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An equivalent definition is sometimes useful: if a and b are integers, then the fraction a / b is irreducible if and only if there is no other equal fraction c / d such that | c | < | a | or | d | < | b |, where | a | means the absolute value of a. [4] (Two fractions a / b and c / d are equal or equivalent if and ...
The earliest form of mathematics was phenomenological. For example, if someone could draw a reasonable picture, or give a convincing description, then that met all the criteria for something to be described as a mathematical “fact”. On occasion, analogical arguments took place, or even by “invoking the gods”. The idea that mathematical ...
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. L'Hôpital's rule
Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms. [5] One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.
In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function.Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier.
Determining whether a λ-calculus formula has a normal form. Conway's Game of Life on whether, given an initial pattern and another pattern, the latter pattern can ever appear from the initial one. Rule 110 - most questions involving "can property X appear later" are undecidable.
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