Null (mathematics)

{{Short description|Mathematical representation of absence of a value}}

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In mathematics, the word null (from {{langx|de|null}}{{citation needed|date=January 2023}} meaning "zero", which is from {{langx|la|nullus}} meaning "none") is often associated with the concept of zero, or with the concept of nothing.{{cite journal |title="null" |journal=The Oxford English Dictionary, Draft Revision March 2004 |url=http://dictionary.oed.com |year=2004 |access-date=2007-04-05}}{{cite journal |title=Definition of "null" adjective from the Oxford Advanced Learner's Dictionary |journal=Oxford Advanced Learner's Dictionary 9th |url=https://en.oxforddictionaries.com/definition/null |archive-url=https://web.archive.org/web/20180621143802/https://en.oxforddictionaries.com/definition/null |url-status=dead |archive-date=June 21, 2018 |year=2016 |access-date=2018-06-21}} It is used in varying contexts from "having zero members in a set" (e.g., null set){{Cite web|url=https://whatis.techtarget.com/definition/null-set|title=What is null set? - Definition from WhatIs.com|website=WhatIs.com|language=en|access-date=2019-12-09}} to "having a value of zero" (e.g., null vector).{{Cite web|url=http://mathworld.wolfram.com/NullVector.html|title=Null Vector|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-09}}

In a vector space, the null vector is the neutral element of vector addition; depending on the context, a null vector may also be a vector mapped to some null by a function under consideration (such as a quadratic form coming with the vector space, see null vector, a linear mapping given as matrix product or dot product, a seminorm in a Minkowski space, etc.). In set theory, the empty set, that is, the set with zero elements, denoted "{}" or "∅", may also be called null set.{{Cite web|url=http://www.solving-math-problems.com/math-symbols-set-null.html|archive-url=https://web.archive.org/web/20090731235502/http://www.solving-math-problems.com/math-symbols-set-null.html|url-status=usurped|archive-date=July 31, 2009|title=Math Symbols: Null Set|website=www.solving-math-problems.com|access-date=2019-12-09}} In measure theory, a null set is a (possibly nonempty) set with zero measure.

A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element). For example, in linear algebra, the null space of a linear mapping, also known as kernel, is the set of vectors which map to the null vector under that mapping.

In statistics, a null hypothesis is a proposition that no effect or relationship exists between populations and phenomena. It is the hypothesis which is presumed true—unless statistical evidence indicates otherwise.{{Cite web|url=https://www.thoughtco.com/definition-of-null-hypothesis-and-examples-605436|title=What Is the Null Hypothesis? Definition and Examples|last=Helmenstine|first=Anne Marie|date=|website=ThoughtCo|language=en|access-date=2019-12-09}}