# 2

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Cardinaltwo
Ordinal2nd (second / twoth)
Numeral systembinary
Factorizationprime
Gaussian integer factorization${\displaystyle (1+i)(1-i)}$
Prime1st
Divisors1, 2
Greek numeralΒ´
Roman numeralII, ii
Greek prefixdi-
Latin prefixduo-/bi-
Old English prefixtwi-
Binary102
Ternary23
Senary26
Octal28
Duodecimal212
Greek numeralβ'
Arabic, Kurdish, Persian, Sindhi, Urdu٢
Ge'ez
Bengali
Chinese numeral二，弍，貳
Devanāgarī
Telugu
Tamil
Hebrewב
ArmenianԲ
Khmer
Maya numerals••
Thai
Georgian Ⴁ/ⴁ/ბ(Bani)
Malayalam
Babylonian numeral𒐖
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.

## 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.[1]

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

## As a word

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.

### Etymology of two

The word two is derived from the Old English words twā (feminine), (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]

## Mathematics

### Divisibility rule

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]

### 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.[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 ${\displaystyle d(n)}$ 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).[6]

Specifically,

• 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.
• 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 ${\displaystyle 2n}$.

In a set-theoretical construction of the natural numbers, ${\displaystyle 2}$ 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. 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 ${\displaystyle \log _{2}}$ ${\displaystyle n}$ 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.[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]

#### Cantor space

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 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). ${\displaystyle 2}$ 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, ${\displaystyle 2^{n}.}$[7][8]

### Integer sequences

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

Inside other important integer sequences,

${\displaystyle 2}$ 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 ${\displaystyle 6}$ — the smallest perfect number, unitary perfect number, and Ore number greater than ${\displaystyle 1}$ — is also ${\displaystyle 2}$. 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 ${\displaystyle 2\times 2}$ magic square that yields a magic constant, and as such they are the only null ${\displaystyle n}$ by ${\displaystyle n}$ magic square set.[27][a] There are only two known sublime numbers, which are numbers with a perfect number of factors, whose sum itself yields a perfect number:[28]

• ${\displaystyle 12=(2^{2})\times (2^{2}-1)}$
• ${\displaystyle 6\;086\;\ldots \;264=(2^{126})\times (2^{61}-1)\times (2^{31}-1)\times (2^{19}-1)\times (2^{7}-1)\times (2^{5}-1)\times (2^{3}-1)}$

The latter is a number that is seventy-six digits long (in decimal representation).

Regarding Bernouilli numbers ${\displaystyle B_{2k}}$, by convention ${\displaystyle 2}$ has an irregularity of ${\displaystyle -1.}$[29]

#### Iterative sequences

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 ${\displaystyle ww.}$[30] 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.[31] In general, the repetition threshold of an infinite binary-rich word will be ${\displaystyle 2+{\tfrac {\sqrt {2}}{2}}.}$[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]

#### 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.[34][35]

### Geometry

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 ${\displaystyle {\tfrac {R}{r}}=2.}$[36]
• The span of an octagon is in silver ratio ${\displaystyle \delta _{s}}$ with its sides, which can be computed with the continued fraction ${\displaystyle [2;2,2,...]=2.414\;235\dots }$[37]

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.

### 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

## Notes

1. ^ Meanwhile, the magic constant of an ${\displaystyle n}$-pointed normal magic star is ${\displaystyle M=4n+2}$.

## References

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2. ^ Huddleston, Rodney D.; Pullum, Geoffrey K.; Reynolds, Brett (2022). A student's introduction to English grammar (2nd ed.). Cambridge, United Kingdom: Cambridge University Press. p. 117. ISBN 978-1-316-51464-1. OCLC 1255524478.
3. ^ a b "two, adj., n., and adv.". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
4. ^ Sloane, N. J. A. (ed.). "Sequence A005843 (The nonnegative even numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
5. ^ "Sloane's A002378: Pronic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 2016-06-09. Retrieved 2020-11-30.
6. ^ Hardy, G. H.; Wright, E. M. (2008), An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, pp. 342–347, §18.1, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001
Also, ${\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)}{\log n/\log \log n}}=\log 2}$.
7. ^ Smith, Karl J. (1973). "Pascal's Triangle". The Two-Year College Mathematics Journal. 4 (1). Washington, D.C.: Mathematical Association of America: 4. doi:10.2307/2698949. JSTOR 2698949. S2CID 265738469.
8. ^ Sloane, N. J. A. (ed.). "Sequence A000079 (Powers of 2: a(n) equal to 2^n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-06.
9. ^ Sloane, N. J. A. (ed.). "Sequence A007510 (Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-05.
10. ^ Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-05.
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"{11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12*k^2-1) and 3*(12*k^2+1) respectively (k is an integer)."
14. ^ Sloane, N. J. A. (ed.). "Sequence A005384 (Sophie Germain primes p: 2p+1 is also prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
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16. ^ Sloane, N. J. A. (ed.). "Sequence A005479 (Prime Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
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18. ^ Sloane, N. J. A. (ed.). "Sequence A001006 (Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
19. ^ Sloane, N. J. A. (ed.). "Sequence A000110 (Bell or exponential numbers: number of ways to partition a set of n labeled elements.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
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22. ^ Sloane, N. J. A. (ed.). "Sequence A000682 (Semi-meanders: number of ways a semi-infinite directed curve can cross a straight line n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
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"Only a(1) = 0 prevents this from being a simple continued fraction. The motivation for this alternate representation is that the simple pattern {1, 2*n, 1} (from n=0) may be more mathematically appealing than the pattern in the corresponding simple continued fraction (at A003417)."
36. ^ Svrtan, Dragutin; Veljan, Darko (2012). "Non-Euclidean versions of some classical triangle inequalities" (PDF). Forum Geometricorum. 12. Boca Raton, FL: Department of Mathematical Sciences, Florida Atlantic University: 198. ISSN 1534-1178. MR 2955631. S2CID 29722079. Zbl 1247.51012. Archived (PDF) from the original on 2023-05-03. Retrieved 2023-04-30.
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