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In arithmetic, and therefore algebra, division by zero is 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 ] :
Quizlet was founded in October 2005 by Andrew Sutherland, who at the time was a 15-year old student, [2] and released to the public in January 2007. [3] Quizlet's primary products include digital flash cards , matching games , practice electronic assessments , and live quizzes.
This is an accepted version of this page This is the latest accepted revision, reviewed on 9 January 2025. Look up undefined in Wiktionary, the free dictionary. Undefined may refer to: Mathematics Undefined (mathematics), with several related meanings Indeterminate form, in calculus Computing Undefined behavior, computer code whose behavior is not specified under certain conditions Undefined ...
Despite these subtle logical problems, it is quite common to use the term definition (without apostrophes) for "definitions" of this kind, for three reasons: It provides a handy shorthand of the two-step approach. The relevant mathematical reasoning (i.e., step 2) is the same in both cases. In mathematical texts, the assertion is "up to 100%" true.
Some authors of abstract algebra textbooks define an indeterminate over a ring R as an element of a larger ring that is transcendental over R. [ 3 ] [ 4 ] [ 5 ] This uncommon definition implies that every transcendental number and every nonconstant polynomial must be considered as indeterminates.
Alfred Tarski explained the role of primitive notions as follows: [4]. When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings.
This is not sharp; the gap between the functions is everywhere at least 1. Among the exponential functions of the form α x, setting α = e 2/e = 2.0870652... results in a sharp upper bound; the slightly smaller choice α = 2 fails to produce an upper bound, since then α 3 = 8 < 3 2. In applied fields the word "tight" is often used with the ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.