# Mathematical terms and definitions pdf

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An intuitive way of looking at absolute value is to consider that the absolute value of a real number is its distance from zero on the number line. For example:. Additive strategies are techniques used to solve addition problems from known facts.

## Origins of Mathematical Words: A Comprehensive Dictionary of Latin, Greek, and Arabic Roots

A definition is a statement of the meaning of a term a word , phrase , or other set of symbols. A term may have many different senses and multiple meanings, and thus require multiple definitions.

In mathematics , a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. In modern usage, a definition is something, typically expressed in words, that attaches a meaning to a word or group of words.

The word or group of words that is to be defined is called the definiendum , and the word, group of words, or action that defines it is called the definiens. The definiens is not the meaning of the word defined, but is instead something that conveys the same meaning as that word.

There are many sub-types of definitions, often specific to a given field of knowledge or study. These include, among many others, lexical definitions , or the common dictionary definitions of words already in a language; demonstrative definitions , which define something by pointing to an example of it "This," [said while pointing to a large grey animal], "is an Asian elephant.

An intensional definition , also called a connotative definition, specifies the necessary and sufficient conditions for a thing to be a member of a specific set. An extensional definition , also called a denotative definition, of a concept or term specifies its extension. It is a list naming every object that is a member of a specific set. Thus, the " seven deadly sins " can be defined intensionally as those singled out by Pope Gregory I as particularly destructive of the life of grace and charity within a person, thus creating the threat of eternal damnation.

An extensional definition, on the other hand, would be the list of wrath, greed, sloth, pride, lust, envy, and gluttony. In contrast, while an intensional definition of " Prime Minister " might be "the most senior minister of a cabinet in the executive branch of parliamentary government", an extensional definition is not possible since it is not known who the future prime ministers will be even though all prime ministers from the past and present can be listed.

A genus—differentia definition is a type of intensional definition that takes a large category the genus and narrows it down to a smaller category by a distinguishing characteristic i. Those definitions can be expressed as a genus "a plane figure" and two differentiae "that has three straight bounding sides" and "that has four straight bounding sides", respectively. It is also possible to have two different genus—differentia definitions that describe the same term, especially when the term describes the overlap of two large categories.

For instance, both of these genus—differentia definitions of "square" are equally acceptable:. Thus, a "square" is a member of both genera the plural of genus : the genus "rectangle" and the genus "rhombus".

One important form of the extensional definition is ostensive definition. This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind.

For example, one can explain who Alice an individual is, by pointing her out to another; or what a rabbit a class is, by pointing at several and expecting another to understand. The process of ostensive definition itself was critically appraised by Ludwig Wittgenstein. An enumerative definition of a concept or a term is an extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets and in fact only practical for relatively small sets.

Divisio and partitio are classical terms for definitions. A partitio is simply an intensional definition. A divisio is not an extensional definition, but an exhaustive list of subsets of a set, in the sense that every member of the "divided" set is a member of one of the subsets.

An extreme form of divisio lists all sets whose only member is a member of the "divided" set. The difference between this and an extensional definition is that extensional definitions list members , and not subsets. In classical thought, a definition was taken to be a statement of the essence of a thing.

Aristotle had it that an object's essential attributes form its "essential nature", and that a definition of the object must include these essential attributes. The idea that a definition should state the essence of a thing led to the distinction between nominal and real essence—a distinction originating with Aristotle. In the Posterior Analytics , [13] he says that the meaning of a made-up name can be known he gives the example "goat stag" without knowing what he calls the "essential nature" of the thing that the name would denote if there were such a thing.

This led medieval logicians to distinguish between what they called the quid nominis , or the "whatness of the name", and the underlying nature common to all the things it names, which they called the quid rei , or the "whatness of the thing". It has a quid nominis , but one could not know the real nature of hobbits, and so the quid rei of hobbits cannot be known. By contrast, the name "man" denotes real things men that have a certain quid rei. The meaning of a name is distinct from the nature that a thing must have in order that the name apply to it.

This leads to a corresponding distinction between nominal and real definitions. A nominal definition is the definition explaining what a word means i. A real definition, by contrast, is one expressing the real nature or quid rei of the thing. This preoccupation with essence dissipated in much of modern philosophy. Analytic philosophy , in particular, is critical of attempts to elucidate the essence of a thing. Russell described essence as "a hopelessly muddle-headed notion". More recently Kripke's formalisation of possible world semantics in modal logic led to a new approach to essentialism.

Insofar as the essential properties of a thing are necessary to it, they are those things that it possesses in all possible worlds. Kripke refers to names used in this way as rigid designators.

