Search results
Results From The WOW.Com Content Network
In mathematics, the additive inverse of an element x, denoted -x, [1] is the element that when added to x, yields the additive identity, 0 (zero). [2] In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element .
In mathematics, −1 (negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0.
The multiplicative identity 1 and its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if r n = 1, then r n−1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R × is not closed under addition.
The additive identity is unique. The additive inverse of each element is unique. The multiplicative identity is unique. For any element x in a ring R, one has x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x.
The inverse or multiplicative inverse (for avoiding confusion with additive inverses) of a unit x is denoted , or, when the multiplication is commutative, . The additive identity 0 is never a unit, except when the ring is the zero ring , which has 0 as its unique element.
The additive inverse of a number is unique, as is shown by the following proof. As mentioned above, an additive inverse of a number is defined as a value which when added to the number yields zero. Let x be a number and let y be its additive inverse. Suppose y′ is another additive inverse of x.
The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0. [41] There are not only inverse elements but also inverse operations. In an informal sense, one operation is the inverse of another operation if it undoes the ...
However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that under multiplication is not a group. All the rules from the above property table (except for the last), when taken together, say that Z {\displaystyle \mathbb {Z} } together with addition and multiplication is a commutative ring with unity .