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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]
Modular addition, defined in this way for the integers from to , forms a group, denoted as or (/, +) , with as the identity element and as the inverse element of . A familiar example is addition of hours on the face of a clock , where 12 rather than 0 is chosen as the representative of the identity.
The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.
Under addition, a ring is an abelian group, which means that addition is commutative and associative; it has an identity, called the additive identity, and denoted 0; and every element x has an inverse, called its additive inverse and denoted −x. Because of commutativity, the concepts of left and right inverses are meaningless since they do ...
Addition. Summation – Answer after adding a sequence of numbers; Additive inverse; Subtraction – Taking away numbers; Multiplication – Repeated addition Multiple – Product of multiplication Least common multiple; Multiplicative inverse; Division – Repeated subtraction Modulo – The remainder of division; Quotient – Result of division
Commutativity of addition and multiplication: a + b = b + a, and a ⋅ b = b ⋅ a. Additive and multiplicative identity: there exist two distinct elements 0 and 1 in F such that a + 0 = a and a ⋅ 1 = a. Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.
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Subtraction is the inverse of addition. [1] The function whose value for any real or complex argument is the additive inverse of that argument. For example, if x = 3, then −x = −3, but if x = −3, then −x = +3. Similarly, −(−x) = x. A prefix of a numeric constant. When it is placed immediately before an unsigned number, the ...
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