In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system.
[1] Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that 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.
[2] Caution must be taken to avoid the use of such undefined values in a deduction or proof.
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.
[3] 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.
In some mathematical contexts, undefined can refer to a primitive notion which is not defined in terms of simpler concepts.
[4] 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".
[5] Although these terms are not further defined, Euclid uses them to construct more complex geometric concepts.
[6] 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.
Many fields of mathematics refer to various kinds of expressions as undefined.
[7] 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[8]: The above "proof" is not meaningful.
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.
[11] Controversy exists as to which definitions are mathematically rigorous, and under what conditions.
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.
This is a consequence of the identities of these functions, which would imply a division by zero at those points.
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.