Definitions

# 9 (number)

9
Cardinal 9
nine
Ordinal 9th
ninth
Numeral system nonary
Factorization $3^2$
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.

## In 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.

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 $9 = 3^\left\{2^1\right\}$, 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

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 $1.overline\left\{285714\right\}$ 1.125 1 0.9 $0.overline\left\{81\right\}$ 0.75 $0.overline\left\{692307\right\}$ $0.6overline\left\{428571\right\}$ 0.6
$x div 9$ $0.overline\left\{1\right\}$ $0.overline\left\{2\right\}$ $0.overline\left\{3\right\}$ $0.overline\left\{4\right\}$ $0.overline\left\{5\right\}$ $0.overline\left\{6\right\}$ $0.overline\left\{7\right\}$ $0.overline\left\{8\right\}$ 1 $1.overline\left\{1\right\}$ $1.overline\left\{2\right\}$ $1.overline\left\{3\right\}$ $1.overline\left\{4\right\}$ $1.overline\left\{5\right\}$ $1.overline\left\{6\right\}$

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
$x_\left\{9\right\}$ 1 5 $11_\left\{9\right\}$ $16_\left\{9\right\}$ $22_\left\{9\right\}$ $27_\left\{9\right\}$ $33_\left\{9\right\}$ $44_\left\{9\right\}$ $55_\left\{9\right\}$ $66_\left\{9\right\}$ $77_\left\{9\right\}$ $88_\left\{9\right\}$ $110_\left\{9\right\}$ $121_\left\{9\right\}$
$132_\left\{9\right\}$ $143_\left\{9\right\}$ $154_\left\{9\right\}$ $165_\left\{9\right\}$ $176_\left\{9\right\}$ $242_\left\{9\right\}$ $307_\left\{9\right\}$ $615_\left\{9\right\}$ $1331_\left\{9\right\}$ $14641_\left\{9\right\}$ $162151_\left\{9\right\}$ $1783661_\left\{9\right\}$

## 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.

## In science

### 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 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.

## Physiology

A human pregnancy normally lasts nine months.

## In religion

• 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" ( pinyin jiǔ). The Japanese consider 9 to be unlucky, however, because it sounds similar to the Japanese word for "pain" or "distress" ( 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.

## In technology

• 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

The Enneagram 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) [2]

## References

Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN-10: 3791319337, ISBN-13: 9783791319339