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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion.

Zero to the zero power
The question of may be the most common point on which branches of mathematics disagree.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion.

Examples and workarounds


The following expressions are undefined in all contexts, but remarks in the analysis
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
 section may apply.



The following are defined in some, but not all contexts, as described in sections of this article.

See division by zero
Division by zero

In mathematics, a division is called a division by zero if the divisor is 0 . Such a division can be formally expressed as a/0 where a is the dividend....
.
zero to the zero power
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
, analysis
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
, and set theory
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
analysis
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
 and set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
analysis
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
 and set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
analysis
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
, set theory
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
, and measure theory
Defined and undefined

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....


Zero to the zero power


The question of may be the most common point on which branches of mathematics disagree. Here we note only two considerations, one from analysis and one from combinatorics
Combinatorics

Combinatorics is a branch of pure mathematics concerning the study of Countable set objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics....
, as an example of the way different approaches may yield different answers.

In 1821, Cauchy also listed 00 as undefined. The function 0x (for x>0) is constantly 0, and the function x0 (for x>0) is constantly 1, so there seems to be no natural value for 00. Indeed, for suitably chosen continuous functions f and g with whose limit as is 0 (with f taking positive values), the limit



can be any nonnegative number, or infinity, or fail to exist.

Modern textbooks often define . For example, Ronald Graham
Ronald Graham

Ronald Lewis Graham is a mathematician credited by the American Mathematical Society with being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years"....
, Donald Knuth
Donald Knuth

Donald Ervin Knuth is a renowned computer science and Emeritus of the Art of Computer Programming at Stanford University.Author of the seminal multi-volume work The Art of Computer Programming , Knuth has been called the "father" of the run-time analysis, contributing to the development of, and systematizing formal mathematical techn...
 and Oren Patashnik
Oren Patashnik

File:Patashnik.jpegOren Patashnik is a computer scientist. He is notable for co-creating BibTeX, and co-writing Concrete Mathematics. He is a researcher at the Center for Communications Research, La Jolla....
 argue in their book Concrete Mathematics
Concrete Mathematics

Concrete Mathematics: A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashnik, is a perennial textbook in university computer science departments....
:

Analysis

In mathematical analysis the domain of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 is usually determined by the limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
 of the function, so as to make the function continuous. This definition makes all of the expressions undefined. In calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
, some of the expressions arise in intermediate calculations, where they are called indeterminate form
Indeterminate form

In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine th...
s and dealt with using techniques such as L'Hôpital's rule
L'Hôpital's rule

In calculus, l'H?pital's rule uses derivatives to help evaluate limit s involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit....
.

Measure theory

In measure theory (which is the common way of treating probability theory
Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
 in mathematics), measures are preserved under countable addition. Taking as countable, .

Notation using ? and ?

In computability theory
Computability theory (computer science)

In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different Model of computation....
, if f is a partial function
Partial function

In mathematics, a partial function is a binary relation that associates each element of a Set , sometimes called its domain , with at most one element of another set, called its codomain....
 on S and a is an element of S, then this is written as f(a)? and is read "f(a) is defined."

If a is not in the domain of f, then f(a)? is written and is read as "f(a) is undefined" .

See also

  • Bottom type
    Bottom type

    In type theory, the bottom type is the type that has no values. It is also called the zero or empty type, and is sometimes denoted ?....
  • Expression (mathematics)
    Expression (mathematics)

    In mathematics, the word expression is a term for any well-formed formula combination of mathematical symbols. For example,is an expression, while...
  • Indeterminate form
    Indeterminate form

    In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine th...
  • Indefinite
    Indefinite

    The word indefinite has these meanings:-*In mathematics:-**When talking about positive or negative indefinite forms in multilinear algebra, see definite bilinear form....
  • L'Hôpital's rule
    L'Hôpital's rule

    In calculus, l'H?pital's rule uses derivatives to help evaluate limit s involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit....
  • Mathematical singularity
    Mathematical singularity

    In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional Set where it fails to be well-behaved in some particular way, such as derivative....
  • Well-defined
    Well-defined

    In mathematics, the term well-defined is used to specify that a certain concept or object is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy....