page_title: Addition
section_title: Notation and terminology
hierarchical_section_title: Addition / Notation and terminology
context_page_description: Addition is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent picture shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5".
  Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.
  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. Repeated addition of 1 is the same as counting; addition of 0 does not change a number.
context_section_description: 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")
   ("one plus two equals three")
   (see "associativity" below)
   (see "multiplication" below)
  There are also situations where addition is "understood", even though no symbol appears:
  A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number. For example,
        3½ = 3 + ½ = 3.5.
  This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead.
  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, the addends or the summands;
  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.
  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 *deh₃- "to give"; thus to add is to give to. Using the gerundive suffix -nd results in "addend", "thing 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.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.
  The plus sign "+" (Unicode:U+002B; ASCII: &#43;) is an abbreviation of the Latin word et, meaning "and". It appears in mathematical works dating back to at least 1489.
caption: Columnar addition – the numbers in the column are to be added, with the sum written below the underlined number.