Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all. 1 is the most common leading digit in many sets of data, a consequence of Benfords law. 1 is the only known Tamagawa number for a simply connected algebraic group over a number field.
In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}. In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields. By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually 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 absolutely or almost certain to occur. In category theory, 1 is sometimes used to denote the terminal object of a category. In number theory, 1 is the value of Legendres constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendres constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.