In mathematics, the word undefined has several different meanings, depending on the context. In geometry, simple words such as "point" and "line" are taken as undefined terms. In arithmetic, some arithmetic operations are called "undefined". The most famous example is that division by zero is undefined. In algebra, a function is said to be "undefined" at points not in its domain. For example, in the real number system, is undefined for negative .
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 recognized that attempting to define every word inevitably led to circular definitions, and in geometry left some words, "point" for example, as undefined. See primitive notion.
The reasoning behind leaving division by zero undefined is as follows. Division is the inverse of multiplication. If , then . But if , then any multiple of is also , and so if , no such exists. On the other hand, if and are both zero, then every real number satisfies . Either way, it is impossible to assign a particular real number to the quotient when the divisor is zero.
In calculus, is sometimes used as a symbol, and is called an indeterminate form, but the symbol does not represent division in the sense the word is used in ordinary arithmetic.
Another common operation that is undefined is that of raising zero to the zero power. On the one hand, if , then . On the other hand, if is any positive number, , while if is negative, implies division by zero, which is undefined. Thus, to make the laws of exponents work in every case where exponents are defined, is left undefined. That said, there are branches of higher mathematics where various definitions of zero to the zero power are given (see: Exponentiation).
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 .
If a is not in the domain of f, then f(a)↑ is written and is read as "f(a) is undefined".
In analysis, measure theory, and other mathematical disciplines, the symbol is frequently used to denote an infinite pseudo-number, in real analysis alongside with its negative . The symbol has no well-defined meaning by itself, but an expression like is a shorthand for a divergent sequence which is eventually larger than any given real number.
Arithmetic with the symbols is undefined. The following conventions of addition and multiplication are in common use:
No sensible extension of addition and multiplication with exist in the following cases:
- (although in measure theory, this is often defined as )
See extended real number line for more information.
In complex analysis, a point where a holomorphic function is undefined is called a singularity. One distinguishes between removable singularities (the function can be extended holomorphically to , poles (the function can be extended meromorphically to ), and essential singularities, where no meromorphic extension to exists.
- James R. Smart, Modern Geometries Third Edition, Brooks/Cole, 1988, ISBN 0-534-08310-2