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Addition is one of the basic operations of arithmetic.

In its simplest form, addition combines two numbers (terms, summands), the augend and addend, into a single number, the sum. Adding more numbers corresponds to repeated addition. By extension, addition of zero, one or infinitely many numbers can be defined, see below.

For a definition of addition in the natural numbers, see Addition in N.

1 Important properties

When adding finitely many numbers, it doesn't matter how you group the numbers and in which order you add them; you will always get the same result. (See Associativity and Commutativity.) If you add zero to any number, the quantity won't change; zero is the identity element for addition. The sum of any number and its additive inverse (in contexts where such a thing exists) is zero.

2 Notation

If the terms are all written out individually, then addition is written using the plus sign ("+"). Thus, the sum of 1, 2, and 4 is 1 + 2 + 4 = 7. If the terms are not written out individually, then the sum may be written with an ellipsis to mark out the missing terms. Thus, the sum of all the natural numbers from 1 to 100 is 1 + 2 + ... + 99 + 100.

Alternatively, the sum can be represented by the summation symbol, which is the capital SigmaSigma may refer to many things: Sigma (upper case Σ, lower case σ, alternative ς) is a letter in the Greek alphabet. See Sigma (letter). The upper-case letter Σ is used as a symbol for the summation operator forming a series in ma. This is defined as:

The subscript gives the symbol for a dummy variableIn computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. In mathematics, a variable often represents an unknown quantity; in computer science, it represents a place where a quantity can be stored. Variabl, i. Here, i represents the index of summation; m is the lower bound of summation, and n is the upper bound of summation. So, for example:

One may also consider sums of infinitely many terms; these are called infinite series. Notationally, we would replace n above by the infinityInfinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. In theology, for instance in the work of Duns Scotus, the infinity of God carries the sense not so much of quantity (leading to the question symbol (∞). The sum of such a series is defined as the limitIn mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets "close" to either some point, or infinity; or the behavior of a sequence's elements, as their index approaches infinity. Limits are used in calculu of the sum of the first n terms, as n grows without bound. That is:

One can similarly replace m with negative infinity, and

for some integer m, provided both limits exist.

3 Relationships to other operations and constants

It's possible to add fewer than 2 numbers:

If you add the single term x, then the sum is x.
If you add zero terms, then the sum is zero, because zero is the identity for addition. This is known as the empty sumIn mathematics, the empty sum or nullary sum is the result of adding no numbers. Its numerical value is zero. This fact is especially useful in discrete mathematics and algebra. A simple, well known case is that 0a 0 multiplication of any number a by zero.

These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if m = n in the definition above, then there is only one term in the sum; if m = n + 1, then there is none.

Many other operations can be thought of as generalised sums. If a single term x appears in a sum n times, then the sum is nx, the result of a multiplicationArithmetic In its simplest form, multiplication is a quick way of adding identical numbers. The result of multiplying numbers is called a product''. The numbers being multiplied are called coefficients or factors and individually as the multiplicand and m. If n is not a natural number, then the multiplication may still make sense, so that we have a sort of notion of adding a term, say, two and a half times.

A special case is multiplication by -1, which leads to the concept of the additive inverse, and to subtraction, the inverse operation to addition.

The most general version of these ideas is the linear combination, where any number of terms are included in the generalised sum any number of times.

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