Term (mathematics)

In mathematics, a term referred to a meaningful expression of numbers, variables, symbols for mathematical operations and parentheses can contain. Terms are the words or phrases syntactically correct educated in the formal language of mathematics.

In practice, the term is often used to talk about individual components of a formula or a larger term. So you can for example speak for the linear function of a linear term and a constant term.

• 5.1 expressions
• 5.2 Example

Colloquial Statement

The term "Term " is colloquially used for anything that carries meaning. In a narrower sense, mathematical shapes are meant that you can calculate in principle, at least if you have assigned to the variables contained in the values ​​. For example, a term, because you add the variables contained therein, and a value, so the term is given a value. Instead of numbers here, other values ​​can also be considered, it is about a term which is given a value when the Boolean variable assigns a truth value. However, the precise mathematical definition makes no reference to the possible value assignments, as will be explained below.

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

With terms usually following operations can be performed:

• Calculate (this is expected only from the "inner" functions and then the outer ):
• According to certain rules of calculation transform: by using the distributive law and some other "permitted " rules.
• Compare with each other if ratios of matching types are defined:
• Insert into each other (often a term is used instead of a variable of a different term). A special form of the establishment is the substitution, in which a term with variables by another expression with variable (usually a single variable ) is replaced: results from by replacing by.

Frequently terms or subterms are named after their substantive significance. The term, which describes the physics, the total energy of a particle is called the first term "Term of kinetic energy " and the second " term in the potential energy." Often characteristics are used for naming. It is meant, as this term is the part that contains the variable in the squared shape with the " quadratic term " in the term portion.

Formal definition

The exact mathematical definition of a term, as given in mathematical logic, renames rules by which terms are established. A term is then any term which arises due to application of such system:

• Each variable is a term
• Each constant symbol is a Term
• Are Terme and is an - ary function symbol, then a Term

Comments

• Looking at the designated Addition, according to the above, a formal definition is a term that is not. Nevertheless, one prefers the more easily readable form, the latter is an alternative, advantageous way of writing the correct term. Thus, the string is a name for a term that is a metalinguistic expression for a Term As long as it is clear that one could such strings at any time translate them back into the formally correct spelling, if you wanted that arise here no difficulties.
• Some functions ( for example, the power function, multiplication with variables) are instead represented by its own symbol function by positioning the terms to each other (eg or )
• In nested parenthesis settlements sometimes [] and { } are used to make the links between the brackets clearly, for example,
• There are also clip- free notations such as the Polish notation, these are usually not so easy to read but.
• From a possible insertion of values ​​to the variables, as occurred in the above colloquial description, is here no question. A term is here a purely syntactic term, since it has to satisfy only certain construction rules. Terms obtained in retrospect a semantic meaning, by restricting the possible values ​​of variables in the so-called models. The terms and are initially different as strings. Considering these terms but in the model of the real numbers, it is clear that they always take the same values. The term equality is to be understood as that equality for all is. For other models that may well be wrong, such as for the set of matrices.

Example

Is a term, because

• X and y are terms (as variables),
• 4 is a term ( a constant ),
• Xy is a term (actually " multiply (x, y) ")
• Is a term (division symbol is the fraction line, actually " divide ( multiply (x, y ), 4) " )

Applications

If one forms an expression with variables, so it is intended for applications often replacing these variables by certain values ​​that come from a certain basic quantity or set of definitions. Specifying an amount according to the above, a formal definition is not necessary to the concept of the term itself. You no longer interested in the abstract term but for a defined by this term role in a specific model.

That is a formula to calculate the stopping distance ( plus braking distance pathway ) of a car in meters. This string is a Term We intend to use for the car's speed in miles per hour to use the value of the term then assumes, as stopping distance in meters. If a car, for example, 160 km / h runs, the formula provides a stopping distance of 304 m.

We use the term here to define the allocation rule of a function.

Terms themselves are neither true nor false and have no values ​​. Only in a model, that is stating a basic amount for the existing variables, terms can take on values ​​.

Algebraic transformations

Long, complicated terms can often be simplified by calculation rules applying to them which leave the value of the term unchanged, such as the commutative, associative law or distributive:

Multiplying

Apply commutative

The term of the term does not provide such forming operations as defined above, are in each case by different terms. With these algebraic transformations has always meant that, do not change the values ​​that can accept a term for choosing a particular base set by these transformations. That depends on the basic quantity from! Thus, the above transformations only in such basic quantities correctly, in which the laws used such as the commutative law apply.

Such algebraic transformations are still called term transformations, as one passes by force in the agreed basic set of rules one term to another without changing its possible values ​​. There are thus the following objectives:

• Simplification of expressions
• Inflating of terms to produce the desired structures, such as, for example, in the quadratic supplement
• Dissect out the desired part, such as in terms of Cardano formula:

Demarcation expressed

Expressions

An expression is a term as a formal string; their structure is defined according to a logic, such as predicate logic. In the first order predicate logic with equality is defined:

• Are terms, it is an expression.

• Are Terme and is a k- ary relation symbol, is an expression.

This can build up by repeated application of this education laws arbitrarily complicated expressions. By this definition, one can roughly Terme than describe what stand on the side of an equation or can be used in a relation terms are exactly those elements of expressions.

The exact definition of the term depends on the considered logic, predicate logic in the second step one takes, for example, still add the insertion of terms in relations variables and quantification over relations.

Example

For a description of the real numbers to use for the multiplication sign and the link for the inequality the relation symbol, also constants such as 0,1,2, ... are variables, so by definition are also

By the definition of the expression are

Expressions, since the first character string is the equality of two terms, the second, a relation, were inserted into the two terms. This also

An expression, and finally

This expression is true in the model of the real numbers. It is important to understand that the above construction of the expression is not a proof; it is merely the formation of a string according to certain rules. True or false can be only in a model of an accompanying statement, and there they may possibly be proved. Although this statement is known to be false in the model of rational numbers.