Editing Addition (section)

=== Integers ===
{{Further|Integer}}
The simplest conception of an integer is that it consists of an [[absolute value]] (which is a natural number) and a [[sign (mathematics)|sign]] (generally either [[positive number|positive]] or [[negative numbers|negative]]). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:{{sfnmp
 | 1a1 = Smith | 1y = 1980 | 1p = 234
 | 2a1 = Sparks | 2a2 = Rees | 2y = 1979 | 2p = 66
}}
{{blockquote|For an integer <math> n </math>, let <math> |n| </math> be its absolute value. Let <math> a </math> and <math> b </math> be integers. If either <math> a </math> or <math> b </math> is zero, treat it as an identity. If <math> a </math> and <math> b </math> are both positive, define <math> a + b = |a| + |b| </math>. If <math> a </math> and <math> b </math> are both negative, define <math> a + b = -(|a| + |b|) </math>. If <math> a </math> and <math> b </math> have different signs, define <math> a + b </math> to be the difference between <math> |a| + |b| </math>, with the sign of the term whose absolute value is larger.
}}
As an example, {{nowrap|1=−6 + 4 = −2}}; because −6 and 4 have different signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative.

Although this definition can be useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. So the following method is commonly used for defining integers. It is based on the remark that every integer is the difference of two natural integers and that two such differences, <math> a - b </math> and <math> c - d </math> are equal if and only if <math> a + d = b + c </math>. So, one can define formally the integers as the [[equivalence class]]es of [[ordered pair]]s of natural numbers under the [[equivalence relation]] <math> (a,b) \sim (c,d) </math> if and only if <math> a + d = b + c </math>.{{sfnp|Campbell|1970|p=[https://archive.org/details/structureofarith00camp/page/83 83]}} The equivalence class of <math> (a,b) </math> contains either <math> (a-b,0) </math> if <math> a \ge b </math>, or <math> (0,b-a) </math> if otherwise. Given that <math> n </math> is a natural number, then one can denote <math> +n </math> the equivalence class of <math> (n,0) </math>, and by <math> -n </math> the equivalence class of <math> (0,n) </math>. This allows identifying the natural number <math> n </math> with the equivalence class <math> +n </math>.

The addition of ordered pairs is done component-wise:{{sfnp|Campbell|1970|p=[https://archive.org/details/structureofarith00camp/page/84 84]}}
<math display="block"> (a,b) + (c,d) = (a+c, b+d).</math>
A straightforward computation shows that the equivalence class of the result depends only on the equivalence classes of the summands, and thus that this defines an addition of equivalence classes, that is, integers.{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA92 92]}} Another straightforward computation shows that this addition is the same as the above case definition.

This way of defining integers as equivalence classes of pairs of natural numbers can be used to embed into a [[group (mathematics)|group]] any commutative [[semigroup]] with [[cancellation property]]. Here, the semigroup is formed by the natural numbers, and the group is the additive group of integers. The rational numbers are constructed similarly, by taking as a semigroup the nonzero integers with multiplication.

This construction has also been generalized under the name of [[Grothendieck group]] to the case of any commutative semigroup. Without the cancellation property, the [[semigroup homomorphism]] from the semigroup into the group may be non-injective. Originally, the Grothendieck group was the result of this construction applied to the equivalence classes under isomorphisms of the objects of an [[abelian category]], with the [[direct sum]] as semigroup operation.