Editing Addition (section)

== Notation and terminology ==
[[File:PlusCM128.svg|upright=0.4|thumb|The plus sign]]
Addition is written using the [[Plus and minus signs|plus sign]] "+" [[infix notation|between the terms]], and the result is expressed with an [[equals sign]]. For example, <math>1 + 2 = 3</math> reads "one plus two equals three".{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87 87]}} Nonetheless, some situations where addition is "understood", even though no symbol appears: a whole number followed immediately by a [[fraction (mathematics)|fraction]] indicates the sum of the two, called a ''mixed number'', with an example,{{sfnp|Devine|Olson|Olson|1991|p=263}}<math display="block">3\frac{1}{2}=3+\frac{1}{2}=3.5.</math> This notation can cause confusion, since in most other contexts, [[Juxtaposition#Mathematics|juxtaposition]] denotes [[multiplication]] instead.{{sfnp|Mazur|2014|p=161}}

{{anchor|summand|addend}}
[[File:Addition1.png|thumb|upright=1|The terms of addends in the operation of an addition]]
The numbers or the objects to be added in general addition are collectively referred to as the '''terms''',{{sfnp|Department of the Army|1961|loc=[https://archive.org/details/TM11-684/page/16/mode/1up?view=theater Section 5.1]}} the '''addends''' or the '''summands'''.{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87 87]}} This terminology carries over to the summation of multiple terms.
This is to be distinguished from ''factors'', which are [[multiplication|multiplied]].
Some authors call the first addend the ''augend''.{{sfnmp
 | 1a1 = Shmerko | 1a2 = Yanushkevich | 1a3 = Lyshevski | 1y = 2009 | 1p = 80
 | 2a1 = Schmid | 2y = 1974
 | 3a1 = Schmid | 3y = 1983
}} In fact, during the [[Renaissance]], many authors did not consider the first addend an "addend" at all. Today, due to the [[commutative property]] of addition, "augend" is rarely used, and both terms are generally called addends.{{sfnp|Schwartzman|1994|p=19}}

All of the above terminology derives from [[Latin]]. "[[wikt:addition|Addition]]" and "[[wikt:add|add]]" are [[English language|English]] words derived from the Latin [[verb]] {{lang|la|addere}}, which is in turn a [[compound (linguistics)|compound]] of {{lang|la|ad}} "to" and {{lang|la|dare}} "to give", from the [[Proto-Indo-European root]] {{lang|ine-x-proto|deh₃-}} "to give"; thus to ''add'' is to ''give to''.{{sfnp|Schwartzman|1994|p=19}} Using the [[gerundive]] [[Affix|suffix]] ''-nd'' results in "addend", "thing to be added".<ref group=lower-alpha>"Addend" is not a Latin word; in Latin it must be further conjugated, as in {{lang|la|numerus addendus}} "the number to be added".</ref> Likewise from {{lang|la|augere}} "to increase", one gets "augend", "thing to be increased".

[[File:AdditionNombryng.svg|thumb|upright=1|Redrawn illustration from ''The Art of Nombryng'', one of the first English arithmetic texts, in the 15th century.<ref>{{harvtxt|Karpinski|1925}}, pp. 56–57, reproduced on p. 104</ref>]]
"Sum" and "summand" derive from the Latin [[noun]] {{lang|la|summa}} "the highest, the top" and associated verb {{lang|la|summare}}. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the [[Ancient Greece|ancient Greeks]] and [[Ancient Rome|Romans]] to add upward, contrary to the modern practice of adding downward, so that a sum was literally at the top of the addends.<ref>{{harvtxt|Schwartzman|1994}}, p. 212 attributes adding upwards to the [[Ancient Greece|Greeks]] and [[Ancient Rome|Romans]], saying it was about as common as adding downwards. On the other hand, {{harvtxt|Karpinski|1925}}, p. 103 writes that [[Leonard of Pisa]] "introduces the novelty of writing the sum above the addends"; it is unclear whether Karpinski is claiming this as an original invention or simply the introduction of the practice to Europe.</ref>
{{lang|la|Addere}} and {{lang|la|summare}} date back at least to [[Anicius Manlius Severinus Boethius|Boethius]], if not to earlier Roman writers such as [[Vitruvius]] and [[Sextus Julius Frontinus|Frontinus]]; Boethius also used several other terms for the addition operation. The later [[Middle English]] terms "adden" and "adding" were popularized by [[Geoffrey Chaucer|Chaucer]].{{sfnp|Karpinski|1925|pp=150–153}}