| ||||
---|---|---|---|---|
Cardinal | nine | |||
Ordinal | 9th (ninth) | |||
Numeral system | nonary | |||
Factorization | 3^{2} | |||
Divisors | 1,3,9 | |||
Greek numeral | Θ´ | |||
Roman numeral | IX, ix | |||
Greek prefix | ennea- | |||
Latin prefix | nona- | |||
Binary | 1001_{2} | |||
Ternary | 100_{3} | |||
Senary | 13_{6} | |||
Octal | 11_{8} | |||
Duodecimal | 9_{12} | |||
Hexadecimal | 9_{16} | |||
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 numeral | ౯ | |||
Thai numeral | ๙ | |||
Malayalam | ൯ |
9 (nine) is the natural number following 8 and preceding 10.
See also: Hindu–Arabic numeral system |
Circa 300 BCE, 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. 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 modern digit resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, they are often underlined. Another distinction from the 6 is that it is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letter q. 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.
Nine is the fourth composite number, and the first composite number that is odd. Nine is the third square number (3^{2}), and the second non-unitary square prime of the form p^{2}, and, the first that is odd, with all subsequent squares of this form odd as well. Nine has the even aliquot sum of 4, and with a composite number sequence of two (9, 4, 3, 1, 0) within the 3-aliquot tree. There are nine Heegner numbers, or square-free positive integers that yield an imaginary quadratic field whose ring of integers has a unique factorization, or class number of 1.^{[1]} By Mihăilescu's theorem, 9 is the only positive perfect power that is one more than another positive perfect power, since the square of 3 is one more than the cube of 2.^{[2]}^{[3]}
9 is the sum of the cubes of the first two non-zero positive integers which makes it the first cube-sum number greater than one.^{[4]} It is also the sum of the first three nonzero factorials and equal to the third exponential factorial, since ^{[5]}
Nine is the number of derangements of 4, or the number of permutations of four elements with no fixed points.^{[6]}
9 is the fourth refactorable number, as it has exactly three positive divisors, and 3 is one of them.^{[7]}
A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.^{[8]}
If an odd perfect number exists, it will have at least nine distinct prime factors.^{[9]}
9 is a Motzkin number, for the number of ways of drawing non-intersecting chords between four points on a circle.^{[10]}
The first non-trivial magic square is a x magic square made of nine cells, with a magic constant of 15.^{[11]} Meanwhile, a x magic square has a magic constant of 369.^{[12]}
A polygon with nine sides is called a nonagon.^{[13]} Since 9 can be written in the form , for any nonnegative natural integers and with a product of Pierpont primes, a regular nonagon can be constructed with a regular compass, straightedge, and angle trisector.^{[14]}
Also an enneagon, a regular nonagon is able to fill a plane-vertex alongside an equilateral triangle and a regular 18-sided octadecagon (3.9.18), and as such, it is one of only nine polygons that are able to fill a plane-vertex without uniformly tiling the plane.^{[15]}
There are nine distinct uniform colorings of the triangular tiling and the square tiling, which are the two simplest regular tilings; the hexagonal tiling, on the other hand, has three distinct uniform colorings.
There are a maximum of nine semiregular Archimedean tilings by convex regular polygons, when including chiral forms of the snub hexagonal tiling.
There are nine uniform edge-transitive convex polyhedra in three dimensions:
Nine distinct stellation's by Miller's rules are produced by the truncated tetrahedron.^{[16]} It is the simplest Archimedean solid, with a total of four equilateral triangular and four hexagonal faces.
In four-dimensional space, there are nine paracompact hyperbolic honeycomb Coxeter groups, as well as nine regular compact hyperbolic honeycombs from regular convex and star polychora.^{[17]} There are also nine uniform demitesseractic () Euclidean honeycombs in the fourth dimension.
