Editing Addition (section)

== Addition of numbers ==
To prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined on the [[natural number]]s. In [[set theory]], addition is then extended to progressively larger sets that include the natural numbers: the [[integer]]s, the [[rational number]]s, and the [[real number]]s.<ref>[[Herbert Enderton|Enderton]] chapters 4 and 5, for example, follow this development.</ref> In [[mathematics education]],<ref>According to a survey of the nations with highest TIMSS mathematics test scores; see {{harvtxt|Schmidt|Houang|Cogan|2002}}, p. 4.</ref> positive fractions are added before negative numbers are even considered; this is also the historical route.<ref>{{harvtxt|Baez|Dolan|2001}}, p. 37 explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"</ref>

=== Natural numbers ===
{{Further|Natural number}}
There are two popular ways to define the sum of two natural numbers <math> a </math> and <math> b </math>. If one defines natural numbers to be the [[Cardinal number|cardinalities]] of finite sets (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows:{{sfnmp
 | 1a1 = Begle | 1y = 1975 | 1p = 49
 | 2a1 = Johnson | 2y = 1975 | 2p = 120
 | 3a1 = Devine | 3a2 = Olson | 3a3 = Olson | 3y = 1991 | 3p = 75
}}
{{blockquote|Let <math> N(S) </math> be the cardinality of a set <math> S </math>. Take two disjoint sets <math> A </math> and <math> B </math>, with <math> N(A) = a </math> and <math> N(B) = b </math>. Then <math> a + b </math> is defined as <math> N(A \cup B)</math>.
}}
Here <math> A \cup B </math> means the [[union (set theory)|union]] of <math> A </math> and <math> B </math>. An alternate version of this definition allows <math> A </math> and <math> B </math> to possibly overlap and then takes their [[disjoint union]], a mechanism that allows common elements to be separated out and therefore counted twice.

The other popular definition is recursive:{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA79 79]}}
{{blockquote|Let <math> n^+ </math> be the successor of <math> n </math>, that is the number following <math> n </math> in the natural numbers, so <math> 0^+ = 1 </math>, <math> 1^+ = 2 </math>. Define <math> a + 0 = a </math>. Define the general sum recursively by <math> a + b^+ = (a + b)^+ </math>. Hence <math> 1 + 1 = 1 + 0^+ = (1 + 0)^+ = 1^+ = 2 </math>.
}}
Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the [[Recursion#The recursion theorem|recursion theorem]] on the [[partially ordered set]] <math> \mathbb{N}^2 </math>.<ref>For a version that applies to any poset with the [[descending chain condition]], see {{harvtxt|Bergman|2005}}, p. 100</ref> On the other hand, some sources prefer to use a restricted recursion theorem that applies only to the set of natural numbers. One then considers <math> a </math> to be temporarily "fixed", applies recursion on <math> b </math> to define a function "<math> a + </math>", and pastes these unary operations for all <math> a </math> together to form the full binary operation.<ref>{{harvtxt|Enderton|1977}}, p. [http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA79 79] observes, "But we want one binary operation <math> + </math>, not all these little one-place functions."</ref>

This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, through [[mathematical induction]].{{sfnp|Ferreirós|1999|p=223}}

=== 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.

=== Rational numbers (fractions) ===
{{main article|Field of fractions}}
Addition of [[rational number]]s involves the [[fraction]]s. The computation can be done by using the [[least common denominator]], but a conceptually simpler definition involves only integer addition and multiplication:
<math display="block"> \frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}.</math>
As an example, the sum <math display="inline">\frac 34 + \frac 18 = \frac{3 \times 8+4 \times 1}{4 \times 8} = \frac{24 + 4}{32} = \frac{28}{32} = \frac78</math>.

Addition of fractions is much simpler when the [[denominator]]s are the same; in this case, one can simply add the numerators while leaving the denominator the same:
<math display="block"> \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}, </math>
so <math display="inline">\frac 14 + \frac 24 = \frac{1 + 2}{4} = \frac 34</math>.{{sfnp|Cameron|Craig|2013|p=29}}

The commutativity and associativity of rational addition are easy consequences of the laws of integer arithmetic.<ref>The verifications are carried out in {{harvtxt|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA104 104]}} and sketched for a general field of fractions over a commutative ring in {{harvtxt|Dummit|Foote|1999}}, p. 263.</ref>

=== 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}}

=== Complex numbers ===
[[File:Vector Addition.svg|right|thumb|Addition of two complex numbers can be done geometrically by constructing a parallelogram.]]
Complex numbers are added by adding the real and imaginary parts of the summands.<ref>{{Citation |last=Conway |first=John B. |title=Functions of One Complex Variable I |year=1986 |publisher=Springer |isbn=978-0-387-90328-6}}</ref><ref>{{Citation |last1=Joshi |first1=Kapil D
|title=Foundations of Discrete Mathematics |publisher=[[John Wiley & Sons]] |location=New York |isbn=978-0-470-21152-6|year=1989}}</ref> That is to say:
:<math>(a+bi) + (c+di) = (a+c) + (b+d)i.</math>
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers ''A'' and ''B'', interpreted as points of the complex plane, is the point ''X'' obtained by building a [[parallelogram]] three of whose vertices are ''O'', ''A'' and ''B''. Equivalently, ''X'' is the point such that the [[triangle]]s with vertices ''O'', ''A'', ''B'', and ''X'', ''B'', ''A'', are [[Congruence (geometry)|congruent]].