- 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.Examples and workarounds
The following expressions are undefined in all contextsFact|date=March 2008, 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 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 , as an example of the way different approaches may yield different answers.In 1821, Cauchy also listed 00 as undefined. The function 0"x" (for "x">0) is constantly 0, and the function "x"0 (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 ,Donald Knuth andOren Patashnik argue in their book "Concrete Mathematics ":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 calledindeterminate form s and dealt with using techniques such asL'Hôpital's rule .Measure theory
In
measure theory (which the common way of treatingprobability theory in mathematics), measures are preserved undercountable 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" .
ee also
*
Bottom type
*Expression (mathematics)
*Indeterminate form
*Indefinite
*L'Hôpital's rule
*Mathematical singularity
*Well-defined
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