Empty sum

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In mathematics, an empty sum, or nullary sum,[citation needed] is a summation where the number of terms is zero. By convention,[1] the value of any empty sum of numbers is the additive identity.

If a1, a2, a3,... is a sequence of numbers, and

is the sum of the first m terms of the sequence, then

for all m = 1,2,... provided that the following conventions are used: and . In other words, a "sum" with only one term evaluates to that one term, while a "sum" 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 convention" is standard practice in mathematics and computer programming.

For the same reason, the empty product is taken to be the multiplicative identity.

For summations defined in terms of addition of other values than numbers (such as vectors, matrices, polynomials), in general of values in some given monoid, the value of an empty summation is taken to be its identity element.

Relevance of defining empty sums[edit]

The notion of an empty sum is useful for the same reason that the number zero and the empty set are useful: while they seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentation of many subjects.

An example: empty linear combinations[edit]

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. Because of the empty sum convention, the zero-dimensional vector space V={0} has a basis, namely the empty set.

See also[edit]


  1. ^ David M. Bloom (1979). Linear Algebra and Geometry. p. 45. ISBN 0521293243.