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# 2  ( Integers )

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2 ( two) is a , , and . It is the following 1 and preceding 3.

In mathematics
An is called even if it is divisible by 2. For integers written in a numeral system based on an even number, such as , , 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 , and the only even prime number (for this reason it is sometimes called "the oddest prime").John Horton Conway & Richard K. Guy, The Book of Numbers. New York: Springer (1996): 25. . "Two is celebrated as the only even prime, which in some sense makes it the oddest prime of all." The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first , the first , the first , and the first Smarandache–Wellin prime.

Two is the third (or fourth) .

Two is the of the binary system, the with the least number of tokens allowing to denote a natural number substantially more concise ( tokens), compared to a direct representation by the corresponding count of a single token ( tokens). This binary number system is used extensively in .

For any number x:

x + x = 2 · x to
x · x = x2 to
x x = x↑↑2 to

Extending this sequence of operations by introducing the notion of , 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 , 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

$\sum_\left\{k=0\right\}^\left\{\infin\right\}\frac \left\{1\right\}\left\{2^k\right\}=1+\frac\left\{1\right\}\left\{2\right\}+\frac\left\{1\right\}\left\{4\right\}+\frac\left\{1\right\}\left\{8\right\}+\frac\left\{1\right\}\left\{16\right\}+\cdots=2.$

This comes from the fact that:

$\sum_\left\{k=0\right\}^\infin \frac \left\{1\right\}\left\{n^k\right\}=1+\frac\left\{1\right\}\left\{n-1\right\} \quad\mbox\left\{for all\right\} \quad n\in\mathbb R > 1.$

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

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

Two also has the unique property such that

$\sum_\left\{k=0\right\}^\left\{n-1\right\} 2^k = 2^\left\{n\right\} - 1$

and also

$\sum_\left\{k=a\right\}^\left\{n-1\right\} 2^k = 2^n - \sum_\left\{k=0\right\}^\left\{a-1\right\} 2^k - 1$
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 , where V is the number of vertices, E is the number of edges, and F is the number of faces.

List of basic calculations
2 × x2468101214161820222426283032343638404244464850 !1002002000

2 ÷ x210.0.40.0. 0.0.20.0.10.0.0.1
x ÷ 20.511.522.533.544.555.566.577.5

2248163264128256512102420484096819232768655361310722621445242881048576
x149162536496481100121144169196225256289324361400

Evolution of the glyph
The used in the modern Western world to represent the number 2 traces its roots back to the Indic , where "2" was written as two horizontal lines. The modern Chinese and Japanese languages still use this method. The 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 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.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

In fonts with , 2 usually is of , for example, .

In science

In technology
• The resin identification code used in recycling to identify high-density polyethylene.

In religion

Judaism
The number 2 is important in , with one of the earliest references being that God ordered Noah to put two of every (Gen. 7:2) in his ark (see Noah's Ark). Later on, the 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 , recalling the two different ways Shabbat is referred to in the two times the are recorded in the . These two expressions are known in as שמור וזכור ("guard" and "remember"), as in "Guard the Shabbat day to sanctify it" ( 5:12) and "Remember the Shabbat day to sanctify it" (Ex. 20:8). Two ( lechem mishneh) are placed on the table for each meal and a blessing made over them, to commemorate the double portion of which fell in the every Friday to cover that day's meals and the Shabbat meals.

In , 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 (i.e., the first and seventh day of , the day of , the first day of , and the day of ) outside the Land of Israel.

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

The ancient Sanskrit language of India, does not only have a and form for , 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 , 2 legs, etc.)

Two (二, 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. people like the number two because it sounds the same as the word "easy" (易) in Cantonese.

In , two 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 orthography, 2 was shorthand for the that forms plurals: orang "person", orang-orang or orang2 "people".

In , Taurus is the second sign of the .

In sports
• In scorekeeping, 2 is the position of the catcher.
• In :
• 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 .
• In :
• A team typically has two 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 competitions (though not the European , which uses static squad numbering), the starting right wing wears number 2.
• In and , the starting hooker wears number 2.

In other fields
• AD 2, the second year of the
• Groups of two:
• Lists of pairs
• List of twins

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