2

Short description: Integer number 2

This article is about the number. For the years, see 2 BC and AD 2. For other uses, see II (disambiguation) and Number Two (disambiguation).

Short description: Natural number

← 1 2 3 →
-1 0 1 2 3 4 5 6 7 8 9 → List of numbers — Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	two
Ordinal	2nd (second / twoth)
Numeral system	binary
Factorization	prime
Gaussian integer factorization	[math]\displaystyle{ (1 + i)(1 - i) }[/math]
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
Quaternary	24
Quinary	25
Senary	26
Octal	28
Duodecimal	212
Hexadecimal	216
Vigesimal	220
Base 36	236
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
Khmer	២
Thai	๒
Georgian	Ⴁ/ⴁ/ბ(Bani)
Malayalam	൨

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.

Evolution

Arabic digit

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.

In fonts with text figures, digit 2 usually is of x-height, for example, .

As a word

Two is most commonly a determiner used with plural countable nouns, as in two days or I'll take these two. Two is a noun when it refers to the number two as in two plus two is four.

Etymology of two

The word two is derived from the Old English words twā (feminine), tū (neuter), and twēġen (masculine, which survives today in the form twain).

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/.

Mathematics

Characterizations

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. 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.

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 [math]\displaystyle{ d(n) }[/math] of positive integers [math]\displaystyle{ n }[/math] satisfies, [math]\displaystyle{ \liminf_{n\to\infty} d(n)=2, }[/math] where [math]\displaystyle{ \liminf }[/math] represents the limit inferior (since there will always exist a larger prime number with a maximum of two divisors).

Specifically,

In a set-theoretical construction of the natural numbers, [math]\displaystyle{ 2 }[/math] is identified with the set [math]\displaystyle{ \{\{\varnothing\},\varnothing\} }[/math], where [math]\displaystyle{ \varnothing }[/math] 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 [math]\displaystyle{ 2^\mathbb{N} }[/math] 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, [math]\displaystyle{ \{0, 1\} }[/math], is the traditional elementary example of a Cantor space. 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 [math]\displaystyle{ \log_{2} }[/math] [math]\displaystyle{ n }[/math] tokens) than a direct representation by the corresponding count of a single token (with [math]\displaystyle{ n }[/math] tokens). This number system is used extensively in computing.

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

Powers of 2

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).

[math]\displaystyle{ 2 }[/math] is the only number such that the sum of the reciprocals of its natural powers equals itself. In symbols,

[math]\displaystyle{ \sum_{n=0}^{\infin}\frac {1}{2^n}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=2. }[/math]

Two also has the unique property that [math]\displaystyle{ 2+2=2\times2=2^{2}=2\uparrow\uparrow 2= 2\uparrow\uparrow\uparrow2=\text{ }... }[/math] up through any level of hyperoperation, here denoted in Knuth's up-arrow notation, all equivalent to [math]\displaystyle{ 4. }[/math]

Notably, row sums in Pascal's triangle are in equivalence with successive powers of two, [math]\displaystyle{ 2^{n}. }[/math]

Properties

The numbers two and three are the only two consecutive prime numbers. 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. In consequence, three and five encase four in-between, which is the square of two, [math]\displaystyle{ 2^2 }[/math]. These are also the two odd prime numbers that lie amongst the only all-Harshad numbers (1, 2, 4, and 6) that are also the first four highly composite numbers, with 2 the only number that is both a prime number and a highly composite number.

Furthermore, [math]\displaystyle{ (3,5) }[/math] are the unique pair of twin primes [math]\displaystyle{ (q, q+2) }[/math] that yield the second and only prime quadruplet [math]\displaystyle{ (11, 13, 17, 19) }[/math] that is of the form [math]\displaystyle{ (d-4, d-2, d+2, d+4) }[/math], where [math]\displaystyle{ d }[/math] is the product of said twin primes.

In particular, the sum of the reciprocals of all non-zero triangular numbers converges to 2.

Integer sequences

The first number to return zero for the Mertens function is 2, and the harmonic mean of the divisors of 6, the smallest Ore number greater than 1, is also 2.

Inside other important integer sequences, Template:Bullet list

Two consecutive twos (as in "22" for "two twos"), or equivalently "2-2", is the only fixed point of John Conway's look-and-say function. On the other hand, there are no [math]\displaystyle{ 2\times2 }[/math] magic squares, and as such they are the only null [math]\displaystyle{ n }[/math] by [math]\displaystyle{ n }[/math] magic square set.

There are only two known sublime numbers, which are numbers with a perfect number of factors, whose sum itself yields a perfect number. 12 is one of the two sublime numbers, with the other being 76 digits long.

Euler's number

[math]\displaystyle{ e }[/math] can be simplified to equal,

[math]\displaystyle{ e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!} = 2 + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots }[/math]

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

Geometry

Regarding regular polygons in two dimensions:

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

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

• The span of an octagon is in silver ratio [math]\displaystyle{ \delta_s }[/math] with its sides, which can be computed with the continued fraction [math]\displaystyle{ [2;2,2,...] = 2.414\;235\dots }[/math]

Whereas a square of unit side length has a diagonal equal to [math]\displaystyle{ \sqrt{2} }[/math], a space diagonal inside a tesseract measures 2 when its side lengths are of unit length.

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.

For any polyhedron homeomorphic to a sphere, the Euler characteristic is [math]\displaystyle{ \chi=V-E+F=2 }[/math], where [math]\displaystyle{ V }[/math] is the number of vertices, [math]\displaystyle{ E }[/math] is the number of edges, and [math]\displaystyle{ F }[/math] is the number of faces. A double torus has an Euler characteristic of [math]\displaystyle{ -2 }[/math], on the other hand, and a non-orientable surface of like genus [math]\displaystyle{ k }[/math] has a characteristic [math]\displaystyle{ \chi=2-k }[/math].

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

List of basic calculations

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

In science

• The number of polynucleotide strands in a DNA double helix.
• The first magic number.
• The atomic number of helium.

See also

• List of highways numbered 2
• Binary number

External links

• Prime curiosities: 2