To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

From Wikipedia, the free encyclopedia

In mathematics, an empty sum, or nullary sum,[1] is a summation where the number of terms is zero. The natural way to extend non-empty sums[2] is to let the empty sum be the additive identity.

Let , , , ... be a sequence of numbers, and let

be the sum of the first m terms of the sequence. This satisfies the recurrence

provided that we use the following natural convention: . In other words, a "sum" with only one term evaluates to that one term, while a "sum" 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" 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 vectors, matrices, polynomials), the value of an empty summation is taken to be its additive identity.

Examples

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. The empty sum convention allows the zero-dimensional vector space V={0} to have a basis, namely the empty set.

See also

References

  1. ^ Harper, Robert (2016). Practical Foundations for Programming Languages. Cambridge University Press. p. 86. ISBN 9781107029576.
  2. ^ David M. Bloom (1979). Linear Algebra and Geometry. pp. 45. ISBN 0521293243.
This page was last edited on 13 March 2022, at 01:52
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.