Empty sum

In mathematics, the empty sum is the special case of a sum with zero summands . The empty sum is assigned the value zero , the neutral element of addition . The counterpart of the empty sum for multiplication is the empty product .

definition

A sum of numbers is called empty if the set of numbers to be summed over is the empty set . The result of the empty sum is defined as the number zero . In the sums notation this means

${\displaystyle \sum _{i\in I}a_{i}=0}$,

when the index set is. In particular, an empty sum is obtained if the start index of a finite sum is greater than the end index . So it applies ${\displaystyle I=\emptyset }$ ${\displaystyle m}$ ${\displaystyle n}$

${\displaystyle \sum _{i=m}^{n}a_{i}=0}$,

whenever is, because the index set is then empty. Since there is exactly one possibility not to add anything, one also speaks of the empty sum. ${\displaystyle m>n}$ ${\displaystyle I=\{j\in \mathbb {Z} \mid m\leq j\leq n\}}$

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 elements of such an algebraic structure then gives the neutral element of the structure with regard to the addition. For example, the empty sum of vectors of a vector space gives the zero vector , that is ${\displaystyle v_{i}}$ ${\displaystyle V}$ ${\displaystyle 0_{V}}$

${\displaystyle \sum _{i\in \emptyset }v_{i}=0_{V}}$,

because the zero vector represents the neutral element in terms of vector addition . ${\displaystyle V}$

literature

• Harro Heuser : Textbook of Analysis - Part 1 . Springer, 2009, ISBN 3-8348-0777-X .
• Jörg Liesen, Volker Mehrmann : Lineare Algebra: A textbook on theory with a view to practice . Gabler, 2012, ISBN 978-3-8348-0081-7 .
• Oliver Deiser, Caroline Lasser, Elmar Vogt, Dirk Werner : 12 × 12 key concepts for mathematics . Springer, 2011, ISBN 3-8274-2298-1 .

Web links

Wikibooks: Math for Non-Freaks: Empty Sum  - Learning and Teaching Materials

• pahio: empty sum . In: PlanetMath . (English)

Individual evidence

1. ^ Heuser: Textbook of Analysis - Part 1 . Springer, 2009, p. 90 f .
2. ↑ Liesen, Mehrmann: Lineare Algebra: A textbook on theory with a view to practice . Gabler, 2012, p. 924 .
3. ↑ Deiser, Lasser, Vogt, Werner: 12 × 12 key concepts for mathematics . Springer, 2011, p. 114 .