Hyperbola degenerating into a pair of diagonal lines. The act of identifying a previously-unidentified mathematical pattern, or establishing the validity or the invalidity of a mathematical claim. This presupposes the existence of a mathematical reality , which is well-articulated by, say, G. Under this viewpoint, mathematics can be thought of as an endeavour encompassing both discovery and creation , in that even if mathematical truths and patterns exist independent of our actions and choices, we are still free to conceptualize them using whichever concepts and statements we can come up with.

A metaphor used to refer to a bottom component of a mathematical expression e.

It is the opposite of upstair , and can be handy when used as an adverb e. An elegant proof of the Pythagorean Theorem. A highly subjective term similar to beauty , but which is usually reserved for solutions to a problem e. According to Gian-Carlo Rota , a founding father of modern combinatorics, elegance is different from beauty in that elegance pertains to the presentation of the content, while beauty pertains directly to the content itself.

Although one cannot strive for mathematical beauty, one can achieve elegance in the presentation of mathematics. Mathematical elegance has to do with the presentation of mathematics, and only tangentially does it relate to its content. Whichever the interpretation, elegance remains one of the holy grails of mathematics, and is reflected in the way one tackles a problem incessantly, until a path of least resistance — such as one that involves a minimal amount of deep results and mathematical machinery — is found.

As such, elegance is often a reflection of a deep understanding of the subject matter. In contrast, certain approaches — such as those that involve laborious calculations and breaking a proof into a dozen or more cases — can be regarded as clumsy and awkward , as it might reflect a lack of understanding or mathematical maturity. A proof of a claim which — while not necessarily a proof from first principles — only involves basic notions and methods within the field without much development into the subject matter.

On the other end of the spectrum, a proof which requires heavy mathematical machinery from multiple fields is typically referred to as a deep result , and would often be considered deep until an elementary proof is found. An elementrary proof in number theory, for example, is simply one which does not resort to the methods of complex analysis or other mathematical machinery.

A statement which proves and can be proved from another statement under a set of axioms and presuppositions. Essential uniqueness at play with triangles of fixed lengths. Platonic solids as an early example of exceptional objects. For more examples, see Wikipedia — Exceptional Object. A rare object usually finite in number with some desirable property that makes it different from most objects of the same class e. A term used to assert the presence of an object satisfying a certain property, and is formalized using existential claims i.

In general, to prove that an existential claim holds is to demonstrate, constructively or non-constructively, that object s with the property demanded in the claim exist — even though such objects might not be unique among those with the property. That whose quantity pertains to a natural number or a real number — which might or might not include zero or the infinitesimals. A term used to refer to an unpublished mathematical result with no clear originator, which is well-circulated and well-understood to be true or useful among the specialists, but which is less so among those who are less entrenched in the field.

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In general, a folklore claim is most often obtained through conferences and word of mouth , though if one chooses to publish it with all the write-up, then it would lose its folklore status as a result. An adverb used to describe a mathematical object which satisfies a certain property repeatedly usually indefinitely — for arbitrarily large arguments or instances.

For example, one could say that:. A form of abstraction where one goes from a set of simpler cases to higher-order instances , and can manifest in mathematical thinking in different ways e. Some examples of generalization in mathematics include:. In general, generalization is both advantageous and disadvantageous, in that while it can make a claim less concrete and less visualizable, it can also make it more powerful and more applicable as a result.

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A term used to indicate that a mathematical object satisfies a property across its entire object e. A cartoon depicting mathematical hand-waving by Sydney Harris. The act of demonstrating the validity of a claim without providing an entirely rigorous argument for its validity, and which typically involves the use of unrepresentative examples, unjustified assumptions, key omissions and faulty logic.

Expectedly, hand-waving is generally regarded as a bad mathematical practice, since it tends to distract one from the legitimate flaws one is making, and can be especially devastating when the unjustified assumptions are false.

A series of general, trial-and-error-based strategies for solving mathematical problems which — while not necessarily optimal — can be cost-effective in yielding a satisfactory solution given the amount of invested time and effort. Some of most used heuristics in mathematics include:. In general, heuristics and algorithms are similar in that both constitute some form of mathematical procedures, but are different in that the former prioritizes higher-order thinking over the actual steps — and as such might require a bit more mental flexibility and creativity.

On the other hand, a heuristic argument is generally one which — while pedagogically useful and seemingly convincing — makes a few key oversimplifications which disqualify it from a rigorous proof e.

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A mathematical claim in the form of an equation — usually with some variables — which has been shown to be true for all values of the variables within a certain range of validity. Some examples of identities in mathematics include:. A term typically used to refer to a mathematical expression capable of assuming multiple values , though its actual meaning is generally quite precise and dependent upon context. A mathematical statement which makes a non-equal comparison between two mathematical expressions, and which can be further categorized into the following three types:. Some prominent inequalities in mathematics include:.

Similar to equations, inequalities can be put into chain notation e. An adjective used to refer to the general quality of being boundless e. In many scenarios, infinity is used solely as a concept e. A term used to indicate that a generalization is under way.

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Despite its instinctive nature, mathematical intuition is often a product of accumulated knowledge and experience , and can be indicative of historical patterns and connections, as an anonymous mathematician puts on Quora:. Like rigor , intuition is often considered to be an integral part of mathematics, as it allows one to find the answer to a problem before solving it a recurring theme in the history of mathematics , and helps imbue mathematics with a series of qualities that set it apart from some meaningless manipulations of symbols.

Odd functions as functions invariant under rotations of degrees. The property of a mathematical object to remain unchanged after an operation or a transformation. In geometry and other visually-based topics, invariance is often known under the name of symmetry e. A minor, relatively-easy-to-prove claim which is often used as a stepping stone for proving major results e.

A term used to indicate that a mathematical object satisfies a property at some limited portions of the object — or from the point of view of some narrow, immediate surrounding.

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Some examples of local in mathematics include:. A mapping, also known as a map, is simply a general function from a source to a target.

A mathematical inference or derivation which violates the condition of its applicability. Some examples of common mathematical fallacies include:. While mathematical fallacies can compromise the validity of an argument, they alone generally do not provide sufficient ground for accepting or rejecting a statement. A colloquial term for the advanced theorems , methods and techniques developed over the years — which can be used to tackle future problems in the same field or a different field with relative ease. While heavy mathematical machinery can be immensely useful in application, they can also contribute to the degradation of elegance in a proof.

As a result, elementary proofs are almost always sought after — even after a claim is proved. The quality of having a general understanding and mastery of the way mathematicians operate — and the language they use to communicate ideas. These include, among others:. In general, mathematical maturity allows one to operate intellectually in an independent, standalone manner, and like mathematical intuition , can be a product of both knowledge and experience. A pentagon with 5 types of mathematical representations. In mathematical education, a representation is one which encodes a mathematical idea or a relationship.

It can occur as both internal e. Some prominent examples of representation in mathematics include:. While representations can come in all shapes and forms, having multiple representations of the same mathematical idea e. A mathematical object consisting of a set along with some additional features on the set e.

Some eminent mathematical structures include, among others:. A term derived from the Latin word modulus i.

Aside from its occurrences in the common mathematical parlance e.