Addition - Notation and Terminology

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,

(verbally, "one plus one equals two")

(verbally, "two plus two equals four")

(verbally, "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. 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 called 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 this 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 once common 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.