The sum of two or more quantities is the result of combining them by the operation called addition. In elementary arithmetic, adding numbers amounts to combining their magnitudes to produce a single value: for example, the sum of 3 and 5 is 8. More generally, a sum can refer to the total of a list of terms, whether they are numbers, vectors, functions, or other objects for which an addition operation is defined. The basic idea—accumulating contributions—remains the same across contexts.
Notation and basic properties
Sums are written in many ways. A simple finite list is written with plus signs, as in 1 + 2 + 3. A compact form uses the sigma symbol (∑) to denote summation of a sequence of terms: ∑_{k=1}^n a_k. The operation of addition satisfies several familiar properties that govern sums:
- Commutativity: a + b = b + a.
- Associativity: (a + b) + c = a + (b + c), which allows grouping without changing the result.
- Identity element: 0 is the additive identity because a + 0 = a.
- Linearity: sums distribute over scalar multiplication and are linear in each argument.
Types of sums
Finite sums add a fixed number of terms and are basic to arithmetic and algebra. Infinite sums, or series, are limits of partial sums; whether such a series has a well-defined sum depends on convergence. Some series converge absolutely, others conditionally, and some diverge. Special patterns of finite sums include arithmetic and geometric series, which have closed-form expressions useful in algebra and finance.
Summation appears in many mathematical constructions: Riemann sums approximate integrals as limits of finite sums; expected value in probability is a weighted sum of outcomes; and discrete convolutions combine sequences by summing products. Algorithms exploit sums too — prefix sums compute running totals efficiently, and telescoping sums simplify expressions by cancellation.
Historically, adding quantities is one of the oldest mathematical operations, with systematic notation and techniques developing over centuries. Modern compact notation like ∑ became standard as mathematics formalized sequences and series. For practical introductions to addition and summation, see elementary texts and references on arithmetic and series; for more advanced treatment consult sources on analysis, algebra, or discrete algorithms. For a basic overview of the operation of addition, start with introductory material and examples.