Editing Empty sum (section)

{{Short description|Summation where the number of terms is zero}}
{{Distinguish|Zero sum}}
{{one source|date=December 2017}}
In [[mathematics]], an '''empty sum''', or '''nullary sum''',<ref>{{Cite book|title=Practical Foundations for Programming Languages|last=Harper|first=Robert|publisher=Cambridge University Press|year=2016|isbn=9781107029576|pages=86}}</ref> is a [[summation]] where the number of terms is zero. 
The natural way to extend non-empty sums<ref>{{cite book |author=David M. Bloom |title=Linear Algebra and Geometry |url=https://archive.org/details/linearalgebrageo0000bloo |url-access=registration |year=1979 |isbn=0521293243 |pages=[https://archive.org/details/linearalgebrageo0000bloo/page/45 45]}}</ref> is to let the empty sum be the [[additive identity]].

Let <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, ... be a sequence of numbers, and let
:<math>s_m = \sum_{i=1}^m a_i = a_1 + \cdots + a_m </math>
be the sum of the first ''m'' terms of the sequence. This satisfies the recurrence
:<math>s_m = s_{m-1} + a_m</math>
provided that we use the following natural convention: <math>s_0=0</math>.
In other words, a "sum" <math>s_1</math> with only one term evaluates to that one term, while a "sum" <math>s_0</math> 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 [[mathematical induction|induction proofs]], as well as in algorithms. For these reasons, the "empty sum is zero" extension is standard practice in mathematics and computer programming (assuming the domain has a [[zero element]]).
For the same reason, the [[empty product]] is taken to be the [[multiplicative identity]].

For sums of other objects (such as [[vector (mathematics)|vector]]s, [[matrix (mathematics)|matrices]], [[polynomial]]s), the value of an empty summation is taken to be its [[additive identity]].