Addition ( $+$ )

Addition combines two numbers or elements of an additive algebraic structure into their sum.

Table of contents
- Additive inverse Addition
- Summation Addition
  - Sum Summation
    - Weighted arithmetic mean Sum
      - Arithmetic mean Weighted arithmetic mean
        Harmonic mean Arithmetic mean
    - Prefix sum Sum

Additive inverse ( $- x$ )

The additive inverse

- x

satisfies

x + (- x) = 0

.

Summation ( $\sum$ )

Summation is repeated addition, conventionally written with the symbol

\sum

.

Sum

A sum combines a finite or infinite sequence of terms by repeated addition. The symbol

\sum

denotes summation over an index.

Weighted arithmetic mean ( $\overset{x}{ˉ}_{w}$ )

For positive weights

w_{i}

, the weighted mean is

\overset{x}{ˉ}_{w} = \frac{\sum _{i} w _{i} x _{i}}{\sum _{i} w _{i}} .

It lies between the smallest and largest

x_{i}

; the same result holds for integrals with a positive weight function.

Arithmetic mean ( $\overset{x}{ˉ}$ )

The arithmetic mean of

x_{1}, \dots, x_{n}

is

\overset{x}{ˉ} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} .

It is the weighted mean with equal weights.

Harmonic mean ( $H = n / (\sum_{i} 1/ x_{i})$ )

The harmonic mean of positive numbers

x_{1}, \dots, x_{n}

is

H = n / (\sum_{i} 1/ x_{i})

. The Jensen inequality for

1/ x

gives

H \leq \overline{x}

, with equality exactly when all entries coincide.

Prefix sum ( $P_{m} = \sum_{i = 1}^{m} a_{i}$ )

The prefix sums

P_{m} = \sum_{i = 1}^{m} a_{i}

can all be computed by the recurrence

P_{m} = P_{m - 1} + a_{m}

in linear time. Any contiguous sum is then

P_{b} - P_{a - 1}

, so it takes constant time after preprocessing.

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Wikipedia Bot  1

Addition is a fundamental mathematical operation that involves combining two or more numbers to obtain a total or sum. It is one of the four basic arithmetic operations, alongside subtraction, multiplication, and division. The symbol used for addition is "+". For example, in the expression \(3 + 2\), the numbers 3 and 2 are added together to yield a result of 5.

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Ciro Santilli  40 Updated 2025-07-16

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