Editing Addition (section)

== Definition and interpretations ==
Addition is one of the four basic [[Operation (mathematics)|operations]] of [[arithmetic]], with the other three being [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]]. This operation works by adding two or more terms.{{sfnp|Lewis|1974|p=1}} An arbitrary of many operation of additions is called the [[summation]].{{sfnp|Martin|2003|p=49}} An infinite summation is a delicate procedure known as a [[series (mathematics)|series]],{{sfnp|Stewart|1999|p=8}} and it can be expressed through [[capital sigma notation]] <math display="inline"> \sum </math>, which compactly denotes [[iteration]] of the operation of addition based on the given indexes.{{sfnp|Apostol|1967|p=37}} For example,
<math display="block">\sum_{k=1}^5 k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55.</math>

Addition is used to model many physical processes. Even for the simple case of adding [[natural number]]s, there are many possible interpretations and even more visual representations.

=== Combining sets ===
[[File:AdditionShapes.svg|right|thumb|upright=0.8|One set has three shapes while the other set has two. The total of shapes is five, which is a consequence of the addition of the objects from the two sets: <math> 3 + 2 = 5 </math>.]]
Possibly the most basic interpretation of addition lies in combining [[Set (mathematics)|sets]], that is:{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87 87]}}
{{blockquote|When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections.}}
This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires, see {{Section link||Natural numbers}} below). However, it is not obvious how one should extend this version of an addition's operation to include fractional or negative numbers.<ref>See {{harvtxt|Viro|2001}} for an example of the sophistication involved in adding with sets of "fractional cardinality".</ref>

One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.{{sfnp|National Research Council|2001|p=[http://books.google.com/books?id=pvI7uDPo0-YC&pg=PA74 74]}}
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=== Extending a length ===
{{multiple image
 | image1 = AdditionLineAlgebraic.svg
 | caption1 = A number-line visualization of the algebraic addition <math> 2 + 4 = 6 </math>. A "jump" that has a distance of <math> 2 </math> followed by another that is as long as <math> 4 </math>, is the same as a translation by <math> 6 </math>.
 | image2 = AdditionLineUnary.svg
 | caption2 = A number-line visualization of the unary addition <math> 2 + 4 = 6 </math>. A translation by <math> 4 </math> is equivalent to four translations by <math> 1 </math>.
 | direction = vertical
 | total_width = 400
}}
A second interpretation of addition comes from extending an initial length by a given length:{{sfnp|Mosley|2001|p=[http://books.google.com/books?id=I-__WcWjemUC&pg=PA8 8]}}
{{blockquote|When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension.}}
The sum <math> a + b </math> can be interpreted as a [[binary operation]] that combines <math> a </math> and <math> b </math> algebraically, or it can be interpreted as the addition of <math> b </math> more units to <math> a </math>. Under the latter interpretation, the parts of a sum <math> a + b </math> play asymmetric roles, and the operation <math> a + b </math> is viewed as applying the [[unary operation]] <math> +b </math> to <math> a </math>.{{sfnp|Li|Lappan|2014|p=204}} Instead of calling both <math> a </math> and <math> b </math> addends, it is more appropriate to call <math> a </math> the "augend" in this case, since <math> a </math> plays a passive role. The unary view is also useful when discussing [[subtraction]], because each unary addition operation has an inverse unary subtraction operation, and vice versa.
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