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An extract on #2

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 first 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, the first Ramanujan prime, and the first Smarandache-Wellin prime. Two is the third Fibonacci number. 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 = x2 multiplication to exponentiation xx = x2 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 = 22 = 22, 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 k = 0 1 2 k = 1 + 1 2 + 1 4 + 1 8 + 1 16 + = 2. {\displaystyle \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: k = 0 1 n k = 1 + 1 n 1 for all n R > 1. {\displaystyle \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 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 also has the unique property such that k = 0 n 1 2 k = 2 n 1 {\displaystyle \sum _{k=0}^{n-1}2^{k}=2^{n}-1} and also k = a n 1 2 k = 2 n k = 0 a 1 2 k 1 {\displaystyle \sum _{k=a}^{n-1}2^{k}=2^{n}-\sum _{k=0}^{a-1}2^{k}-1} for a not equal to zero 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 = V E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces.