Editing Addition (section)

=== Real numbers ===
{{Further|Construction of the real numbers}}
A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a [[Dedekind cut]] of rationals: a [[non-empty set]] of rationals that is closed downward and has no [[greatest element]]. The sum of real numbers ''a'' and ''b'' is defined element by element:{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA114 114]}}
<math display="block"> a+b = \{q+r \mid q\in a, r\in b\}.</math>
This definition was first published, in a slightly modified form, by [[Richard Dedekind]] in 1872.<ref>{{harvtxt|Ferreirós|1999}}, p. 135; see section 6 of ''[http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html Stetigkeit und irrationale Zahlen] {{webarchive |url=https://web.archive.org/web/20051031071536/http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html |date=2005-10-31 }}''.</ref>
The commutativity and associativity of real addition are immediate; defining the real number 0 as the set of negative rationals, it is easily seen as the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.<ref>The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see {{harvtxt|Enderton|1977}}, p. [http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA117 117] for details.</ref>

[[File:AdditionRealCauchy.svg|right|thumb|Adding <math> \pi^2/6 </math> and <math> e </math> using Cauchy sequences of rationals.]]
Unfortunately, dealing with the multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers.<ref>Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. "Higher Order Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, volume 971 of." ''Lecture Notes in Computer Science'' (1995).</ref> Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of a [[Cauchy sequence]] of rationals, lim&nbsp;''a''<sub>''n''</sub>. Addition is defined term by term:<ref>Textbook constructions are usually not so cavalier with the "lim" symbol; see {{harvtxt|Burrill|1967}}, p. 138 for a more careful, drawn-out development of addition with Cauchy sequences.</ref>
<math display="block">\lim_n a_n + \lim_n b_n = \lim_n (a_n + b_n).</math>
This definition was first published by [[Georg Cantor]], also in 1872, although his formalism was slightly different.{{sfnp|Ferreirós|1999|p=128}}
One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions.{{sfnp|Burrill|1967|p=140}}