1.
1 (number)
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1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few

2.
2 (number)
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2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set

3.
4 (number)
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4 is a number, numeral, and glyph. It is the number following 3 and preceding 5. Four is the only cardinal numeral in the English language that has the number of letters as its number value. Four is the smallest composite number, its divisors being 1 and 2. Four is also a composite number. The next highly composite number is 6, Four is the second square number, the second centered triangular number. 4 is the smallest squared prime and the even number in this form. It has a sum of 3 which is itself prime. The aliquot sequence of 4 has 4 members and is accordingly the first member of the 3-aliquot tree, a number is a multiple of 4 if its last two digits are a multiple of 4. For example,1092 is a multiple of 4 because 92 =4 ×23, only one number has an aliquot sum of 4 and that is squared prime 9. Four is the smallest composite number that is equal to the sum of its prime factors, however, it is the only composite number n for which. It is also a Motzkin number, in bases 6 and 12,4 is a 1-automorphic number. In addition,2 +2 =2 ×2 =22 =4, continuing the pattern in Knuths up-arrow notation,2 ↑↑2 =2 ↑↑↑2 =4, and so on, for any number of up arrows. A four-sided plane figure is a quadrilateral which include kites, rhombi, a circle divided by 4 makes right angles and four quadrants. Because of it, four is the number of plane. Four cardinal directions, four seasons, duodecimal system, and vigesimal system are based on four, a solid figure with four faces as well as four vertices is a tetrahedron, and 4 is the smallest possible number of faces of a polyhedron. The regular tetrahedron is the simplest Platonic solid, a tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only regular polyhedron

4.
5 (number)
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5 is a number, numeral, and glyph. It is the number following 4 and preceding 6. Five is the prime number. Because it can be written as 221 +1, five is classified as a Fermat prime, therefore a regular polygon with 5 sides is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, Five is the first Wilson prime and the third factorial prime, also an alternating factorial. Five is the first good prime and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also the number that is part of more than one pair of twin primes. Five is conjectured to be the only odd number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. Five is also the only prime that is the sum of two primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being 2 plus 3,5 is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation. Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers,5 is the length of the hypotenuse of the smallest integer-sided right triangle. In bases 10 and 20,5 is a 1-automorphic number,5 and 6 form a Ruth–Aaron pair under either definition. There are five solutions to Známs problem of length 6 and this is related to the fact that the symmetric group Sn is a solvable group for n ≤4 and not solvable for n ≥5. While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar, K5, Five is also the number of Platonic solids. A polygon with five sides is a pentagon, figurate numbers representing pentagons are called pentagonal numbers. Five is also a square pyramidal number, Five is the only prime number to end in the digit 5, because all other numbers written with a 5 in the ones-place under the decimal system are multiples of five. As a consequence of this,5 is in base 10 a 1-automorphic number, vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions, unlike expansions with all other prime denominators, because they are prime factors of ten, the base. When written in the system, all multiples of 5 will end in either 5 or 0

5.
6 (number)
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6 is the natural number following 5 and preceding 7. The SI prefix for 10006 is exa-, and for its reciprocal atto-,6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number, its proper divisors are 1,2 and 3, since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and S -perfect number. As a perfect number,6 is related to the Mersenne prime 3,6 is the only even perfect number that is not the sum of successive odd cubes. As a perfect number,6 is the root of the 6-aliquot tree, and is itself the sum of only one number. Six is the number that is both the sum and the product of three consecutive positive numbers. Unrelated to 6 being a number, a Golomb ruler of length 6 is a perfect ruler. Six is the first discrete biprime and the first member of the discrete biprime family, Six is the smallest natural number that can be written as the sum of two positive rational cubes which are not integers,6 =3 +3. Six is a perfect number, a harmonic divisor number and a superior highly composite number. The next superior highly composite number is 12,5 and 6 form a Ruth-Aaron pair under either definition. There are no Graeco-Latin squares with order 6, if n is a natural number that is not 2 or 6, then there is a Graeco-Latin square with order n. The smallest non-abelian group is the symmetric group S3 which has 3, s6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of mathematical objects such as the S Steiner system, the projective plane of order 4. This can also be expressed category theoretically, consider the category whose objects are the n element sets and this category has a non-trivial functor to itself only for n =6. 6 similar coins can be arranged around a central coin of the radius so that each coin makes contact with the central one. This makes 6 the answer to the kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the lattice in which each circle touches just six others. 6 is the largest of the four all-Harshad numbers, a six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane

