Undefined (mathematics)

In some mathematical contexts, undefined can refer to a primitive notion which is not defined in terms of simpler concepts. For example, in Elements, Euclid defines a point merely as "that of which there is no part", and a line merely as "length without breadth". Although these terms are not further defined, Euclid uses them to construct more complex geometric concepts. Contrast also the term undefined behavior in computer science, in which the term indicates that a function may produce or return any result, which may or may not be correct. == Common examples of undefined expressions ==

Many fields of mathematics refer to various kinds of expressions as undefined. Therefore, the following examples of undefined expressions are not exhaustive. Division by zero In arithmetic, and therefore algebra, division by zero is undefined. Use of a division by zero in an arithmetical calculation or proof, can produce absurd or meaningless results. Assuming that division by zero exists, can produce inconsistent logical results, such as the following fallacious "proof" that one is equal to two: The above "proof" is not meaningful. Since we know that x=y, if we divide both sides of the equation by x-y, we divide both sides of the equation by zero. This operation is undefined in arithmetic, and therefore deductions based on division by zero can be contradictory. If we assume that a non-zero answer n exists, when some number k \mid k \neq 0 is divided by zero, then that would imply that k = n \times 0. But there is no number, which when multiplied by zero, produces a number that is not zero. Therefore, our assumption is incorrect. indefinite, or equal to 1. Controversy exists as to which definitions are mathematically rigorous, and under what conditions. The square root of a negative number When restricted to the field of real numbers, the square root of a negative number is undefined, as no real number exists which, when squared, equals a negative number. Mathematicians, including Gerolamo Cardano, John Wallis, Leonhard Euler, and Carl Friedrich Gauss, explored formal definitions for the square roots of negative numbers, giving rise to the field of complex analysis. In trigonometry In trigonometry, for all n \in \mathbb{Z}, the functions \tan \theta and \sec \theta are undefined for \theta = \pi \left(n - \frac{1}{2}\right), while the functions \cot \theta and \csc \theta are undefined for all \theta = \pi n. This is a consequence of the identities of these functions, which would imply a division by zero at those points. Also, \arcsin k and \arccos k are both undefined when k > 1 or k, because the range of the \sin and \cos functions is between -1 and 1 inclusive. In complex analysis In complex analysis, a point z on the complex plane where a holomorphic function is undefined, is called a singularity. Some different types of singularities include: • Removable singularities - in which the function can be extended holomorphically to z • Poles - in which the function can be extended meromorphically to z • Essential singularities - in which no meromorphic extension to z can exist == Related terms ==

Indeterminate The term undefined should be contrasted with the term indeterminate. In the first case, undefined generally indicates that a value or property can have no meaningful definition. In the second case, indeterminate generally indicates that a value or property can have many meaningful definitions. Additionally, it seems to be generally accepted that undefined values may not be safely used within a particular formal system, whereas indeterminate values might be, depending on the relevant rules of the particular formal system. ==See also==