Addition

Addition (English: summation, sum) is the result obtained when two or more numbers are combined. The operation of addition is denoted by the plus sign (+), and the numbers being added are called addends or terms. For example, 3 + 5 = 8; here 8 is the sum.

The concept of addition is one of the foundations of arithmetic with whole numbers, but it extends beyond ordinary numbers: fractions, integers, real numbers, complex numbers, vectors, and matrices can be added, provided that addition operations are defined for them.

Properties of addition

Addition of ordinary numbers has two important properties. Commutativity: a + b = b + a, meaning that the order of the addends does not change the sum. Associativity: (a + b) + c = a + (b + c), allowing multiple numbers to be added without requiring parentheses. Zero is the additive identity: a + 0 = a.

The sigma notation

When many numbers with a regular pattern are added together, the Greek capital letter sigma (Σ) is used. Example: Σ (i = 1 to n) of i means 1 + 2 + ... + n. Below the symbol is written the starting index, and above it the ending index. The sum of consecutive integers from 1 to n equals n(n + 1) / 2.

Infinite sums

When the addends are infinite in number, this is called a series. Some series converge to a definite value—for example, 1/2 + 1/4 + 1/8 + ... approaches 1—while others do not have a definite sum. Distinguishing between these two types is done using convergence criteria from mathematical analysis.

Applications

Addition is used in counting (tallying, cumulative totals), finance (budget calculations), and computer programming, where loops accumulate successive values.