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{{pp-protected|small=yes}}{{About|the number|the year|AD 2|other uses|2 (disambiguation)|and|II (disambiguation)|and|Number Two (disambiguation){{!}}Number Two}}- the content below is remote from Wikipedia
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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").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." 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.WEB,weblink Sloane's A104272 : Ramanujan primes, The On-Line Encyclopedia of Integer Sequences, OEIS Foundation, 2016-06-01, 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 {{math|n}} substantially more concise ({{math|log2 n }} tokens), compared to a direct representation by the corresponding count of a single token ({{math|n}} tokens). This binary number system is used extensively in computing.For any number x:
x + x = 2 Â· x addition to multiplication
x Â· x = x2 multiplication to exponentiation
xx = xâ†‘â†‘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 {{nowrap|1=2 + 2 = 2 Â· 2 = 2{{sup|2}} = 2â†‘â†‘2 = 2â†‘â†‘â†‘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}^{infin}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}^infin frac {1}{n^k}=1+frac{1}{n-1} quadmbox{for all} quad ninmathbb 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 {{âˆ…},âˆ…}. 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 zeroIn any n-dimensional, euclidean space two distinct points determine a line.For any polyhedron homeomorphic to a sphere, the Euler characteristic is {{nowrap|1=Ï‡ = V âˆ’ E + 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{|class"wikitable" style"text-align: center; background: white"
!width="105px"|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!width="5px"|!50!100!1000|2 Ã— x|24 (number)>46 (number)>68 (number)>810 (number)>1012 (number)>1214 (number)>1416 (number)>1618 (number)>1820 (number)>2022 (number)>2224 (number)>2426 (number)>2628 (number)>2830 (number)>3032 (number)>3234 (number)>3436 (number)>3638 (number)>3840 (number)>4042 (number)>4244 (number)>4446 (number)>4648 (number)>4850 (number)>50!100 (number)>100200 (number)>2002000 (number)>2000{|class="wikitable" style="text-align: center; background: white"!width="105px"|Division!1!2!3!4!5!6!7!8!9!10!11!12!13!14!15|2 Ã· x|21 (number)>16}}one half>0.5|0.43}}285714}}0.25)2}}|0.218}}6}}153846}}142857}}3}}|x Ã· 2|0.51 (number)>1|1.5|2|2.53 (number)>3|3.54 (number)>4|4.55 (number)>5|5.56 (number)>6|6.57 (number)>7|7.5{|class="wikitable" style="text-align: center; background: white"!width="105px"|Exponentiation!1!2!3!4!5!6!7!8!9!10!11!12!13!14!15!16!17!18!19!202{{sup>x}}|24 (number)>48 (number)>816 (number)>1632 (number)>3264 (number)>64128 (number)>128256 (number)>256512 (number)>5121024 (number)>10242048 (number)>20484096 (number)>40968192 (number)>819216384)32768 (number)>3276865536 (number)>65536|131072|262144|524288|1048576x{{sup>2}}1 (number)>14 (number)>49 (number)>916 (number)>1625 (number)>2536 (number)>3649 (number)>4964 (number)>6481 (number)>81100 (number)>100121 (number)>121144 (number)>144169 (number)>169196 (number)>196225 (number)>225256 (number)>256289 (number)>289324 (number)>324361 (number)>361400 (number)>400{{anchor|glyph}}Evolution of the glyph
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- Evolution2glyph.png -
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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.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. 24.62In fonts with text figures, 2 usually is of x-height, for example, Image:Text figures 256.svg.- Evolution2glyph.png -
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In science
25px|right- The number of polynucleotide strands in a DNA double helix.
- The first magic number.
- The atomic number of helium.
- The atomic mass of deuterium, an isotope of hydrogen.
- The ASCII code of "Start of Text".
- Group 2 in the Periodic table of the elements consists of the alkaline earth metals whose usual valence is +2.
- Period 2 in the Periodic table consists of the eight elements lithium through neon.
- 2 Pallas, a large asteroid in the main belt and the second asteroid ever to be discovered.
- The Roman numeral II (usually) stands for the second-discovered satellite of a planet or minor planet (e.g. Pluto II or (87) Sylvia II Remus).
- A binary star is a stellar system consisting of two stars orbiting around their center of mass.
- The number of brain and cerebellar hemispheres.
In technology
50px|right|2 as a resin identification code, used in recycling.- The resin identification code used in recycling to identify high-density polyethylene.
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 ("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
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- 2 playing cards.jpg -
The twos of all four suits in playing cards
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.
- 2 playing cards.jpg -
The twos of all four suits in playing cards
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.
In film
- 2 Fast 2 Furious (2003), starring Paul Walker, Tyrese Gibson, Cole Hauser, and Ludacris.
- Two for the Money (2005), starring Al Pacino, Matthew McConaughey, Rene Russo, Armand Assante, and Carly Pope.
- Two Lovers (2008), starring Joaquin Phoenix, Gwyneth Paltrow, and Vinessa Shaw.
In other fields
{{multiple image| align = right| image1 = ICS Pennant Two.svg| width1 = 100| alt1 = Pennant (commissioning)>International maritime pennant for 2| image2 = ICS Two.svg| width2 = 100| alt2 = | caption2 = International maritime signal flag for 2| footer = }}- AD 2, the second year of the Common Era
- Groups of two:
See also
- Square (algebra) â€“ (2 superscript)
References
{{Reflist}}External links
- weblink" title="web.archive.org/web/20140201161638weblink">The Number 2
- The Positive Integer 2
- Prime curiosities: 2
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