There are only three types of Coxeter groups of uniform figures in dimensions nine and thereafter, aside from the many families of prisms and proprisms: the simplex groups, the hypercube groups, and the demihypercube groups. The ninth dimension is also the final dimension that contains Coxeter-Dynkin diagrams as uniform solutions in hyperbolic space. Inclusive of compact hyperbolic solutions, there are a total of 238 compact and paracompact Coxeter-Dynkin diagrams between dimensions two and nine, or equivalently between ranks three and ten. The most important of the last paracompact groups is the group with 1023 total honeycombs, the simplest of which is 6_{21} whose vertex figure is the 5_{21} honeycomb: the vertex arrangement of the densest-possible packing of spheres in 8 dimensions which forms the lattice. The 6_{21} honeycomb is made of 9-simplexes and 9-orthoplexes, with 1023 total polytope elements making up each 9-simplex. It is the final honeycomb figure with infinite facets and vertex figures in the k_{21} family of semiregular polytopes, first defined by Thorold Gosset in 1900.
9 is the highest single-digit number in the decimal system.
A positive number is divisible by nine if and only if its digital root is nine:
That is, if any natural number is multiplied by 9, and the digits of the answer are repeatedly added until it is just one digit, the sum will be nine.^{[18]}
In base-, the divisors of have such a property, which makes 3 the only other number aside from 9 in decimal that shares this property. Another consequence of 9 being 10 − 1 is that it is a Kaprekar number.
There are other interesting patterns involving multiples of nine:
The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples:
If dividing a number by the amount of 9s corresponding to its number of digits, the number is turned into a repeating decimal. (e.g. 274/999 = 0.274274274274...)
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.^{[19]}
Six recurring nines appear in the decimal places 762 through 767 of π. (See six nines in pi).
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 |
---|---|---|---|---|---|---|---|---|---|---|
9^{x} | 9 | 81 | 729 | 6561 | 59049 | 531441 | 4782969 | 43046721 | 387420489 | 3486784401 |
x^{9} | 1 | 512 | 19683 | 262144 | 1953125 | 10077696 | 40353607 | 134217728 | 387420489 | 1000000000 |
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_{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} |
132_{9} | 143_{9} | 154_{9} | 165_{9} | 176_{9} | 242_{9} | 307_{9} | 615_{9} | 1331_{9} | 14641_{9} | 162151_{9} | 1783661_{9} |
Nine is a number that appears often in Indian culture and mythology. Some instances are enumerated below.
The Pintupi Nine, a group of 9 Aboriginal Australian women who remained unaware of European colonisation of Australia and lived a traditional desert-dwelling life in Australia's Gibson Desert until 1984.
There are three verses that refer to nine in the Quran.
We surely gave Moses nine clear signs.^{1} ˹You, O Prophet, can˺ ask the Children of Israel. When Moses came to them, Pharaoh said to him, “I really think that you, O Moses, are bewitched.”
Note 1: The nine signs of Moses are: the staff, the hand (both mentioned in Surah Ta-Ha 20:17-22), famine, shortage of crops, floods, locusts, lice, frogs, and blood (all mentioned in Surah Al-A'raf 7:130-133). These signs came as proofs for Pharaoh and the Egyptians. Otherwise, Moses had some other signs such as water gushing out of the rock after he hit it with his staff, and splitting the sea.
Now put your hand through ˹the opening of˺ your collar, it will come out ˹shining˺ white, unblemished.^{2} ˹These are two˺ of nine signs for Pharaoh and his people. They have truly been a rebellious people.”
— Surah Al-Naml (The Ant):12^{[28]}
Note 2: Moses, who was dark-skinned, was asked to put his hand under his armpit. When he took it out it was shining white, but not out of a skin condition like melanoma.
And there were in the city nine ˹elite˺ men who spread corruption in the land, never doing what is right.
— Surah Al-Naml (The Ant):48^{[29]}
A human pregnancy normally lasts nine months, the basis of Naegele's rule.
Common terminal digit in psychological pricing.