Undefined (mathematics)
In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system.In practice, mathematicians may use the term undefined to warn that a particular calculation or property can produce mathematically inconsistent results, and therefore, it should be avoided.In some mathematical contexts, however, undefined can refer to a primitive notion which is not defined in terms of simpler concepts.[3] 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".[4] Although these terms are not further defined, Euclid uses them to construct more complex geometric concepts.[5] Whether a particular function or value is undefined, depends on the rules of the formal system in which it is used.So it is meaningless to reason about the value, solely within the discourse of real numbers., allows there to be a consistent set of mathematics referred to as the complex number plane.Many new fields of mathematics have been created, by taking previously undefined functions and values, and assigning them new meanings.[6] Most mathematicians generally consider these innovations significant, to the extent that they are both internally consistent and practically useful.For example, Ramanujan summation may seem unintuitive, as it works upon divergent series that assign finite values to apparently infinite sums such as 1 + 2 + 3 + 4 + ⋯.However, Ramanujan summation is useful for modelling a number of real-world phenomena, including the Casimir effect and bosonic string theory.[7] Use of a division by zero in an arithmetical calculation or proof, can produce absurd or meaningless results.This operation is undefined in arithmetic, and therefore deductions based on division by zero can be contradictory.[7] Depending on the particular context, mathematicians may refer to zero to the power of zero as undefined,[9] indefinite,[10] or equal to 1.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.on the complex plane where a holomorphic function is undefined, is called a singularity.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.