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In mathematics, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the propensity of assuming different values).^{[1]}^{[2]} The term can take on several different meanings depending on the context. For example:
In ancient times, geometers attempted to define every term. For example, Euclid defined a point as "that which has no part". In modern times, mathematicians recognize that attempting to define every word inevitably leads to circular definitions, and therefore leave some terms (such as "point") undefined (see primitive notion for more).
This more abstract approach allows for fruitful generalizations. In topology, a topological space may be defined as a set of points endowed with certain properties, but in the general setting, the nature of these "points" is left entirely undefined. Likewise, in category theory, a category consists of "objects" and "arrows", which are again primitive, undefined terms. This allows such abstract mathematical theories to be applied to very diverse concrete situations.
The expression 0/0 is undefined in arithmetic, as explained in division by zero (the same expression is used in calculus to represent an indeterminate form).
Mathematicians have different opinions as to whether 0^{0} should be defined to equal 1, or be left undefined.
The set of numbers for which a function is defined is called the domain of the function. If a number is not in the domain of a function, the function is said to be "undefined" for that number. Two common examples are , which is undefined for , and , which is undefined (in the real number system) for negative .
In trigonometry, the functions and are undefined for all , while the functions and are undefined for all .
In computability theory, if is a partial function on and is an element of , then this is written as , and is read as "f(a) is defined."^{[4]}
If is not in the domain of , then this is written as , and is read as " is undefined".
In analysis, measure theory and other mathematical disciplines, the symbol is frequently used to denote an infinite pseudonumber, along with its negative, . The symbol has no welldefined meaning by itself, but an expression like is shorthand for a divergent sequence, which at some point is eventually larger than any given real number.
Performing standard arithmetic operations with the symbols is undefined. Some extensions, though, define the following conventions of addition and multiplication:
No sensible extension of addition and multiplication with exists in the following cases:
For more detail, see extended real number line.
In complex analysis, a point where a holomorphic function is undefined is called a singularity. One distinguishes between removable singularities (i.e., the function can be extended holomorphically to ), poles (i.e., the function can be extended meromorphically to ), and essential singularities (i.e., no meromorphic extension to can exist).