2 (number)
Template:Redirect10 Template:Infobox number 2 (Two; Template:IPAc-en) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3.
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In mathematics
The number two has many properties in mathematics.^{[1]} 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 and hexadecimal, 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 first prime number, and the only even prime number ^{[2]} (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, the first Ramanujan prime, and the first Smarandache-Wellin prime. It is an Eisenstein prime with no imaginary part and real part of the form . It is also a Stern prime, a Pell number, the first Fibonacci prime, and a Markov number—appearing in infinitely many solutions to the Markov Diophantine equation involving odd-indexed Pell numbers.
It is the third Fibonacci number, and the third and fifth Perrin numbers.
Despite being prime, two is also a superior highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six.
Vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base.
Two is the base of the simplest numeral system in which natural numbers can be written concisely, being the length of the number a logarithm of the value of the number (whereas in base 1 the length of the number is the value of the number itself); the binary system is used in computers.
For any number x:
- x+x = 2·x addition to multiplication
- x·x = x^{2} multiplication to exponentiation
- x^{x} = x↑↑2 exponentiation to tetration
In general:
- hyper(x,n,x) = hyper(x,n+1,2)
Two also has the unique property that 2+2 = 2·2 = 2²=2↑↑2=2↑↑↑2, and so on, no matter how high the operation is.
Two is the only number x such that the sum of the reciprocals of the powers of x equals itself. In symbols
This comes from the fact that:
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 roots and other such roots, is left blank for square roots, as it is considered tacit.
The square root of 2 was the first known irrational number.
The smallest field has two elements.
In the 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 is a primorial, as well as its own factorial. Two often occurs in numerical sequences, such as the Fibonacci number sequence, but not quite as often as one does. Two is also a Motzkin number, a Bell number, an all-Harshad number, a meandric number, a semi-meandric number, and an open meandric number.
Two is the number of n-Queens Problem solutions for n = 4. With one exception, all known solutions to Znám's problem start with 2.
Two also has the unique property such that
and also
for a not equal to zero
The number of domino tilings of a 2×2 checkerboard is 2.
In n-dimensional space for any n, any 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.
As of 2008, there are only two known Wieferich primes.
With the exception of the sequence 3, 5, 7, the maximum number of consecutive odd numbers that are prime is two.
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 | 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 | 1 | 0.Template:Overline | 0.5 | 0.4 | 0.Template:Overline | 0.Template:Overline | 0.25 | 0.Template:Overline | 0.2 | 0.Template:Overline | 0.1Template:Overline | 0.Template:Overline | 0.Template:Overline | 0.1Template:Overline | |
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 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | 2048 | 4096 | 8192 | 16384 | |
1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 |
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Evolution of the glyph
The glyph used in the modern Western world to represent the number 2 traces its roots back to the Brahmin Indians, who wrote "2" as two horizontal lines. The modern Chinese and Japanese languages still use this method. The Gupta rotated the two lines 45 degrees, making them diagonal, and sometimes also made the top line shorter and made its bottom end curve towards the center of the bottom line. Apparently for speed, the Nagari started making the top line more like a curve and connecting to the bottom line. The Ghubar Arabs made the bottom line completely vertical, and now the glyph looks 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.^{[4]}
In fonts with text figures, 2 usually is of x-height, for example, .
In science
- The number of polynucleotide strands in a DNA double helix.
- The first magic number.
- The atomic number of helium.
- 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.
Astronomy
- Messier object M2, a magnitude 6.5 globular cluster in the constellation Aquarius.
- The New General Catalogue object NGC 2, a magnitude 14.2 spiral galaxy in the constellation Pegasus
- The Saros number of the solar eclipse series which began on May 4, 2861 BC and ended on June 21, 1563 BC . The duration of Saros series 2 was 1298.1 years, and it contained 73 solar eclipses.
- The Saros number of the lunar eclipse series which began on February 21, 2541 BC and ended on April 22, 1225 BC. The duration of Saros series 2 was 1316.2 years, and it contained 74 lunar eclipses.
- The Roman numeral II stands for bright giant in the Yerkes spectral classification scheme.
- 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.
In technology
- The resin identification code used in recycling to identify high-density polyethylene.
In religion
The number 2 is important in Judaism, with one of the earliest reference 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 challahs (lechem mishnah) 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 testimony 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 creates two perspectives from one.
The ancient Sanskrit language (संस्कृत भाषा) of India, does not only have a singular and plural form for nouns, as do most 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 (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 brandnames, e.g. double happiness, double coin, double elephants etc. Cantonese people like the number two because it sounds the same as the word "easy" (易) in Cantonese.
In Finland, two candles are lit on Independence Day. Putting them on the windowsill invokes the symbolical meaning of division, and thus independence. {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
In pre-1972 Indonesian and Malay orthography, 2 was shorthand for the reduplication that forms plurals: orang "person", orang-orang or orang2 "people". {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
In Astrology, Taurus is the second sign of the Zodiac.
In sports
- In baseball scorekeeping, 2 is the position of the catcher.
- In American football, a safety has a two-point value. Also, a two-point conversion is a point after touchdown (PAT) attempt where the ball crosses the goal line via run or pass. (In six-man football, however, the traditional PAT kick is worth two points, whereas a PAT via pass or run is only one point.)
- In Association football, the scoring of two goals by one individual in a single match is referred to as a brace.
- The successor of a brace is the "hat-trick", three goals scored by one player.
- In standard basketball, the value of any made shot taken from inside the three-point arc in normal play is 2 points.
- In the half-court 3x3 variant, made shots taken from outside the "three-point" arc are worth 2 points.
In other fields
Template:Double image Groups of two:
- The name of several fictional characters: Number Two.
- The designation of the Trans-Canada Highway in most of the province of New Brunswick.
- U.S. Route 2, two separated highways in the northern tier of the United States, the western segment connecting Everett, Washington to St. Ignace, Michigan and the eastern route connecting Rouses Point, New York to Houlton, Maine.
- The lowest channel of television in the United States, Canada, Argentina and Mexico on which television signals are broadcast.
- Brace is also used in hunting to refer to a pair. For example, "He shot a brace of pheasants".
- "Two turtle doves" is the gift on the second day of Christmas in the carol "The Twelve Days of Christmas"
In North American educational systems, the number 2.00 denotes a grade-point average of "C", which in some colleges and universities is the minimum required for good academic standing at the undergraduate level.^{[5]}
See also
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- Square (algebra) — (2 superscript)
References
- ↑ Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 41–44
- ↑ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 31
- ↑ 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."
- ↑ 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
- ↑ For a typical example, see the University of Oklahoma grading regulations.