In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an
indeterminate form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...
, 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 (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
). As these terms are not defined in terms of other concepts, they may be referred to as "undefined terms".
* A
function is said to be "undefined" at points outside of its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
for example, the real-valued function
is undefined for negative
(i.e., it assigns no value to negative arguments).
* In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, some
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
operations may not assign a meaning to certain values of its operands (e.g.,
division by zero). 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; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
defined a
point 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 description that uses the term(s) being defined as part of the description or assumes that the term(s) being described are already known. There are several kinds of circular definition, and several ways of character ...
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, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
may be defined as a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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 is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
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 (the same expression is used
in calculus to represent an
indeterminate form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...
).
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 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
, which is undefined for
, and
, which is undefined (in the real number system) for negative
.
In trigonometry
In trigonometry, for all
, the functions
and
are undefined for all
, while the functions
and
are undefined for all
.
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 that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, a point
where a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
is undefined is called a
singularity. One distinguishes between
removable singularities (i.e., the function can be extended holomorphically to
),
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in ...
(i.e., the function can be extended
meromorphically to
), and
essential singularities (i.e., no meromorphic extension to
can exist).
In computer science
Notation using ↓ and ↑
In
computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has sinc ...
, if
is a
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
on
and
is an element of
, then this is written as
, and is read as "''f''(''a'') is ''defined''."
If
is not in the domain of
, then this is written as
, and is read as "
is ''undefined''".
The symbols of infinity
In
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
,
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
and other mathematical disciplines, the symbol
is frequently used to denote an infinite pseudo-number, along with its negative,
. The symbol has no well-defined meaning by itself, but an expression like
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
is undefined. Some extensions, though, define the following conventions of addition and multiplication:
*
for all
.
*
for all
.
*
for all
.
No sensible extension of addition and multiplication with
exists in the following cases:
*
*
(although in
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
, this is often defined as
)
*
*
For more detail, see extended real number line.
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