Empty sum

Not to be confused with Zero sum.

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.

Let a₁, a₂, a₃,... be a sequence of numbers, and let

$s_m = \sum_{i=1}^m a_i = a_1 + \ldots + a_m$

be the sum of the first m terms of the sequence. Then

$s_m = a_m + s_{m-1}$

for all m = 1,2,... provided that we use the following conventions: $s_1 = a_1$ and $s_0=0$. 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 convention" is standard practice in mathematics and computer programming. For the same reason, the empty product is taken to be one, the neutral element for multiplication.

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.

See also

• Empty product