9 (nine) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following and preceding .

Evolution of the Hindu–Arabic digit

Circa 300 BC, as part of the

Brahmi numerals Brahmi numerals are a numeral system attested in the Indian subcontinent from the 3rd century BCE. It is the direct graphic ancestor of the modern Hindu–Arabic numeral system. However, the Brahmi numeral system was conceptually distinct from ...

, various Indians wrote a digit 9 similar in shape to the modern closing

question mark The question mark (also known as interrogation point, query, or eroteme in journalism) is a punctuation, punctuation mark that indicates a question or interrogative clause or phrase in many languages. History The history of the question mark is ...

without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a -look-alike. 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

typeface A typeface (or font family) is a design of Letter (alphabet), letters, Numerical digit, numbers and other symbols, to be used in printing or for electronic display. Most typefaces include variations in size (e.g., 24 point), weight (e.g., light, ...

s, in typefaces with

text figures Text figures (also known as non-lining, lowercase, old style, ranging, hanging, medieval, billing, or antique figures or numerals) are numerals designed with varying heights in a fashion that resembles a typical line of running text, hence the ...

the character usually has a

descender In typography and handwriting, a descender is the portion of a grapheme that extends below the Baseline (typography), baseline of a typeface, font. For example, in the letter ''y'', the descender is the "tail", or that portion of the diagonal li ...

, as, for example, in TextFigs196

. 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. Seven-segment 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 A seven-segment display is a display device for Arabic numerals, less complex than a device that can show more characters such as dot matrix displays. Seven-segment displays are widely used in digital clocks, electronic meters, basic calculators, ...

, the number 9 can be constructed either with a hook at the end of its stem or without one. Most

LCD A liquid-crystal display (LCD) is a flat-panel display or other electronically modulated optical device that uses the light-modulating properties of liquid crystals combined with polarizers to display information. Liquid crystals do not em ...

calculators use the former, but some VFD models use the latter.

Mathematics

9 is the fourth

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...

, and the first odd composite number. 9 is also a

refactorable number A refactorable number or tau number is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that \tau(n)\mid n with \tau(n)=\sigma_0(n)=\prod_^(e_i+1) for n=\prod_^np_i^. The first few refact ...

Casting out nines Casting out nines is any of three arithmetical procedures: *Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to 9 or a multiple of 9. The result of this procedure is a number which is small ...

is a quick way of testing the calculations of sums, differences, products, and

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

s of

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s in

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

, a method known as long ago as the 12th century. 9 is the only square number that is the sum of two consecutive, positive cubes:

3^2 = 9 = 1^3 + 2^3

If an odd

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...

exists, it will have at least nine distinct

prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

9 is the sum of the

cubes A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

of the first two non-zero positive integers

1^ + 2^

which makes it the first cube-sum number greater than

one 1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sp ...

. A number that is 4 or 5

modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...

9 cannot be represented as the sum of three cubes. Yang Hui magic circle

There are nine

Heegner number In number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from int ...

s, or square-free positive integers

n

that yield an imaginary

quadratic field In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...

\Q\left sqrt\right /math> whose

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...

has a

unique factorization In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...

, or class number of 1.

Geometry

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

with nine sides is called a

nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix Hybrid word, hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in Fre ...

. A regular nonagon can be constructed with a regular

compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with No ...

, straightedge, and angle trisector. The lowest number of squares needed for a Rectangle#Squared, perfect, and other tiled rectangles, perfect tiling of a rectangle is 9. 9 is the largest single-digit number in the decimal, decimal system.

List of basic calculations

Culture and mythology

Indian culture

Nine is a number that appears often in Indian culture and mythology. For example, there are nine Navagraha, influencers attested to in Indian astrology. In the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements: Prithvi, Earth, Ap (water), Water, Vayu, Air, Agni, Fire, Akasha, Ether, Kāla (time), Time, Vaisheshika#The Categories or Padārtha, Space, Ātman (Hinduism), Soul, and Manas (early Buddhism), Mind. And Navaratri is a nine-day festival dedicated to the Navadurga, nine forms of Durga.

Chinese culture

* Nine (; ) is considered a good Numbers in Chinese culture, number in Chinese culture because it sounds the same as the word "long-lasting" (; ). * 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 yin and yang, yang (masculine, heavenly) and 36 yin and yang, yin (feminine, earthly). All three numbers are multiples of 9 (, , ).

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. ICS Niner

Religion and philosophy

* Nine, as the largest single-digit number (in decimal, base ten), symbolizes completeness in the Baháʼí Faith. In addition, the word Baháʼ in the Abjad numerals, Abjad notation has a value of 9, and a 9-pointed star is used to Baháʼí symbols, 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

system, which originated from the Indian subcontinent as early as 30th century BC, 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 (Freudian), ego since it "always returns to itself" even after being multiplied by any number.

Science

Chemistry

Nine (purity), Nines are a notation for expressing the purity (chemistry), purity of a chemical.