2 (Two; ) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3.
The number two has many properties in mathematics.〔Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987): 41–44〕 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 〔Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 31〕 (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, 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
: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 level of the hyperoperation 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
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 Jun. 2015, there are only two known Wieferich primes in base 2.
With the exception of the sequence 3, 5, 7, the maximum number of consecutive odd numbers that are prime is two.
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