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 ${\displaystyle {\frac {n}{0}},n\neq 0}$  is undefined in arithmetic, as explained in division by zero (the ${\displaystyle {\frac {0}{0}}}$  expression is used in calculus to represent an indeterminate form).

Mathematicians have different opinions as to whether 0⁰ 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 ${\textstyle f(x)={\frac {1}{x}}}$ , which is undefined for ${\displaystyle x=0}$ , and ${\displaystyle f(x)={\sqrt {x}}}$ , which is undefined (in the real number system) for negative ${\displaystyle x}$ .

In trigonometry edit

In trigonometry, for all ${\displaystyle n\in \mathbb {Z} }$ , the functions ${\displaystyle \tan \theta }$  and ${\displaystyle \sec \theta }$  are undefined for all ${\textstyle \theta =\pi \left(n-{\frac {1}{2}}\right)}$ , while the functions ${\displaystyle \cot \theta }$  and ${\displaystyle \csc \theta }$  are undefined for all ${\displaystyle \theta =\pi n}$ .

In complex analysis edit

In complex analysis, a point ${\displaystyle z\in \mathbb {C} }$  where a holomorphic function is undefined is called a singularity. One distinguishes between removable singularities (i.e., the function can be extended holomorphically to ${\displaystyle z}$ ), poles (i.e., the function can be extended meromorphically to ${\displaystyle z}$ ), and essential singularities (i.e., no meromorphic extension to ${\displaystyle z}$  can exist).

Notation using ↓ and ↑ edit

In computability theory, if ${\displaystyle f}$  is a partial function on ${\displaystyle S}$  and ${\displaystyle a}$  is an element of ${\displaystyle S}$ , then this is written as ${\displaystyle f(a)\downarrow }$ , and is read as "f(a) is defined."^[3]

If ${\displaystyle a}$  is not in the domain of ${\displaystyle f}$ , then this is written as ${\displaystyle f(a)\uparrow }$ , and is read as "${\displaystyle f(a)}$  is undefined".

In analysis, measure theory and other mathematical disciplines, the symbol ${\displaystyle \infty }$  is frequently used to denote an infinite pseudo-number, along with its negative, ${\displaystyle -\infty }$ . The symbol has no well-defined meaning by itself, but an expression like ${\displaystyle \left\{a_{n}\right\}\rightarrow \infty }$  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 ${\displaystyle \pm \infty }$  is undefined. Some extensions, though, define the following conventions of addition and multiplication:

• ${\displaystyle x+\infty =\infty }$    for all ${\displaystyle x\in \mathbb {R} \cup \{\infty \}}$ .
• ${\displaystyle -\infty +x=-\infty }$    for all ${\displaystyle x\in \mathbb {R} \cup \{-\infty \}}$ .
• ${\displaystyle x\cdot \infty =\infty }$    for all ${\displaystyle x\in \mathbb {R} ^{+}}$ .

No sensible extension of addition and multiplication with ${\displaystyle \infty }$  exists in the following cases:

• ${\displaystyle \infty -\infty }$ 
• ${\displaystyle 0\cdot \infty }$  (although in measure theory, this is often defined as ${\displaystyle 0}$ )
• ${\displaystyle {\frac {\infty }{\infty }}}$ 
• ${\displaystyle -\infty [{1(0)}]}$ 

For more detail, see extended real number line.

• Smart, James R. (1988). Modern Geometries (Third ed.). Brooks/Cole. ISBN 0-534-08310-2.