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.
Contents |
Examples and workarounds
The following expressions are undefined in all contexts[citation needed], but remarks in the analysis section may apply.
The following are defined in some, but not all contexts, as described in sections of this article.
| Expression | Context in which it is used |
|---|---|
| See division by zero. | |
| 00 | zero to the zero power, analysis, and set theory |
| analysis and set theory | |
| analysis and set theory | |
| analysis, set theory, and measure theory |
Zero to the zero power
Further information: Exponentiation: Zero to the zero powerThe question of 00 may be the most common point on which branches of mathematics disagree[citation needed]. Here we note only two considerations, one from analysis and one from combinatorics, 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 ƒ and g with whose limit as x → 0< is 0 (with ƒ taking positive values), the limit
can be any nonnegative number, or infinity, or fail to exist.
Modern textbooks often define 00 = 1. For example, Ronald Graham, Donald Knuth and Oren Patashnik argue in their book Concrete Mathematics:
| “ | Some textbooks leave the quantity 00 undefined, because the functions 0x and x0 have different limiting values when x decreases to 0. But this is a mistake. We must define x0 = 1 for all x, if the binomial theorem is to be valid when x = 0 , y = 0, and/or x = −y . The theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant. | ” |
Analysis
In mathematical analysis the domain of a function is usually determined by the limit of the function, so as to make the function continuous. This definition makes all of the expressions undefined. In calculus, some of the expressions arise in intermediate calculations, where they are called indeterminate forms and dealt with using techniques such as L'Hôpital's rule.
Measure theory
In measure theory (which is the common way of treating probability theory in mathematics), measures are preserved under countable addition. Taking as countable, .
Notation using ↓ and ↑
In computability theory, if f is a partial function 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
| Look up defined or undefined in Wiktionary, the free dictionary. |
- Bottom type
- Expression (mathematics)
- Indeterminate form
- Indefinite
- L'Hôpital's rule
- Mathematical singularity
- Well-defined
Categories: Mathematical terminology
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