mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an empty sum, or nullary sum, is a

summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...

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 In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from element ...

. Let

a_1

a_2

a_3

, ... be a sequence of numbers, and let :

s_m = \sum_^m a_i = a_1 + \cdots + a_m

be the sum of the first ''m'' terms of the sequence. This satisfies the recurrence :

s_m = s_ + a_m

provided that we use the following natural convention:

s_0=0

. In other words, a "sum"

s_1

with only one term evaluates to that one term, while a "sum"

s_0

with no terms evaluates to 0. Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas. Such "sums" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty sum is zero" extension is standard practice in mathematics and computer programming (assuming the domain has a

zero element In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ...

). For the same reason, the

empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...

is taken to be the

multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

. For sums of other objects (such as

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

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 exa ...

s), the value of an empty summation is taken to be its

Examples

Empty linear combinations

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

, a basis of a vector space ''V'' is a linearly independent subset ''B'' such that every element of ''V'' is a linear combination of ''B''. The empty sum convention allows the zero-dimensional vector space ''V''= to have a basis, namely the empty set.

References

{{DEFAULTSORT:Empty Sum Operations on numbers 0 (number)

Examples

Empty linear combinations

See also

References