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Empty sum

Summation where the number of terms is zero

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated April 14, 2025

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

s_{m}=\sum _{i=1}^{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_{m-1}+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). For the same reason, the empty product is taken to be the multiplicative identity.

For sums of other objects (such as vectors, matrices, polynomials), the value of an empty summation is taken to be its additive identity.

01Examples

Empty linear combinations

In linear algebra, 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={0} to have a basis, namely the empty set.

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Sources and credits

This article is adapted from the Wikipedia article “Empty sum”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

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, just as the empty sum, the result of adding no numbers, is by convention equal to the additive identity (zero).

Iterated binary operation

In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the multiplication operation to the product operation.

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In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity.