Search results
Results from the WOW.Com Content Network
The definition requires closure, that the additive element be found in . This is why despite addition being defined over the natural numbers, it does not an additive inverse for its members. The associated inverses would be negative numbers, which is why the integers do have an additive inverse.
The negations or additive inverses of the positive natural numbers are referred to as negative integers. [2] The set of all integers is often denoted by the boldface Z or blackboard bold Z {\displaystyle \mathbb {Z} } .
In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x.One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.
An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). [3] These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary.
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 axioms of modules imply that (−1)x = −x, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
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.
This includes the existence of an additive inverse −a for all elements a and of a multiplicative inverse b −1 for every nonzero element b. This allows the definition of the so-called inverse operations, subtraction a − b and division a / b, as a − b = a + (−b) and a / b = a ⋅ b −1. Often the product a ⋅ b is represented by ...