| Additive Inverse |
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| CATEGORIES ABOUT ADDITIVE INVERSE | |
| abstract algebra | |
| arithmetic | |
| elementary algebra | |
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The additive inverse of ''n'' is denoted −''n''. For example:
Thus by the last example, −(−0.3) = 0.3. The additive inverse of a number is its Inverse Element under the Binary Operation of addition. It can be calculated using Multiplication by −1; that is, −''n'' = −1 × ''n''. Types of numbers with additive inverses include:
Types of numbers without additive inverses (of the same type) include:
But note that we can construct the integers out of the natural numbers by formally including additive inverses. Thus we can say that natural numbers ''do'' have additive inverses, but because these additive inverses are not themselves natural numbers, the Set of natural numbers is not ''closed'' under taking additive inverses. GENERAL DEFINITION The notation '+' is reserved for Commutative Binary Operation s, i.e. such that ''x + y = y + x'', for all ''x,y''. If such an operation admits a neutral element ''o'' (such that ''x + o (= o + x) = x'' for all ''x''), then this element is unique (''o' = o' + o = o''). If then, for a given ''x'', there exists '' x' '' such that ''x + x' (= x' + x) = o'', then '' x' '' is called an additive inverse of ''x''. If '+' is associative (''(x+y)+z = x+(y+z)'' for all ''x,y,z''), then an additive inverse is unique ('' x" = x" + o = x" + (x + x') = (x" + x) + x' = o + x' = x' '') and denoted by ''(– x)'', and one can write ''x – y'' instead of ''x + (– y)''. OTHER EXAMPLES All the following examples are in fact Abelian Groups :
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