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Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together; making a total of 5 apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" i.e., "3 ''add'' 2 is equal to 5". Besides counting fruits, addition can also represent combining other physical objects. Using systematic generalizations, addition can also be defined on more abstract quantities, such as integers, rational numbers, real numbers and complex numbers and other abstract objects such as vectors and matrices. In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others. In algebra, addition is studied more abstractly. Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see ''Summation''). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication. Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some non-human animals. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day. ==Notation and terminology== Addition is written using the plus sign "+" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example, : ("one plus one equals two") : ("two plus two equals four") : ("three plus three equals six") : (see "associativity" below) : (see "multiplication" below) There are also situations where addition is "understood" even though no symbol appears: *A column of numbers, with the last number in the column underlined, usually indicates that the numbers in the column are to be added, with the sum written below the underlined number. *A whole number followed immediately by a fraction indicates the sum of the two, called a ''mixed number''.〔Devine et al. p.263〕 For example, 3½ = 3 + ½ = 3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead.〔Mazur, Joseph. ''Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers''. Princeton University Press, 2014. p. 161〕 The sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example, : The numbers or the objects to be added in general addition are collectively referred to as the terms,〔Department of the Army (1961) Army Technical Manual TM 11-684: Principles and Applications of Mathematics for Communications-Electronics. Section 5.1〕 the addends〔Shmerko, V. P., Yanushkevich, S. N., & Lyshevski, S. E. (2009). Computer arithmetics for nanoelectronics. CRC Press. p.80〕 or the summands;〔Hosch, W. L. (Ed.). (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. p.38〕 this terminology carries over to the summation of multiple terms. This is to be distinguished from ''factors'', which are multiplied. Some authors call the first addend the ''augend''.〔 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.〔Schwartzman p.19〕 All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb ''addere'', which is in turn a compound of ''ad'' "to" and ''dare'' "to give", from the Proto-Indo-European root "to give"; thus to ''add'' is to ''give to''.〔 Using the gerundive suffix ''-nd'' results in "addend", "thing to be added".〔"Addend" is not a Latin word; in Latin it must be further conjugated, as in ''numerus addendus'' "the number to be added".〕 Likewise from ''augere'' "to increase", one gets "augend", "thing to be increased". "Sum" and "summand" derive from the Latin noun ''summa'' "the highest, the top" and associated verb ''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 Greeks and Romans to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.〔Schwartzman (p.212) attributes adding upwards to the Greeks and Romans, saying it was about as common as adding downwards. On the other hand, Karpinski (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.〕 ''Addere'' and ''summare'' date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus; Boethius also used several other terms for the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer.〔Karpinski pp.150–153〕 The plus sign "+" (Unicode:U+002B; ASCII: +) is an abbreviation of the Latin word ''et'', meaning "and". It appears in mathematical works dating back to at least 1489.抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「addition」の詳細全文を読む スポンサード リンク
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