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Term (mathematics)

A mathematical term is a single element of an expression—such as a number, variable, or product of them—used in algebra, polynomials, sequences, and formal logic; terms can be combined, compared, and classified.

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Overview: In mathematics a term is a single, self-contained expression that appears as a component of a larger expression. In elementary algebra a term is usually an operand: a number, a variable, or a product of numbers and variables (for example, 3x^2, -5, or a b c). When terms are joined by addition or subtraction they are often called summands or addends; when they appear in a product they are regarded as factors.

Characteristics and common forms

Terms may include a coefficient (a numerical factor), one or more variables, and exponents. In polynomials each term is a monomial: for instance, in 4x^3 - 2xy + 7 the three terms are 4x^3, -2xy, and 7. Terms are classified by degree (the sum of exponents on variables) and by whether they are like terms: like terms have identical variable parts and may be combined by addition or subtraction.

Uses and operations

Common operations on terms include combining like terms, factoring (expressing a term as a product of simpler factors), and collecting terms to simplify expressions. In a series or sequence, the word term denotes a particular element, often indexed (a_n denotes the n-th term). In summation notation the index runs over terms to produce a total.

In formal logic and symbolic systems

In first-order logic and formal languages a term denotes an object-level expression built from variables, constants, and function symbols (for example f(x,y)). These logical terms differ from algebraic terms because they represent syntactic names for objects rather than numerical values.

Distinctions and notable facts

  • Term vs factor: a term can be a product of factors; factors multiply to form terms.
  • Term vs coefficient: the coefficient multiplies the variable part of a term.
  • Like terms can be combined; unlike terms cannot be added algebraically without further manipulation.

The word "term" has long usage in mathematics and logic; for concise definitions and examples see an introductory algebra text or an entry on operands and operators, for example operand.

Colloquial explanation

The term "term" is used colloquially for everything that carries a meaning. In the narrower sense, it refers to mathematical entities that can in principle be calculated, at least if you have assigned values to the variables they contain. For example, is (x+y)^{2}a term, because if you assign a value to the variables xand contained in yit, the term also gets a value. Instead of numbers, other mathematical objects may be considered here, for example is (p_{1}\vee \neg p_{2})\wedge p_{3}a term that obtains a value if one assigns a truth value to the Boolean variables .p_{1},p_{2},p_{3} However, in the normal case (one-sort logic), the exact mathematical definition makes no reference to the possible value assignments, as explained below.

Roughly, we can say that a term is a side of an equation or relation, such as an inequality. The equation or relation itself is not a term, it consists of terms.

The following operations can usually be performed with terms:

  • (to do this, first calculate the "inner" functions and then the outer): (2+3)^{2}=5^{2}=25
  • transform according to certain calculation rules: (x+y)^{2}=x^{2}+2xy+y^{2}by applying the distributive law and some other "allowed" rules.
  • with each other if relations are defined for the matching types: 2xy\leq x^{2}+y^{2}
  • into each other (often a term is substituted for a variable of another term). A special form of substitution is substitution, where a term with variables is replaced by another term with variables (usually a single variable): (x+y)^{2}arises from z^{2}by substituting zfor x+y.

Terms or subterms are often named according to their substantive meaning. In the term {\tfrac {1}{2}}mv^{2}+mgh, which in physics describes the total energy of a mass point, the first summand is called the "kinetic energy term" and the second the "potential energy term". Characteristic properties are also often used for naming. For example, the "quadratic term" in x^{3}+7x^{2}-2x+1means the subterm , 7x^{2}because this is the subterm containing the variable xin squared form.

Applications

If one forms a term with variables, then one often intends in applications a replacement of these variables by certain values, which originate from a certain basic set or definition set. For the notion of the term itself, the specification of such a set according to the above formal definition is not necessary. Then one is no longer interested in the abstract term, but in a function defined by this term in a certain model.

Thus, a rule of thumb for calculating the stopping distance (braking distance plus reaction distance) of a car in meters \left({\tfrac {x}{10}}\right)^{2}+\left({\tfrac {x}{10}}\cdot 3\right). This string is a term. We intend to substitute for x the speed of the car in km per hour, to use the value that the term then takes as the braking distance in meters. For example, if a car is traveling 160 km/h, the formula yields \left({\tfrac {160}{10}}\right)^{2}+\left({\tfrac {160}{10}}\cdot 3\right)a stopping distance of 304 m.

We use the term here to define the assignment rule of a function f\colon \mathbb{R} _{0}^{+}\to \mathbb{R} _{0}^{+}, x\mapsto \left({\tfrac {x}{10}}\right)^{2}+\left({\tfrac {x}{10}}\cdot 3\right).

Terms themselves are neither true nor false and have no values. Only in a model, i.e. with specification of a basic set for the occurring variables, terms can assume values.

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