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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an
indeterminate formIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
, which has the propensity of assuming different values). The term can take on several different meanings depending on the context. For example: * In various branches of mathematics, certain concepts are introduced as primitive notions (e.g., the terms "point", "line" and "angle" in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
). As these terms are not defined in terms of other concepts, they may be referred to as "undefined terms". * A
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
is said to be "undefined" at points outside of its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
for example, the real-valued function f(x)=\sqrt is undefined for negative x (i.e., it assigns no value to negative arguments). * In
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, some
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
operations may not assign a meaning to certain values of its operands (e.g.,
division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
). In which case, the expressions involving such operands are termed "undefined".


Undefined terms

In ancient times, geometers attempted to define every term. For example,
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...
defined a
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
as "that which has no part". In modern times, mathematicians recognize that attempting to define every word inevitably leads to
circular definition A circular definition is a definition A definition is a statement of the meaning of a term (a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with s ...
s, and therefore leave some terms (such as "point") undefined (see primitive notion for more). This more abstract approach allows for fruitful generalizations. In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
may be defined as a set of points endowed with certain properties, but in the general setting, the nature of these "points" is left entirely undefined. Likewise, in
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
consists of "objects" and "arrows", which are again primitive, undefined terms. This allows such abstract mathematical theories to be applied to very diverse concrete situations.


In arithmetic

The expression is undefined in arithmetic, as explained in
division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
(the same expression is used in calculus to represent an
indeterminate formIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
). Mathematicians have different opinions as to whether should be defined to equal 1, or be left undefined.


Values for which functions are undefined

The set of numbers for which a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
is defined is called the ''domain'' of the function. If a number is not in the domain of a function, the function is said to be "undefined" for that number. Two common examples are f(x)=\frac, which is undefined for x=0, and f(x)=\sqrt, which is undefined (in the real number system) for negative  x .


In trigonometry

In trigonometry, the functions \tan \theta and \sec \theta are undefined for all \theta = 180^\circ \left(n - \frac\right), while the functions \cot \theta and \csc \theta are undefined for all \theta = 180^\circ(n).


In computer science


Notation using ↓ and ↑

In
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. ...
, if f is a
partial function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

partial function
on S and a is an element of S, then this is written as f(a)\downarrow, and is read as "''f''(''a'') is ''defined''." If a is not in the domain of f, then this is written as f(a)\uparrow, and is read as " f(a) is ''undefined''".


The symbols of infinity

In
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...

analysis
,
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
and other mathematical disciplines, the symbol \infty is frequently used to denote an infinite pseudo-number, along with its negative, -\infty. The symbol has no well-defined meaning by itself, but an expression like \left\\rightarrow\infty is shorthand for a divergent sequence, which at some point is eventually larger than any given real number. Performing standard arithmetic operations with the symbols \pm\infty is undefined. Some extensions, though, define the following conventions of addition and multiplication: * x+\infty=\inftyfor all x \in\R\cup\. * -\infty+x=-\inftyfor all x\in\R\cup\. * x\cdot\infty=\inftyfor all x\in\R^. No sensible extension of addition and multiplication with \infty exists in the following cases: * \infty-\infty * 0\cdot\infty (although in
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
, this is often defined as 0) * \frac For more detail, see
extended real number line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
.


Singularities in complex analysis

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
, a point z\in\mathbb where a
holomorphic function In mathematics, a holomorphic function is a complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), ...
is undefined is called a
singularity Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiabl ...
. One distinguishes between
removable singularities
removable singularities
(i.e., the function can be extended holomorphically to z),
poles The Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a nation A nation is a community A community is a social unitThe term "level of analysis" is used in the social sciences to point to the loc ...
(i.e., the function can be extended meromorphically to z), and
essential singularities
essential singularities
(i.e., no meromorphic extension to z can exist).


References


Further reading

*{{cite book , first=James R. , last=Smart , title=Modern Geometries , edition=Third , publisher=Brooks/Cole , year=1988 , isbn=0-534-08310-2 Mathematical terminology Calculus