In mathematics, an empty sum, or nullary sum, is a summation involving no terms at all. The value of any empty sum of numbers is conventionally taken to be zero. 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.
An empty sum can arise in particular for expressions of the form
when ; in such case the summation has value 0 (or the zero element of the additive group in which the terms ti live).
A situation similar to an empty summation can arise for other operations than addition; notably, its counterpart for multiplication is an empty product, a product of no factors at all. In such cases the convention of a zero result does not apply; indeed the value of an empty product of numbers is taken to be one, the neutral element for multiplication.
Read more about Empty Sum: Summation Convention, Relevance of Defining Empty Sums, Significance of "terms" of An Empty Sum
Famous quotes containing the words empty and/or sum:
“Each say following another, either hastening or putting off our death—what pleasure does it bring? I count that man worthless who is cheered by empty hopes. No, a noble man must either live or die well.”
—Sophocles (497–406/5 B.C.)
“but Overall is beyond me: is the sum of these events
I cannot draw, the ledger I cannot keep, the accounting
beyond the account:”
—Archie Randolph Ammons (b. 1926)