Mathematical Term
A Mathematical Term is Mathematical Object that is a component of a Mathematical Formula.
- AKA: Mathematical Expression Term.
- Context:
- Example(s):
- an Addend,
- a Ground Term,
- a Linear Term,
- a Monomial,
- a Summand.
- …
- Counter-Example(s):
- See: Mathematical Equation, Mathematical Expression, Logic, Term Rewriting System, Polynomial.
References
2020a
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Term_(logic) Retrieved:2020-3-26.
- In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact. In particular, terms appear as components of a formula.
A first-order term is recursively constructed from constant symbols, variables and function symbols.
An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation.
For example, is a term built from the constant 1, the variable , and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of .
Besides in logic, terms play important roles in universal algebra, and rewriting systems.
- In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact. In particular, terms appear as components of a formula.
2020b
- (MathWorld-Wolfram, 2020) ⇒ Eric W. Weisstein, "Term." From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/Term.html Retrieved:2020-03-26.
- QUOTE: In logic, a term is a variable, constant, or the result of acting on variables and constants by function symbols.
In algebra, a term is a product of the form $x^n$ (in the univariate case), or more generally of the form $x_1^{a_1} \cdots x_n^{a_n}$ (in the multivariate case) in a polynomial (Becker and Weispfenning 1993, p. 188[1]).
The word “term” is also used commonly to mean a summand of a polynomial including its coefficient (more properly called a monomial), or the corresponding quantity in a series (i.e., a series term).
One term is said to divide another if the powers of its variables are no greater than the corresponding powers in the second monomial. For example, $x^2y$ divides $x^3y$ but does not divide $xy^3$. A term $m$ is said to reduce with respect to a polynomial if the leading term of that polynomial divides $m$. For example, $x^2y$ reduces with respect to $2xy+x+3$ because $xy$ divides $x^2y$, and the result of this reduction is $x^2y-x(2xy+x+3)/2$, or $-x^2/2-3x/2$. A polynomial can therefore be reduced by reducing its terms beginning with the greatest and proceeding downward. Similarly, a polynomial can be reduced with respect to a set of polynomials by reducing in turn with respect to each element in that set. A polynomial is fully reduced if none of its terms can be reduced (Lichtblau 1996[2]).
- QUOTE: In logic, a term is a variable, constant, or the result of acting on variables and constants by function symbols.
- ↑ Becker, T. and Weispfenning, V. Gröbner Bases: "A Computational Approach to Commutative Algebra". New York: Springer-Verlag, 1993.
- ↑ Lichtblau, D. "Grobner Bases in Mathematica 3.0." Mathematica J. 6, 81-88, 1996.
2020c
- (Encyclopedia of Mathematics, 2020) ⇒ https://www.encyclopediaofmath.org/index.php/Term Retrieved: 2020-03-26.
- QUOTE: A linguistic expression used to denote objects. For example, the expressions $1,0+1,\lim_{x\to0}(\sin x)/x$ are distinct terms denoting the same object. A term can contain free variables (parameters) (cf. Free variable), fixation of whose values uniquely defines some object according to the semantic laws of the language — the value of the term for the given values of its free variables. Thus, if $f$ is a variable with as values integrable real-valued functions, and $x$, $a$, $b$ are variables whose values are real numbers, then the expression $\int_a^bf(x)dx$ is a term with three parameters $a$, $b$, $f$, which denotes a well-defined real number for each set of values of the parameters ($x$ in this term is a bound variable). Syntactically, terms are characterized by the fact that they can be substituted for variables in other expressions of the language — terms or formulas, yielding new terms or formulas, respectively.
In a formalized language there exist formal rules, independent of the semantics of the language, for constructing terms and distinguishing free variables in them. In many-sorted languages there are also rules for determining the sorts of the terms which occur.
- QUOTE: A linguistic expression used to denote objects. For example, the expressions $1,0+1,\lim_{x\to0}(\sin x)/x$ are distinct terms denoting the same object. A term can contain free variables (parameters) (cf. Free variable), fixation of whose values uniquely defines some object according to the semantic laws of the language — the value of the term for the given values of its free variables. Thus, if $f$ is a variable with as values integrable real-valued functions, and $x$, $a$, $b$ are variables whose values are real numbers, then the expression $\int_a^bf(x)dx$ is a term with three parameters $a$, $b$, $f$, which denotes a well-defined real number for each set of values of the parameters ($x$ in this term is a bound variable). Syntactically, terms are characterized by the fact that they can be substituted for variables in other expressions of the language — terms or formulas, yielding new terms or formulas, respectively.
2020d
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Addition#addendy Retrieved:2020-3-26.
- The sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example,
[math]\displaystyle{ \sum_{k=1}^5 k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55. }[/math]
The numbers or the objects to be added in general addition are collectively referred to as the terms [1], the addends[2] or the summands; [3] this terminology carries over to the summation of multiple terms. This is to be distinguished from factors, which are multiplied. Some authors call the first addend the augend. [4] In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is rarely used, and both terms are generally called addends.[5] All of the above terminology derives from Latin. “Addition” and “add” are English words derived from the Latin verb addere, which is in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root "to give"; thus to add is to give to. Using the gerundive suffix -nd results in "addend", "thing to be added".[6] Likewise from augere "to increase", one gets "augend", "thing to be increased".
- The sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example,
- ↑ Department of the Army (1961) Army Technical Manual TM 11-684: Principles and Applications of Mathematics for Communications-Electronics. Section 5.1
- ↑ Shmerko, V.P.; Yanushkevich [Ânuškevič], Svetlana N. [Svitlana N.]; Lyshevski, S.E. (2009). Computer arithmetics for nanoelectronics. CRC Press. p. 80.
- ↑ Hosch, W.L. (Ed.). (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. p. 38
- ↑ and
- ↑ Schwartzman p. 19
- ↑ "Addend" is not a Latin word; in Latin it must be further conjugated, as in numerus addendus "the number to be added".
2020e
- (Simple Wikipedia, 2020) ⇒ https://simple.wikipedia.org/wiki/Term_(mathematics) Retrieved:2020-3-26.
- In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables. Terms are separated by a + or - sign in an overall expression. For example, in
$3 + 4x + 5yzw$
$3$, $4x$, and $5yzw$ are three separate terms.
In the context of polynomials, term can mean a monomial with a coefficient. To 'combine like terms' in a polynomial is the basic operation of making it a linear combination of distinct monomials. For example,
$3 x + 2x^2 + 5x + 1 = 2x^2 + (3+5)x + 1 = 2x^2 + 8x + 1$, with like terms collected.
A series is often represented as the sum of a sequence of terms.
In general mathematical use, however, term is not limited to additive expressions. Individual factors in an expression representing a product are multiplicative terms. Indeed, individual elements of any mathematical expression may be referred to as "terms".
- In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables. Terms are separated by a + or - sign in an overall expression. For example, in
2009
- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Term_(mathematics)
- In Elementary Mathematics, a term is either a single number or variable, or the product of several numbers and/or variables separated from another term by a + or - sign in an overall expression.