In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. By convention, the value of any empty sum of numbers is zero.
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Let a1, a2, a3,... be a sequence of numbers, and let
be the sum of the first m terms of the sequence. Then
for all m = 1,2,... provided that we use the following conventions:
For summations defined in terms of addition of other values than numbers (such as vectors, matrices, polynomials), in general of values in some given abelian group, the value of an empty summation is taken to be the zero element of that group.
Relevance of defining empty sums
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
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.