort was a 1995 computer game.
- 9 is the name of Shane Acker's short film, and its 2008 adaption.
- Seven of Nine, fictional character from the Star Trek universe
- Magic: The Gathering has a set of nine rare cards, widely regarded as overpowered, known as the Power Nine.
- Nine West is a brand for clothing.
- The number of hostages in the TV show The Nine
- Ninth Avenue is a major avenue in Manhattan.
- Nine Unknown Men are, in occult legend, the custodians of the sciences of the world since ancient times.
- The 9, a show on the Internet, hosted by Maria Sansone The 9
- The Nines is a film written and directed by John August, starring Ryan Reynolds, Hope Davis and Melissa McCarthy. It premiered at the 2007 Sundance Film Festival, and is now out on DVD.
- In the game of Craps, 9 is known as the center field because it is in the middle of the seven numbers on the field bet.
- In Middle Earth, there are nine rings of power given to men, and consequently, nine ringwraiths.
- "A cat-o'-nine-tails suggests perfect punishment and atonement." --Robert Ripley.
- Wally Cleaver's locker number was #9.
- The Nine Bright Shiners, characters in Garth Nix's Old Kingdom trilogy. The Nine Bright Shiners was a 1930s book of poems by Anne Ridler and a 1988 fiction book by Anthea Fraser; the name derives from "a very curious old semi-pagan, semi-Christian" song.
- "Nine" is a song title on the Neurosis album Given to the Rising.
- Many mathematical magic tricks have a connection to the number nine, that because every multiplication of nine adds up to nine, any number minus the sum of all digits becomes a multiplication of nine, and any three digit number minus the same number in reverse becomes a multiplication of nine
- Nine was the client number of Eliot Spitzer at The Emperor's Club.
- The character Cirno from the Touhou series is often called "nine," "⑨," "circle-nine," or "nineball," because in the game manual for "Phantasmagoria of Flower View," she was labeled .
Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN-10: 3791319337, ISBN-13: 9783791319339