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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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
The following expressions are undefined in all contexts, but remarks in the analysis section may apply. In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
The following are defined in some, but not all contexts, as described in sections of this article. This article or section does not cite its references or sources. ...
| 00 | zero to the zero power, analysis, and set theory |  | analysis and set theory |  | analysis and set theory |  | analysis, set theory, and measure theory | In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...
Zero to the zero power
The question of 00 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. A more complete discussion on 00 is given at Zero to the zero power in the article on exponentiation. Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ...
Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ...
Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ...
Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ...
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 Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...
 can be any nonnegative number.
Modern textbooks often define 00 = 1. For example, Ronald Graham, Donald Knuth and Oren Patashnik argue in their book Concrete mathematics: Ronald L. Graham (born October 31, 1935) is a mathematician credited by the American Mathematical Society with being one of the principle architects of the rapid development worldwide of discrete mathematics in recent years[1]. He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. ...
Donald Ervin Knuth ( or Ka-NOOTH[1], Chinese: [2]) (b. ...
Oren Patashnik (born 1954) is a computer scientist. ...
Concrete Mathematics by Ronald L. Graham, Donald E. Knuth and Oren Patashnik is a textbook that provides its readers with mathematical background that can be especially useful in computer science. ...
| “ | 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. | ” | In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...
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. Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ...
Partial plot of a function f. ...
In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
Calculus [from Latin, literally chalk pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ...
In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression whose limit cannot be evaluated by substituting the limits of the subexpressions. ...
In calculus, lHôpitals rule (often incorrectly lHospitals rule) uses derivatives to help compute limits with indeterminate forms. ...
Set theory In set theory, X Y is defined as the set of functions from Y to X, and for cardinal numbers, we take the cardinality of the set of functions, for arbitrary sets of cardinality Y and X. In this sense,  . The set-theoretic product is similarly taken as the Cartesian product, and so  in set theory. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
Alternative meaning: number of pitch classes in a set. ...
In mathematics, the Cartesian product is a direct product of sets. ...
Measure theory In measure theory (which the common way of treating probability theory in mathematics), measures are preserved under countable addition. Taking  as countable,  . In mathematics, a measure is a function that assigns a number, e. ...
It has been suggested that this article or section be merged with Probability axioms. ...
In mathematics the term countable set is used to describe the size of a set, e. ...
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." In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. ...
In mathematics, a partial function is a relation that associates each element of a set (sometimes called its domain) with at most one element of another (possibly the same) set, called the codomain. ...
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, undefined in Wiktionary, the free dictionary. |
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