Empty sum

The empty sum is in the mathematics of the special case of a sum of zero summands. The empty sum is zero, then the neutral element of addition, assigned. The counterpart of the sum of the multiplication is empty, the empty product.

Definition

A sum of numbers is empty when the amount of the numbers of the summing, the empty set. The result of the blank amount is defined as the number zero. In sum notation, this means

If the index set is. In particular, it receives an empty amount if greater than the last index for a finite sum of the start index. It is therefore

When is forever, for the index set is then empty. Precisely because there is a possibility to add anything, one also speaks of the empty sum.

Generalizations

Sums are not only defined for numbers, but also in more general algebraic structures, such as vector spaces, bodies, rings or abelian groups. The empty sum of the elements of an algebraic structure then yields the neutral element of the structure with respect to addition. For example, yields the empty sum of vectors of a vector space the zero vector, ie

Because the zero vector is in just the neutral element with respect to the vector addition dar.