A definition may also be classified as an operational definition or theoretical definition. A homonym is, in the strict sense, one of a group of words that share the same spelling and pronunciation but have different meanings. The state of being a homonym is called homonymy.

A distinction is sometimes made between "true" homonyms, which are unrelated in origin, such as skate glide on ice and skate the fish , and polysemous homonyms, or polysemes , which have a shared origin, such as mouth of a river and mouth of an animal.

Polysemy is the capacity for a sign such as a word , phrase , or symbol to have multiple meanings that is, multiple semes or sememes and thus multiple senses , usually related by contiguity of meaning within a semantic field. It is thus usually regarded as distinct from homonymy , in which the multiple meanings of a word may be unconnected or unrelated. In mathematics, definitions are generally not used to describe existing terms, but to describe or characterize a concept.

The precise meaning of a term given by a mathematical definition is often different than the English definition of the word used, [20] which can lead to confusion, particularly when the meanings are close.

For example a set is not exactly the same thing in mathematics and in common language. In some case, the word used can be misleading; for example, a real number has nothing more or less real than an imaginary number. Frequently, a definition uses a phrase built with common English words, which has no meaning outside mathematics, such as primitive group or irreducible variety.

Authors have used different terms to classify definitions used in formal languages like mathematics. Norman Swartz classifies a definition as "stipulative" if it is intended to guide a specific discussion. A stipulative definition might be considered a temporary, working definition, and can only be disproved by showing a logical contradiction.

Swartz defines a precising definition as one that extends the descriptive dictionary definition lexical definition for a specific purpose by including additional criteria. A precising definition narrows the set of things that meet the definition. Stevenson has identified persuasive definition as a form of stipulative definition which purports to state the "true" or "commonly accepted" meaning of a term, while in reality stipulating an altered use perhaps as an argument for some specific belief.

A recursive definition , sometimes also called an inductive definition, is one that defines a word in terms of itself, so to speak, albeit in a useful way.

Normally this consists of three steps:. For instance, we could define a natural number as follows after Peano :. So "0" will have exactly one successor, which for convenience can be called "1". In turn, "1" will have exactly one successor, which could be called "2", and so on. Notice that the second condition in the definition itself refers to natural numbers, and hence involves self-reference.

Although this sort of definition involves a form of circularity , it is not vicious , and the definition has been quite successful. In the same way, we can define ancestor as follows:. In medical dictionaries , guidelines and other consensus statements and classifications , definitions should as far as possible be:.

Certain rules have traditionally been given for definitions in particular, genus-differentia definitions. Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions. If every term of every definiens must itself be defined, "where at last should we stop?

Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers claimed that the highest genera called the ten generalissima cannot be defined, since a higher genus cannot be assigned under which they may fall. Thus being , unity and similar concepts cannot be defined. More recently Bertrand Russell sought to develop a formal language based on logical atoms.

Other philosophers, notably Wittgenstein , rejected the need for any undefined simples. Wittgenstein pointed out in his Philosophical Investigations that what counts as a "simple" in one circumstance might not do so in another.

Locke and Mill also argued that individuals cannot be defined. Names are learned by connecting an idea with a sound, so that speaker and hearer have the same idea when the same word is used. Saul Kripke pointed to difficulties with this approach, especially in relation to modality , in his book Naming and Necessity.

There is a presumption in the classic example of a definition that the definiens can be stated. Wittgenstein argued that for some terms this is not the case. In such cases, he argued, there is no fixed boundary that can be used to provide a definition. Rather, the items are grouped together because of a family resemblance.

For terms such as these it is not possible and indeed not necessary to state a definition; rather, one simply comes to understand the use of the term. From Wikipedia, the free encyclopedia. For the definition of another word, see Wiktionary. For other uses of the word "definition" itself, see Definition disambiguation. Statement that attaches a meaning to a term. For the company, see Definiens company. Main articles: Intension and Extension semantics. Main article: Genus—differentia definition.

Main article: Essence. Main article: Homonym.

## Basic Math Definitions

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Account Options Sign in. Top charts. New releases. Add to Wishlist. The Concise Oxford Dictionary of Mathematics provides jargon-free definitions for over 3, of even the most technical mathematical terms. This dictionary covers all commonly encountered terms and concepts from pure and applied mathematics and statistics, such as linear algebra, optimization, nonlinear equations, and differential equations. In addition, there are entries on major mathematicians and topics of more general interest, such as fractals, game theory, and chaos.

GLOSSARY AND EXAMPLES OF MATHEMATICS TERMS. Acute triangle. A triangle with all acute angles (acute means measuring less than 90°).

## Glossary and Examples of Mathematics Terms

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