2007

in arithmetic (algebra) and calculus, the natural number that follows 0 and precedes 2 and the multiplicative identity element of the integers, real numbers and complex numbers; more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring. Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation. Since the base 1 exponential function (1x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist). There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... Formalizations of the natural numbers have their own representations of 1: in the Peano axioms, 1 is the successor of 0; in Principia Msually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different. By definition, 1 is the probability of an event that is almost certain to occur. 1 is the most common leading digit in many sets of data, a consequence of Benford's law. 1 is the only known Tamagawa number for a simply connected algebraic group over a number field. The generating function that has all coefficients 1 is given by {\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots } {\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots } This power series converges and has finite value if and only if, {\displaystyle |x|<1} |x|<1. In category theory, 1 is sometimes used to denote the terminal object of a category. In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899. Table 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 1 × x 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 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ÷ x 1 0.5 0.3 0.25 0.2 0.16 0.142857 0.125 0.1 0.1 0.09 0.083 0.076923 0.0714285 0.06 x ÷ 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1x 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20