| ||||
---|---|---|---|---|
Cardinal | two | |||
Ordinal | 2nd (second / twoth) | |||
Numeral system | binary | |||
Factorization | prime | |||
Gaussian integer factorization | ||||
Prime | 1st | |||
Divisors | 1, 2 | |||
Greek numeral | Β´ | |||
Roman numeral | II, ii | |||
Greek prefix | di- | |||
Latin prefix | duo-/bi- | |||
Old English prefix | twi- | |||
Binary | 102 | |||
Ternary | 23 | |||
Senary | 26 | |||
Octal | 28 | |||
Duodecimal | 212 | |||
Hexadecimal | 216 | |||
Greek numeral | β' | |||
Arabic, Kurdish, Persian, Sindhi, Urdu | ٢ | |||
Ge'ez | ፪ | |||
Bengali | ২ | |||
Chinese numeral | 二,弍,貳 | |||
Devanāgarī | २ | |||
Telugu | ౨ | |||
Tamil | ௨ | |||
Kannada | ೨ | |||
Hebrew | ב | |||
Armenian | Բ | |||
Khmer | ២ | |||
Maya numerals | •• | |||
Thai | ๒ | |||
Georgian | Ⴁ/ⴁ/ბ(Bani) | |||
Malayalam | ൨ | |||
Babylonian numeral | 𒐖 | |||
Egyptian hieroglyph, Aegean numeral, Chinese counting rod | || | |||
Morse code | .._ _ _ |
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.
The digit used in the modern Western world to represent the number 2 traces its roots back to the Indic Brahmic script, where "2" was written as two horizontal lines. The modern Chinese and Japanese languages (and Korean Hanja) still use this method. The Gupta script rotated the two lines 45 degrees, making them diagonal. The top line was sometimes also shortened and had its bottom end curve towards the center of the bottom line. In the Nagari script, the top line was written more like a curve connecting to the bottom line. In the Arabic Ghubar writing, the bottom line was completely vertical, and the digit looked like a dotless closing question mark. Restoring the bottom line to its original horizontal position, but keeping the top line as a curve that connects to the bottom line leads to our modern digit. [1]
In fonts with text figures, digit 2 usually is of x-height, for example, .[ citation needed ]
Two is most commonly a determiner used with plural countable nouns, as in two days or I'll take these two. [2] Two is a noun when it refers to the number two as in two plus two is four.
The word two is derived from the Old English words twā (feminine), tū (neuter), and twēġen (masculine, which survives today in the form twain). [3]
The pronunciation /tuː/, like that of who is due to the labialization of the vowel by the w, which then disappeared before the related sound. The successive stages of pronunciation for the Old English twā would thus be /twɑː/, /twɔː/, /twoː/, /twuː/, and finally /tuː/. [3]
An integer is determined to be even if it is divisible by 2. For integers written in a numeral system based on an even number such as decimal, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. When written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8. [4]
The number two is the smallest, and only even, prime number. As the smallest prime number, two is also the smallest non-zero pronic number, and the only pronic prime. [5]
Every integer greater than 1 will have at least two distinct factors; by definition, a prime number only has two distinct factors (itself and 1). Therefore, the number-of-divisors function of positive integers satisfies,
where represents the limit inferior (since there will always exist a larger prime number with a maximum of two divisors). [6]
Specifically,
In a set-theoretical construction of the natural numbers, is identified with the set , where denotes the empty set. This latter set is important in category theory: it is a subobject classifier in the category of sets. More broadly, a set that is a field has a minimum of two elements.
The binary system has a radix of two, and it is the numeral system with the fewest tokens that allows denoting a natural number substantially more concisely (with tokens) than a direct representation by the corresponding count of a single token (with tokens). This number system is used extensively in computing.[ citation needed ]
In a Euclidean space of any dimension greater than zero, two distinct points in a plane are always sufficient to define a unique line.[ citation needed ]
A Cantor space is a topological space homeomorphic to the Cantor set, whose general set is a closed set consisting purely of boundary points. The countably infinite product topology of the simplest discrete two-point space, , is the traditional elementary example of a Cantor space. Points whose initial conditions remain on a boundary in the logistic map form a Cantor set, where values begin to diverge beyond Between and , the population approaches oscillations among values before chaos ensues.
