2

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 ${\displaystyle k}$ there would be ${\displaystyle 0=2k+1}$ that yields a ${\displaystyle k}$ of ${\displaystyle -{\tfrac {1}{2}}}$ , which is a contradiction (however, for a function, the zero function ${\displaystyle f(x)=0}$ 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 ${\displaystyle d(n)}$ of positive integers ${\displaystyle n}$ satisfies,$${\displaystyle \liminf _{n\to \infty }d(n)=2,}$$ where ${\displaystyle \liminf }$ 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 ${\displaystyle d(n)}$ that is a multiple of ${\displaystyle 2}$ . The two smallest natural numbers ${\displaystyle (0,1)\in \mathbb {N} _{0}}$ have unique properties in this regard: ${\displaystyle 1}$ is the only number with a single divisor (itself), where on the other hand, ${\displaystyle 0}$ is the only number to have an infinite number of divisors, since dividing zero by any strictly positive or negative integer yields ${\displaystyle 0}$ (i.e., aside from division of zero by zero, ${\displaystyle {\tfrac {0}{0}}}$ ).

${\displaystyle (0,2)\in \mathbb {N} _{0}}$ is the only set of numbers whose distinct divisors (with more than one) are also consecutive integers, when excluding negative integers.^[a]

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, ${\displaystyle 2^{2}}$ . 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 ${\displaystyle 2}$ the only number that is both a prime number and a "highly composite number".^[b]

Twin primes are the smallest type of prime k-tuples, that represent patterns of repeating differences between prime numbers. A difference of two in prime k-tuples exists inside prime quintuplets, and in some types of prime triplets and prime quadruplets (et cetera).

${\displaystyle 2}$ is the first Ramanujan prime satisfying ${\displaystyle \pi (x)-\pi \left({\frac {x}{2}}\right)\geq 1,2,3,\ldots {\text{ for all }}x\geq 2,11,17,\ldots }$ where ${\displaystyle \pi (x)}$ is the prime-counting function, equal to the number of primes less than or equal to ${\displaystyle x}$ .^[11][c]

A set that is a field has a minimum of two elements. In a set-theoretical construction of the natural numbers ${\displaystyle \mathbb {N} }$ , two is identified with the set ${\displaystyle \{\{\varnothing \},\varnothing \}}$ , where ${\displaystyle \varnothing }$ 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 ${\displaystyle 2^{\mathbb {N} }}$ 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, ${\displaystyle \{0,1\}}$ , is the traditional elementary example of a Cantor space. Points whose initial conditions remain on a ${\displaystyle [0,1]}$ boundary in the logistic map ${\displaystyle x_{n+1}=rx_{n}(1-x_{n})}$ form a Cantor set, where values begin to diverge beyond ${\displaystyle r=4.}$ Between ${\displaystyle r\approx 3.45}$ and ${\displaystyle 3.57}$ , the population approaches oscillations among ${\displaystyle 8,16,...,2^{n},\ldots ,2^{\infty }}$ 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). ${\displaystyle 2}$ is the only number such that the sum of the reciprocals of its natural powers equals itself. In symbols,

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =2.}$

Two also has the unique property that ${\displaystyle 2+2=2\times 2=2^{2}=2\uparrow \uparrow 2=2\uparrow \uparrow \uparrow 2={\text{ }}...}$ up through any level of hyperoperation, here denoted in Knuth's up-arrow notation, all equivalent to ${\displaystyle 4.}$

Notably, row sums in Pascal's triangle are in equivalence with successive powers of two, ${\displaystyle 2^{n}.}$ ^[12][13] 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 ${\displaystyle n}$ as having a sum of divisors ${\displaystyle \sigma (n)}$ equal to ${\displaystyle 2n.}$ The harmonic mean of the divisors of ${\displaystyle 6}$ — the smallest perfect number, unitary perfect number, and Ore number greater than ${\displaystyle 1}$ — is ${\displaystyle 2}$ . Two itself is the smallest primary pseudoperfect number ${\displaystyle n}$ such that the reciprocal of ${\displaystyle n}$ plus the sum of reciprocals of prime factors of ${\displaystyle n}$ is ${\displaystyle 1.}$ ^[14] There are only two known sublime numbers, which are numbers with a perfect number of factors, whose sum itself yields a perfect number:^[15]

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 ${\displaystyle e}$ can be simplified to equal,

${\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=2+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots }$

A continued fraction for ${\displaystyle e=[2;1,2,1,1,4,1,1,8,...]}$ repeats a ${\displaystyle \{1,2n,1\}}$ pattern from the second term onward.^[16][17]

For a simple calculation involving pi, Brouncker's formula using the generalized continued fraction^[18][19]

${\displaystyle {\frac {4}{\pi }}=1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+\ddots }}}}}}}}}$

contains ${\displaystyle 2}$ as a constant partial denominator, where partial numerators are successive odd squares.

