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).

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.

← 1 2 3 →
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Cardinal	two
Ordinal	2nd (second / twoth)
Numeral system	binary
Factorization	prime
Gaussian integer factorization	${\displaystyle (1+i)(1-i)}$
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
Khmer	២
Thai	๒
Georgian	Ⴁ/ⴁ/ბ(Bani)
Malayalam	൨

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), 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]

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.^[4] 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.^[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. 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. 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]}

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]

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.^[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]

In particular, the sum of the reciprocals of all non-zero triangular numbers converges to 2.^[14]

Integer sequences

The first number to return zero for the Mertens function is 2,^[15] and the harmonic mean of the divisors of 6, the smallest Ore number greater than 1, is also 2.

Inside other important integer sequences,

• 2 is the first Sophie Germain prime,^[16] the first factorial prime,^[17] the first Lucas prime,^[18] and the first Ramanujan prime.^[19]
• 2 is also a Motzkin number,^[20] a Bell number,^[21] and the third (or fourth) Fibonacci number.^[22]
• 2 is a meandric number,^[23] a semi-meandric number,^[24] and an open meandric number.^[25]

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.^[26] On the other hand, there are no ${\displaystyle 2\times 2}$  magic squares, 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. 12 is one of the two sublime numbers, with the other being 76 digits long.^[28]

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.^[29][30]

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 ${\displaystyle {\tfrac {R}{r}}=2.}$ ^[31]

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

• The span of an octagon is in silver ratio ${\displaystyle \delta _{s}}$  with its sides, which can be computed with the continued fraction ${\displaystyle [2;2,2,...]=2.414\;235\dots }$ ^[32]

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]}

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
2^x	2	4	8	16	32	64	128	256	512	1024		2048	4096	8192	16384	32768	65536	131072	262144	524288	1048576
x^2	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.^[33]
• The first magic number.^[34]
• The atomic number of helium.^[35]

See also

• List of highways numbered 2
• Binary number

Notes

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

References

1. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 393, Fig. 24.62
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. "two, adj., n., and adv.". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
4. "Sloane's A002378: Pronic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. from the original on 2016-06-09. Retrieved 2020-11-30.
5. Sloane, N. J. A. (ed.). "Sequence A005843 (The nonnegative even numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
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. Washington, D.C.: Mathematical Association of America. 4 (1): 4. doi:10.1080/00494925.1973.11974228. 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.
11. PrimeFan (2013-03-22). "Harshad number". PlanetMath. Retrieved 2023-12-18.
12. Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-18.
13. Sloane, N. J. A. (ed.). "Sequence A136162 (List of prime quadruplets {p, p+2, p+6, p+8}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-09.

    "{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. Grabowski, Adam (2013). "Polygonal numbers". Formalized Mathematics. Sciendo (De Gruyter). 21 (2): 103–113. doi:10.2478/forma-2013-0012. S2CID 15643540. Zbl 1298.11029.
15. Sloane, N. J. A. (ed.). "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
16. 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.
17. Sloane, N. J. A. (ed.). "Sequence A088054 (Factorial primes: primes which are within 1 of a factorial number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
18. Sloane, N. J. A. (ed.). "Sequence A005479 (Prime Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
19. . The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 2011-04-28. Retrieved 2016-06-01.
20. 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.
21. 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.
22. Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
23. Sloane, N. J. A. (ed.). "Sequence A005315 (Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
24. 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.
25. Sloane, N. J. A. (ed.). "Sequence A005316 (Meandric numbers: number of ways a river can cross a road n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15.
26. Martin, Oscar (2006). (PDF). American Mathematical Monthly. Mathematical association of America. 113 (4): 289–307. doi:10.2307/27641915. ISSN 0002-9890. JSTOR 27641915. Archived from the original (PDF) on 2006-12-24. Retrieved 2022-07-21.
27. Sloane, N. J. A. (ed.). "Sequence A006052 (Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21.
28. Sloane, N. J. A. (ed.). "Sequence A081357 (Sublime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-13.
29. Cohn, Henry (2006). "A Short Proof of the Simple Continued Fraction Expansion of e". The American Mathematical Monthly. Taylor & Francis, Ltd. 113 (1): 57–62. doi:10.1080/00029890.2006.11920278. JSTOR 27641837. MR 2202921. S2CID 43879696. Zbl 1145.11012. from the original on 2023-04-30. Retrieved 2023-04-30.
30. Sloane, N. J. A. (ed.). "Sequence A005131 (A generalized continued fraction for Euler's number e.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-30.

    "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)."

31. Svrtan, Dragutin; Veljan, Darko (2012). "Non-Euclidean versions of some classical triangle inequalities" (PDF). Forum Geometricorum. Boca Raton, FL: Department of Mathematical Sciences, Florida Atlantic University. 12: 198. ISSN 1534-1178. MR 2955631. S2CID 29722079. Zbl 1247.51012. (PDF) from the original on 2023-05-03. Retrieved 2023-04-30.
32. Vera W. de Spinadel (1999). "The Family of Metallic Means". Visual Mathematics. Belgrade: Mathematical Institute of the Serbian Academy of Sciences. 1 (3). eISSN 1821-1437. S2CID 125705375. Zbl 1016.11005. from the original on 2023-03-26. Retrieved 2023-02-25.
33. "Double-stranded DNA". Scitable. Nature Education. from the original on 2020-07-24. Retrieved 2019-12-22.
34. . www.sjsu.edu. Archived from the original on 2019-12-02. Retrieved 2019-12-22.
35. Bezdenezhnyi, V. P. (2004). "Nuclear Isotopes and Magic Numbers". Odessa Astronomical Publications. 17: 11. Bibcode:2004OAP....17...11B.

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