6.
9 (number)
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9 is the natural number following 8 and preceding 10. In the NATO phonetic alphabet, the digit 9 is called Niner, five-digit produce PLU codes that begin with 9 are organic. Common terminal digit in psychological pricing, Nine is a number that appears often in Indian Culture and mythology. Nine influencers are attested 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. Navaratri is a festival dedicated to the nine forms of Durga. Navaratna, meaning 9 jewels may also refer to Navaratnas - accomplished courtiers, Navratan - a kind of dish, according to Yoga, the human body has nine doors - two eyes, two ears, the mouth, two nostrils, and the openings for defecation and procreation. In Indian aesthetics, there are nine kinds of Rasa, Nine is considered a good 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, 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 and 36 yin, all three numbers are multiples of 9 as well as having the same digital root of 9. The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City, the name of the area called Kowloon in Hong Kong literally means, nine dragons. The nine-dotted line delimits certain island claims by China in the South China Sea, the nine-rank system was a civil service nomination system used during certain Chinese dynasties. 9 Points of the Heart / Heart Master Channels in Traditional Chinese Medicine, the nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. The Ennead is a group of nine Egyptian deities, who, in versions of the Osiris myth. The Nine Worthies are nine historical, or semi-legendary figures who, in Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil. The nine Muses in Greek mythology are Calliope, Clio, Erato, Euterpe, Melpomene, Polyhymnia, Terpsichore, Thalia and it takes nine days to fall from heaven to earth, and nine more to fall from earth to Tartarus—a place of torment in the underworld. Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo, according to Georges Ifrah, the origin of the 9 integers can be attributed to 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 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, as time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller

7.
10 (number)
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10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten, to reduce something by one tenth is to decimate. A theoretical highest number in topics that require a rating, by contrast having 0 or 1 as the lowest number, Ten is a composite number, its proper divisors being 1,2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the number of coprimes below it. Ten is the discrete semiprime and the second member of the discrete semiprime family. Ten has an aliquot sum σ of 8 and is accordingly the first discrete semiprime to be in deficit, all subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the second member of the 7-aliquot tree. Ten is the smallest semiprime that is the sum of all the prime numbers from its lower factor through its higher factor Only three other small semiprimes share this attribute. It is the sum of only one number the discrete semiprime 14. Ten is the sum of the first three numbers, of the four first numbers, of the square of the two first odd numbers and also of the first four factorials. Ten is the eighth Perrin number, preceded in the sequence by 5,5,7, a polygon with ten sides is a decagon, and 10 is a decagonal number. Because 10 is the product of a power of 2 with nothing but distinct Fermat primes, Ten is also a triangular number, a centered triangular number, and a tetrahedral number. Ten is the number of n queens problem solutions for n =5, Ten is the smallest number whose status as a possible friendly number is unknown. As is the case for any base in its system, ten is the first two-digit number in decimal, any integer written in the decimal system can be multiplied by ten by adding a zero to the end. The Roman numeral for ten is X, it is thought that the V for five is derived from an open hand, incidentally, the Chinese word numeral for ten, is also a cross, 十

8.
12 (number)
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12 is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a highly composite number, divisible by 2,3,4. It is central to systems of counting, including the Western calendar and units of time. The word twelve is the largest number with a name in English. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance and it derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels Book of John. It has cognates in every Germanic language, whose Proto-Germanic ancestor has been reconstructed as *twaliƀi, from *twa and suffix *-lif- or *-liƀ- of uncertain meaning. It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19, every other Indo-European language instead uses a form of two+ten, such as the Latin duōdecim. The usual ordinal form is twelfth but dozenth or duodecimal is also used in some contexts, similarly, a group of twelve things is usually a dozen but may also be referred to as a duodecad. The adjective referring to a group of twelve is duodecuple, as with eleven, the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan, with the implicit meaning that two is left after having already counted to ten. The Lithuanian suffix is also considered to share a similar development, the suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten. While, as mentioned above,12 has its own name in Germanic languages such as English and German, it is a number in many other languages, e. g. Italian dodici. In Germany, according to an old tradition, the numbers 0 through 12 were spelt out, the Duden now calls this tradition outdated and no longer valid, but many writers still follow it. Another system spells out all numbers written in one or two words, Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1,2,3,4,6 and 12. Twelve is also a composite number, the next one being twenty-four. Twelve is also a highly composite number, the next one being sixty. It is the first composite number of the form p2q, a square-prime,12 has an aliquot sum of 16. Accordingly,12 is the first abundant number and demonstrates an 8-member aliquot sequence,12 is the 3rd composite number in the 3-aliquot tree, the only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant, Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number