• The sum of the reciprocals of all non-zero triangular numbers converges to 2.^[20]
• Numbers also cannot be laid out in a ${\displaystyle 2\times 2}$ magic square that yields a magic constant, and as such they are the only null ${\displaystyle n}$ by ${\displaystyle n}$ magic square set.^[21][d]
• Every number ${\displaystyle n}$ is polygonal by being ${\displaystyle 2}$ -gonal (i.e., a natural number), as well as the root of some type of ${\displaystyle n}$ -gonal number. For ${\displaystyle n=2}$ , being ${\displaystyle 2}$ -gonal and ${\displaystyle n}$ -gonal is the same, which make two the only number that is polygonal in only one way.
• 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.^[22]

Regarding Bernouilli numbers ${\displaystyle B_{2k}}$ , by convention ${\displaystyle 2}$ has an irregularity of ${\displaystyle -1.}$ ^[23] Two is also the first number to return zero for the Mertens function.^[24]

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 ${\displaystyle \{2,3,5\}}$ and ${\displaystyle \{2,5,11,23,47\}}$ .

The first fifteen prime numbers between ${\displaystyle 2}$ and ${\displaystyle 47}$ 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).^[25] The seventh square number, ${\displaystyle 49=7^{2}}$ , is in equivalence with the sum of the first and fifteenth primes.^[e]

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 ${\displaystyle \log _{2}}$ ${\displaystyle n}$ tokens) than a direct representation by the corresponding count of a single token (with ${\displaystyle n}$ tokens). This number system is used extensively in computing.^[26]

In the Thue-Morse sequence ${\displaystyle T}$ , that successively adjoins the binary Boolean complement from ${\displaystyle \{0\}}$ onward (in succession), the critical exponent, or largest number of times an adjoining subsequence repeats, is ${\displaystyle 2}$ , where there exist a vast amount of square words of the form ${\displaystyle ww.}$ ^[27] Furthermore, in ${\displaystyle c}$ , which counts the instances of ${\displaystyle 1}$ between consecutive occurrences of ${\displaystyle 0}$ in ${\displaystyle T}$ that is instead square-free, the critical exponent is also ${\displaystyle 2}$ , since ${\displaystyle c=\{210201210120\ldots \}}$ contains factors of exponents close to ${\displaystyle 2}$ due to ${\displaystyle T}$ containing a large factor of squares.^[28] In general, the repetition threshold of an infinite binary-rich word will be ${\displaystyle 2+{\tfrac {\sqrt {2}}{2}}\approx 2.707\ldots }$ ^[29]

In a Euclidean space of any dimension greater than zero, two distinct points in a plane are always sufficient to define a unique line.^[30]

Regarding regular polygons in two dimensions:

• The equilateral triangle has the smallest ratio of the circumradius ${\displaystyle R}$ to the inradius ${\displaystyle r}$ of any triangle by Euler's inequality, with ${\displaystyle {\tfrac {R}{r}}=2.}$ ^[31]

• The long diagonal of a regular hexagon is of length 2 when its sides are of unit length.^[32]

• The span of an octagon is in silver ratio ${\displaystyle \delta _{s}}$ with its sides, which can be computed with the continued fraction ${\displaystyle [2;2,2,...]=2.414\;235\dots }$ ^[33]

Whereas a square of unit side length has a diagonal equal to ${\displaystyle {\sqrt {2}}}$ , 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 ${\displaystyle \chi =V-E+F=2}$ , where ${\displaystyle V}$ is the number of vertices, ${\displaystyle E}$ is the number of edges, and ${\displaystyle F}$ is the number of faces. A double torus has an Euler characteristic of ${\displaystyle -2}$ , on the other hand, and a non-orientable surface of like genus ${\displaystyle k}$ has a characteristic ${\displaystyle \chi =2-k}$ .^{[citation needed]}

The simplest tessellation in two-dimensional space, though an improper tessellation, is that of two ${\displaystyle \infty }$ -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 ${\displaystyle \{p,2\}}$ .^{[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.

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.^[34]

In fonts with text figures, digit 2 usually is of x-height, for example, .^{[citation needed]}

• The number of polynucleotide strands in a DNA double helix.^[35]
• The first magic number.^[36]
• The atomic number of helium.^[37]

• Binary number

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• Prime curiosities: 2

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