# 2 Information

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Cardinaltwo
Ordinal2nd (second / twoth)
Numeral system binary
Factorization prime
Gaussian integer factorization${\displaystyle (1+i)(1-i)}$
Prime1st
Divisors1, 2
Greek numeralΒ´
Roman numeralII, ii
Greek prefix di-
Latin prefix duo-/ bi-
Old English prefix twi-
Binary102
Ternary23
Octal28
Duodecimal212
Hexadecimal216
Greek numeralβ'
Arabic, Kurdish, Persian, Sindhi, Urdu٢
Ge'ez
Bengali
Chinese numeral二，弍，貳
Devanāgarī
Telugu
Tamil
Kannada
Hebrewב
Khmer
Thai
Georgian Ⴁ/ⴁ/ბ( Bani)
Malayalam

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, .

### 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). [2]

The pronunciation /tuː/, like that of who is due to the labialization of the vowel by the w (combare from womb), 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]

## In mathematics

An integer is called even if it is divisible by 2. For integers written in a numeral system based on an even number, such as decimal, hexadecimal, or in any other base that is even, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8.

Two is the smallest prime number, and the only even prime number (for this reason it is sometimes called "the oddest prime"). [3] The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, and the first Ramanujan prime. [4]

Two is a pronic number and the only pronic prime. [5]

Two is the third (or fourth) Fibonacci number.

Two is the base of the binary system, the numeral system with the fewest tokens that allows denoting a natural number n substantially more concisely (with log2 n tokens) than a direct representation by the corresponding count of a single token (with n tokens). This binary number system is used extensively in computing.

For any number x:

x + x = 2 · x addition to multiplication
x · x = x2 multiplication to exponentiation
xx = x↑↑2 exponentiation to tetration

Extending this sequence of operations by introducing the notion of hyperoperations, here denoted by "hyper(a,b,c)" with a and c being the first and second operand, and b being the level in the above sketched sequence of operations, the following holds in general:

hyper(x,n,x) = hyper(x,(n + 1),2).

Two has therefore the unique property that 2 + 2 = 2 · 2 = 22 = 2↑↑2 = 2↑↑↑2 = ..., disregarding the level of the hyperoperation, here denoted by Knuth's up-arrow notation. The number of up-arrows refers to the level of the hyperoperation.

Two is the only number x such that the sum of the reciprocals of the natural powers of x equals itself. In symbols

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

This comes from the fact that:

${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{n^{k}}}=1+{\frac {1}{n-1}}\quad {\mbox{for all}}\quad n\in \mathbb {R} >1.}$

A Cantor space is a topological space ${\displaystyle 2^{\mathbb {N} }}$ homeomorphic to the Cantor set. The countably infinite product topology of the simplest discrete two-point space, { 0, 1}, is the traditional elementary example.

Powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent.

Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the index would normally be written for cubic and other roots, may simply be left blank for square roots, as it is tacitly understood.

The square root of 2 was the first known irrational number.

The smallest field has two elements.

In a set-theoretical construction of the natural numbers, 2 is identified with the set {{∅},∅}. This latter set is important in category theory: it is a subobject classifier in the category of sets.

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. [6] This in contrast, for example, with "1211", which would read as "one 1, one 2, and two 1s" or "111221".

There are no 2 x 2 magic squares; they also can be defined as the only null ${\displaystyle n}$ by ${\displaystyle n}$ magic square set. [7]

Two also has the unique property such that

${\displaystyle \sum _{k=0}^{n-1}2^{k}=2^{n}-1}$

and also

${\displaystyle \sum _{k=a}^{n-1}2^{k}=2^{n}-\sum _{k=0}^{a-1}2^{k}-1}$

for a not equal to zero

In any n-dimensional, euclidean space two distinct points determine a line.

In two dimensions, 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.

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 {p, 2}.

For any polyhedron homeomorphic to a sphere, the Euler characteristic is χ = VE + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces.

There are 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. [8]

## Other

In pre-1972 Indonesian and Malay orthography, 2 was shorthand for the reduplication that forms plurals: orang (person), orang-orang or orang2 (people).[ citation needed] In Astrology, Taurus is the second sign of the Zodiac. For Pythagorean numerology (a pseudoscience) the number 2 represents duality, the positive and negative poles that come into balance and seek harmony.

## 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. ^ a b . Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
3. ^ John Horton Conway & Richard K. Guy, The Book of Numbers. New York: Springer (1996): 25. ISBN  0-387-97993-X. "Two is celebrated as the only even prime, which in some sense makes it the oddest prime of all."
4. ^ "Sloane's A104272 : Ramanujan primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 2011-04-28. Retrieved 2016-06-01.
5. ^ "Sloane's A002378: Pronic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2020-11-30.
6. ^ Martin, Oscar (2006). "Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA" (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.
7. ^ 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.
8. ^ Sloane, N. J. A. (ed.). "Sequence A081357 (Sublime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-13.
9. ^ "Double-stranded DNA". Scitable. Nature Education. Retrieved 2019-12-22.
10. ^ "The Complete Explanation of the Nuclear Magic Numbers Which Indicate the Filling of Nucleonic Shells and the Revelation of Special Numbers Indicating the Filling of Subshells Within Those Shells". www.sjsu.edu. Archived from the original on 2019-12-02. Retrieved 2019-12-22.
11. ^ Bezdenezhnyi, V. P. (2004). "Nuclear Isotopes and Magic Numbers". Odessa Astronomical Publications. 17: 11. Bibcode: 2004OAP....17...11B.
12. ^ "Asteroid Fact Sheet". nssdc.gsfc.nasa.gov. Retrieved 2019-12-22.
13. ^ Staff (2018-01-17). "Binary Star Systems: Classification and Evolution". Space.com. Retrieved 2019-12-22.
14. ^ Lewis, Tanya (2018-09-28). "Human Brain: Facts, Functions & Anatomy". livescience.com. Retrieved 2019-12-22.