# 2

2 (two) is a number, numeral, and glyph. 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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Cardinaltwo
Ordinal2nd (second / twoth)
Numeral systembinary
Factorizationprime
Gaussian integer factorization$(1+i)(1-i)$ Prime1st
Divisors1, 2
Greek numeralΒ´
Roman numeralII
Roman numeral (unicode)Ⅱ, ⅱ
Greek prefixdi-
Latin prefixduo- bi-
Old English prefixtwi-
Binary102
Ternary23
Quaternary24
Quinary25
Senary26
Octal28
Duodecimal212
Hexadecimal216
Vigesimal220
Base 36236
Greek numeralβ'
Arabic & Kurdish٢
Urdu
Ge'ez
Bengali
Chinese numeral二，弍，貳
Devanāgarī
Telugu
Tamil
Hebrewב
Japanese numeral二/弐
Khmer
Korean이，둘
Thai

## 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"). 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.

Two is the third (or fourth) Fibonacci number.

Two is the base of the binary system, the numeral system with the fewest tokens allowing to denote a natural number n substantially more concise (log2 n tokens), compared to a direct representation by the corresponding count of a single token (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 powers of x equals itself. In symbols

$\sum _{k=0}^{\infty }{\frac {1}{2^{k}}}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =2.$ This comes from the fact that:

$\sum _{k=0}^{\infty }{\frac {1}{n^{k}}}=1+{\frac {1}{n-1}}\quad {\mbox{for all}}\quad n\in \mathbb {R} >1.$ 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 exponent 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 also has the unique property such that

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

$\sum _{k=a}^{n-1}2^{k}=2^{n}-\sum _{k=0}^{a-1}2^{k}-1$ for a not equal to zero

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

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.

### 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 1000
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 2000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
x ÷ 2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5
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 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400

## Evolution of the glyph

The glyph 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 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 glyph 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 glyph.

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

## In religion

### Judaism

The number 2 is important in Judaism, with one of the earliest references being that God ordered Noah to put two of every unclean animal (Gen. 7:2) in his ark (see Noah's Ark). Later on, the Ten Commandments were given in the form of two tablets. The number also has ceremonial importance, such as the two candles that are traditionally kindled to usher in the Shabbat, recalling the two different ways Shabbat is referred to in the two times the Ten Commandments are recorded in the Torah. These two expressions are known in Hebrew as שמור וזכור ("guard" and "remember"), as in "Guard the Shabbat day to sanctify it" (Deut. 5:12) and "Remember the Shabbat day to sanctify it" (Ex. 20:8). Two challot (lechem mishneh) are placed on the table for each Shabbat meal and a blessing made over them, to commemorate the double portion of manna which fell in the desert every Friday to cover that day's meals and the Shabbat meals.

In Jewish law, the testimonies of two witnesses are required to verify and validate events, such as marriage, divorce, and a crime that warrants capital punishment.

"Second-Day Yom Tov" (Yom Tov Sheini Shebegaliyot) is a rabbinical enactment that mandates a two-day celebration for each of the one-day Jewish festivals (i.e., the first and seventh day of Passover, the day of Shavuot, the first day of Sukkot, and the day of Shemini Atzeret) outside the Land of Israel.

## Numerological significance

The most common philosophical dichotomy is perhaps the one of good and evil, but there are many others. See dualism for an overview. In Hegelian dialectic, the process of synthesis reconciles two different perspectives into one.

The ancient Sanskrit language of India, does not only have a singular and plural form for nouns, as do many other languages, but instead has, a singular (1) form, a dual (2) form, and a plural (everything above 2) form, for all nouns, due to the significance of 2. It is viewed as important because of the anatomical significance of 2 (2 hands, 2 nostrils, 2 eyes, 2 legs, etc.)

Two (, èr) is a good number in Chinese culture. There is a Chinese saying, "good things come in pairs". It is common to use double symbols in product brand names, e.g. double happiness, double coin, double elephants etc. Cantonese people like the number two because it sounds the same as the word "easy" () in Cantonese.

In Finland, two candles are lit on Independence Day and put on a windowsill, to remind passersby of the sacrifices of past generations in the struggle for independence and democracy.

In pre-1972 Indonesian and Malay orthography, 2 was shorthand for the reduplication that forms plurals: orang "person", orang-orang or orang2 "people".

In Astrology, Taurus is the second sign of the Zodiac.

## In sports

• In baseball scorekeeping, 2 is the position of the catcher.
• In basketball:
• A standard basket, known in the rules as a "field goal", is worth 2 points.
• In the 3x3 variant, successful shots from behind the "three-point" arc are instead worth 2 points (all other successful shots are worth 1 point).
• In play diagrams, "2" typically denotes the shooting guard.
• In ice hockey:
• A team typically has two defencemen on the ice at any given time.
• Minor penalties last for 2 minutes or until the non-penalized team scores a goal, whichever comes first.
• In most rugby league competitions (though not the Super League, which uses static squad numbering), the starting right wing wears number 2.
• In rugby union and its sevens variant, the starting hooker wears number 2.
• In association football, a player scoring two goals in one match is said to have recorded a brace.

## See also

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