In
mathematics, summation is the
addition of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of any kind of
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well:
functions,
vectors,
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
,
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s and, in general, elements of any type of
mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
s on which an
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
denoted "+" is defined.
Summations of
infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
s are called
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
. They involve the concept of
limit, and are not considered in this article.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted , and results in 9, that is, . Because addition is
associative and
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.
Very often, the elements of a sequence are defined, through a regular pattern, as a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s may be written as . Otherwise, summation is denoted by using
Σ notation, where
is an enlarged capital
Greek letter
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...
sigma. For example, the sum of the first natural numbers can be denoted as
For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find
closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
s for the result. For example,
:
Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.
Notation
Capital-sigma notation
![Greek uc sigma](https://upload.wikimedia.org/wikipedia/commons/e/e7/Greek_uc_sigma.svg)
Mathematical notation uses a symbol that compactly represents summation of many similar terms: the ''summation symbol'',
, an enlarged form of the upright capital Greek letter
sigma. This is defined as
:
where is the index of summation; is an indexed variable representing each term of the sum; is the lower bound of summation, and is the upper bound of summation. The "" under the summation symbol means that the index starts out equal to . The index, , is incremented by one for each successive term, stopping when .
This is read as "sum of , from to ".
Here is an example showing the summation of squares:
:
In general, while any variable can be used as the index of summation (provided that no ambiguity is incurred), some of the most common ones include letters such as
,
,
, and
; the latter is also often used for the upper bound of a summation.
Alternatively, index and bounds of summation are sometimes omitted from the definition of summation if the context is sufficiently clear. This applies particularly when the index runs from 1 to ''n''. For example, one might write that:
:
Generalizations of this notation are often used, in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example:
:
is an alternative notation for
the sum of
over all (
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s)
in the specified range. Similarly,
:
is the sum of
over all elements
in the set
, and
:
is the sum of
over all positive integers
dividing .
There are also ways to generalize the use of many sigma signs. For example,
:
is the same as
:
A similar notation is used for the
product of a sequence
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additio ...
, where
, an enlarged form of the Greek capital letter
pi, is used instead of
Special cases
It is possible to sum fewer than 2 numbers:
* If the summation has one summand
, then the evaluated sum is
.
* If the summation has no summands, then the evaluated sum is
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
, because zero is the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
for addition. This is known as the ''
empty sum
In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero.
The natural way to extend non-empty sums is to let the empty sum be the additive identity.
Let a_1, a_2, a_3, ... be a sequence of numbers, and let
...
''.
These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case.
For example, if
in the definition above, then there is only one term in the sum; if
, then there is none.
Formal definition
Summation may be defined recursively as follows:
:
, for ''b'' < ''a'';
:
:
, for ''b'' ≥ ''a''.
Measure theory notation
In the notation of
measure and
integration theory, a sum can be expressed as a
definite integral,
:
where