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In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]
For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of − + 1 / 12 , which is expressed by a famous formula: [2] + + + + =,
In algebra, a unit or invertible element [a] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that = =, where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u.
In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a 2 = a. [1] [a] That is, the element is idempotent under the ring's multiplication.
Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence; Additive inverse, the inverse of a number that, when added to the original number, yields zero; Compositional inverse, a function that "reverses" another function; Inverse element
The Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio at the confidence level is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss, = + + where is the inverse of the normal cumulative distribution function at probability level , , are the ...
For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in + =. Defined more formally, the operation " ⋆ {\displaystyle \star } " is an inverse of the operation " ∘ {\displaystyle \circ } " if it fulfills the following condition: t ⋆ ...
A simple example of the use of this formula is counting the number of reduced fractions 0 < a / b < 1, where a and b are coprime and b ≤ n. If we let f ( n ) be this number, then g ( n ) is the total number of fractions 0 < a / b < 1 with b ≤ n , where a and b are not necessarily coprime.