Two is the first Mersenne prime exponent, and it is the difference between the first two Fermat primes (3 and 5). Powers of two are essential in computer science, and important in the constructability of regular polygons using basic tools (e.g., through the use of Fermat or Pierpont primes). is the only number such that the sum of the reciprocals of its natural powers equals itself. In symbols,
Two also has the unique property that up through any level of hyperoperation, here denoted in Knuth's up-arrow notation, all equivalent to
Notably, row sums in Pascal's triangle are in equivalence with successive powers of two, [7] [8]
The numbers two and three are the only two prime numbers that are also consecutive integers. Two is the first prime number that does not have a proper twin prime with a difference two, while three is the first such prime number to have a twin prime, five. [9] [10] In consequence, three and five encase four in-between, which is the square of two, . These are also the two odd prime numbers that lie amongst the only all-Harshad numbers (1, 2, 4, and 6) [11] that are also the first four highly composite numbers, [12] with 2 the only number that is both a prime number and a highly composite number. Furthermore, are the unique pair of twin primes that yield the second and only prime quadruplet that is of the form , where is the product of said twin primes. [13]
Inside other important integer sequences,
is the smallest primary pseudoperfect number, [24] and it is the first number to return zero for the Mertens function. [25] The harmonic mean of the divisors of — the smallest perfect number, unitary perfect number, and Ore number greater than — is also . In particular, the sum of the reciprocals of all non-zero triangular numbers converges to 2. [26] On the other hand, numbers cannot be laid out in a magic square that yields a magic constant, and as such they are the only null by magic square set. [27] [lower-alpha 1] There are only two known sublime numbers, which are numbers with a perfect number of factors, whose sum itself yields a perfect number: [28]
The latter is a number that is seventy-six digits long (in decimal representation).
Regarding Bernouilli numbers , by convention has an irregularity of [29]
In the Thue-Morse sequence , that successively adjoins the binary Boolean complement from onward (in succession), the critical exponent, or largest number of times an adjoining subsequence repeats, is , where there exist a vast amount of square words of the form [30] Furthermore, in , which counts the instances of between consecutive occurrences of in that is instead square-free, the critical exponent is also , since contains factors of exponents close to due to containing a large factor of squares. [31] In general, the repetition threshold of an infinite binary-rich word will be [32]
In John Conway's look-and-say function, which can be represented faithfully with a quaternary numeral system, two consecutive twos (as in "22" for "two twos"), or equivalently "2 - 2", is the only fixed point. [33]
can be simplified to equal,
A continued fraction for repeats a pattern from the second term onward. [34] [35]
Regarding regular polygons in two dimensions:
Whereas a square of unit side length has a diagonal equal to , a space diagonal inside a tesseract measures 2 when its side lengths are of unit length.[ citation needed ]
A digon is a polygon with two sides (or edges) and two vertices. On a circle, it is a tessellation with two antipodal points and 180° arc edges.[ citation needed ]
For any polyhedron homeomorphic to a sphere, the Euler characteristic is , where is the number of vertices, is the number of edges, and is the number of faces. A double torus has an Euler characteristic of , on the other hand, and a non-orientable surface of like genus has a characteristic .[ citation needed ]
The simplest tessellation in two-dimensional space, though an improper tessellation, is that of two -sided apeirogons joined along all their edges, coincident about a line that divides the plane in two. This order-2 apeirogonal tiling is the arithmetic limit of the family of dihedra .[ citation needed ] The second dimension is also the only dimension where there are both an infinite number of Euclidean and hyperbolic regular polytopes (as polygons), and an infinite number of regular hyperbolic paracompact tesselations.
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 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 × x | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 | 40 | 42 | 44 | 46 | 48 | 50 | 100 | 200 |
Division | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 ÷ x | 2 | 1 | 0.6 | 0.5 | 0.4 | 0.3 | 0.285714 | 0.25 | 0.2 | 0.2 | 0.18 | 0.16 | 0.153846 | 0.142857 | 0.13 | 0.125 | 0.1176470588235294 | 0.1 | 0.105263157894736842 | 0.1 | |
x ÷ 2 | 0.5 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | 5.5 | 6 | 6.5 | 7 | 7.5 | 8 | 8.5 | 9 | 9.5 | 10 |
Exponentiation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2x | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | 2048 | 4096 | 8192 | 16384 | 32768 | 65536 | 131072 | 262144 | 524288 | 1048576 | |
x2 | 1 | 9 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 | 256 | 289 | 324 | 361 | 400 |
42 (forty-two) is the natural number that follows 41 and precedes 43.
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.
21 (twenty-one) is the natural number following 20 and preceding 22.
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.
90 (ninety) is the natural number following 89 and preceding 91.
84 (eighty-four) is the natural number following 83 and preceding 85.
57 (fifty-seven) is the natural number following 56 and preceding 58.
58 (fifty-eight) is the natural number following 57 and preceding 59.
63 (sixty-three) is the natural number following 62 and preceding 64.
104 is the natural number following 103 and preceding 105.
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
144 is the natural number following 143 and preceding 145.
177 is the natural number following 176 and preceding 178.
1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.
240 is the natural number following 239 and preceding 241.
288 is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen".
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
744 is the natural number following 743 and preceding 745.
14 (fourteen) is a natural number following 13 and preceding 15.
262 is a natural number preceded by the number 261 and followed by 263. It has the prime factorization 2·131.