| ||||
---|---|---|---|---|
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.
Two is most commonly a determiner used with plural countable nouns, as in two days or I'll take these two. [1] 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). [2]
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/. [2]
An integer is determined to be even if it is divisible by two. For integers written in a numeral system based on an even number such as decimal, divisibility by two 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 multiple of 2 will end in 0, 2, 4, 6, or 8. [3]
1 is neither prime nor composite yet odd. 0, which is an origin to the integers in the real line, especially when considered alongside negative integers, is neither prime nor composite, however it is distinctively even (as a multiple of two) since if it were to be odd, then for some integer there would be that yields a of , which is a contradiction (however, for a function, the zero function is the only function to both be even and odd).
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. [4]
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). [5] Aside from square numbers and prime powers raised to an even exponent, or integers that are the product of an even number of prime powers with even exponents, an integer will have a that is a multiple of . The two smallest natural numbers have unique properties in this regard: is the only number with a single divisor (itself), where on the other hand, is the only number to have an infinite number of divisors, since dividing zero by any strictly positive or negative integer yields (i.e., aside from division of zero by zero, ).
is the only set of numbers whose distinct divisors (with more than one) are also consecutive integers, when excluding negative integers. [lower-alpha 1]
Meanwhile, the numbers two and three are the only two prime numbers that are consecutive integers, where the number two is also adjacent to the unit. 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. [6] [7] 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) [8] that are also the first four highly composite numbers, [9] with the only number that is both a prime number and a "highly composite number". [lower-alpha 2]
is the first Ramanujan prime satisfying where is the prime-counting function, equal to the number of primes less than or equal to . [11] [lower-alpha 3]
In the smallest Cunningham chains of nearly doubled primes (of the first and second kind) two is the first member, as part of the sets and .
The first fifteen prime numbers between and are also consecutive primes that are part of Bhargava’s seventeen-integer quadratic matrix representative of all prime numbers (only two other numbers are part of this set of prime integers, namely the nineteenth and twenty-first prime numbers 67 and 73). [12] The seventh square number, , is in equivalence with the sum of the first and fifteenth primes. [lower-alpha 4]
A set that is a field has a minimum of two elements. In a set-theoretical construction of the natural numbers , two 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.
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.
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, [13] [14] Two is the first Mersenne prime exponent, and it is the difference between the first two Fermat primes (3 and 5).
A number is perfect if it is equal to its aliquot sum, or the sum of all of its positive divisors excluding the number itself. This is equivalent to describing a perfect number as having a sum of divisors equal to The harmonic mean of the divisors of — the smallest perfect number, unitary perfect number, and Ore number greater than — is . Two itself is the smallest primary pseudoperfect number such that the reciprocal of plus the sum of reciprocals of prime factors of is [15] There are only two known sublime numbers, which are numbers with a perfect number of factors, whose sum itself yields a perfect number: [16]
The latter is a number that is seventy-six digits long (in decimal representation).
Otherwise, a number is deficient when the sum of its divisors is less than twice the number, whereas an abundant number has a sum of its proper divisors that is larger than the number itself. Primitive abundant numbers are abundant numbers whose proper divisors are all deficient.
Euler's number can be simplified to equal,
A continued fraction for repeats a pattern from the second term onward. [17] [18]
In decimal representation, after the first two, three, four and five digits in the approximation of pi () the number is the only digit greater than zero not yet represented (overall, up to the largest appearing digit). [lower-alpha 5]
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. [23]
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. [24]
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 [25] 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. [26] In general, the repetition threshold of an infinite binary-rich word will be [27]
In a Euclidean space of any dimension greater than zero, two distinct points in a plane are always sufficient to define a unique line. [28]
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 |
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. [32]
In fonts with text figures, digit 2 usually is of x-height, for example, .[ citation needed ]
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32. The smallest positive square-free numbers are
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.
33 (thirty-three) is the natural number following 32 and preceding 34.
27 is the natural number following 26 and preceding 28.
84 (eighty-four) is the natural number following 83 and preceding 85.
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.
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.
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.
360 is the natural number following 359 and preceding 361.
144 is the natural number following 143 and preceding 145.
168 is the natural number following 167 and preceding 169.
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.
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.
744 is the natural number following 743 and preceding 745.
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic ; if this sequence consists only of zeros, the decimal is said to be terminating, and is not considered as repeating. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. For example, the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is 3227/555, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... Another example of this is 593/53, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830....
14 (fourteen) is a natural number following 13 and preceding 15.