Addition is the mathematical process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples, since 3 + 2 = 5. Besides counts of fruit, addition can also represent combining other physical and abstract quantities using different kinds of numbers: negative numbers, fractions, irrational numbers, vectors, and more.

As a mathematical operation, addition follows several important patterns. It is commutative, meaning that order does not matter, and it is associative, meaning that one can add more than two numbers (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. All of these rules can be proven, starting with the addition of natural numbers and generalizing up through the real numbers and beyond. General binary operations that continue these patterns are studied in abstract algebra.

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 animals. In primary education, children learn 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,

$1\; +\; 1\; =\; 2$ (verbally, "one plus one equals two")

$2\; +\; 2\; =\; 4$ (verbally, "two plus two equals four")

$5\; +\; 4\; +\; 2\; =\; 11$ (see "associativity" below)

$3\; +\; 3\; +\; 3\; +\; 3\; =\; 12$ (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 numbers or the objects to be added are generally 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 symmetry of addition, "augend" is rarely used, and both terms are generally called addends.Schwartzman p.19