1 clearly characterized or delimited; "lost in a maze of words both defined and undefined"; "each child has clearly defined duties" [ant: undefined]
2 showing clearly the outline or profile or boundary; "hills defined against the evening sky"; "the setting sun showed the outlined figure of a man standing on the hill" [syn: outlined]
- past of define
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.
Examples and workarounds
The following expressions are undefined in all contexts, 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.
Zero to the zero powerThe question of 0^0 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, 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 x\to 0 is 0 (with f taking positive values), the limit
- \lim_ f(x)^
can be any nonnegative number, or infinity, or fail to exist.
Modern textbooks often define 0^0=1. For example, Ronald Graham, Donald Knuth and Oren Patashnik argue in their book Concrete mathematics:
AnalysisIn 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 theoryIn measure theory (which the common way of treating probability theory in mathematics), measures are preserved under countable addition. Taking \infty as countable, 0 \cdot \infty = \sum_^ 0 = 0.
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" .
defined in German: Unbestimmter Ausdruck
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