variable
In algebra, a symbol (usually a letter) standing in for an unknown numerical value in an equation. Commonly used variables include x and y (real-number unknowns), z (complex-number unknowns), t (time), r (radius), and s (arc length). Variables should be distinguished from coefficients, fixed values that multiply powers of variables in polynomials and algebraic equations. In the quadratic equation ax2 + bx + c = 0, x is the variable, and a, b, and c are coefficients whose values must be specified to solve the equation. In translating word problems into algebraic equations, quantities to be determined can be represented by variables.
Control Values
Variables are widely used to hold control values that keep track of something. For example, the C statement FOR (X=0; X<5; X++) performs the instructions following the statement within open and closed curly braces ({ and }) five times, and X is keeping track of that number of iterations. X is a variable set to zero (x=0), incremented by 1 (x++) and compared to 5 (x<5). The reason it is less than 5 (<5) is because we started with 0.
The Equals Sign
Variables are usually assigned with an equal sign. Numbers are unquoted; for example: COUNTER = 1 places the digit 1 in the variable COUNTER. Character data (text) requires quotes; for example: PRODUCT = "abc1234". In some languages, the type of data must be declared before it is assigned; for example, in C/C++, the statement INT COUNTER creates a variable that holds only whole numbers (integers).
Local and Global
A local variable is one that is referenced only within the subprogram, function or procedure it was defined in. A global variable can be used by the entire program. See undefined variable and local variable.
variable
variable [′ver·ē·ə·bəl]
| (programming) | variable - (Sometimes "var" /veir/ or /var/) A named memory
location in which a program can store intermediate results and
from which it can read it them. Each programming language
has different rules about how variables can be named, typed,
and used. Typically, a value is "assigned" to a variable in
an assignment statement. The value is obtained by
evaluating an expression and then stored in the variable. For
example, the assignment x = y + 1 means "add one to y and store the result in x". This may look like a mathematical equation but the mathematical equality is only true in the program until the value of x or y changes. Furthermore, statements like x = x + 1 are common. This means "add one to x", which only makes sense as a state changing operation, not as a mathematical equality. The simplest form of variable corresponds to a single-word of memory or a CPU register and an assignment to a load or store machine code operation. A variable is usually defined to have a type, which never changes, and which defines the set of values the variable can hold. A type may specify a single ("atomic") value or a collection ("aggregate") of values of the same or different types. A common aggregate type is the array - a set of values, one of which can be selected by supplying a numerical index. Languages may be untyped, weakly typed, strongly typed, or some combination. Object-oriented programming languages extend this to object types or classes. A variable's scope is the region of the program source within which it represents a certain thing. Scoping rules are also highly language dependent but most serious languages support both local variables and global variables. Subroutine and function formal arguments are special variables which are set automatically by the language runtime on entry to the subroutine. In a functional programming language, a variable's value never changes and change of state is handled as recursion over lists of values. |
Variable
a fundamental concept of mathematics and logic. Beginning with the works of P. de Fermat, R. Descartes, I. Newton, G. W. von Leibniz, and other founders of higher mathematics, a variable was understood as a quantity that can vary—in the process of this variation it takes on different values. Variables were thus contrasted with constants—numbers or other quantities each of which has a unique, completely defined value. As mathematics developed and work proceeded on its foundations, great pains were taken to exclude such concepts as processes and variation of quantities from the vocabulary of mathematics as being extramathematical. The variable consequently came to be interpreted as a symbol for an arbitrary element of the domain of objects under consideration—for example, the domain of natural numbers or of real numbers. In other words, a variable was now seen as a generic name for the entire domain; constants, on the other hand, were proper names for numbers or other specific objects in the domain under consideration.
This revision of the concept of variable was closely connected with the reorganization of mathematics on the foundation of set theory, a process that was completed in the late 19th century. For all the simplicity and naturalness of this reorganization, it was essentially based on the abstraction of actual infinity. This abstraction permitted arbitrary infinite sets to be considered as “given” (“completed,” “ready,” “actual”) objects, to which all the methods of classical logic could be applied without regard for the incompletedness and the fundamental impossibility of completing the process of the formation of such a set. The difficulties in solving the logical problems associated with the adoption of this abstraction make understandable the partial rehabilitation of the old concepts of variable quantities. In constructing mathematical theories, the representatives of some schools, such as mathematical intuitionism and the constructive trend, have preferred to make do with the weaker, but logically less vulnerable, abstraction of potential realizability. This point of view associates infinite sets with the concepts of the processes of the sets’ generation—processes that go as far as desired but never end. When investigating the problem of the consistency of different fields of mathematics, most mathematicians and logicians in effect adopt this position.
In formalized languages (calculi, or formal systems) of mathematical logic, the term “variable” is applied to any symbol of a strictly fixed type that under certain conditions permits an expression of the given calculus to be substituted for it. Such variables are called free variables. An example is the variable x in the inequality x > 5. If, say, the numeral (that is, the symbol of the number) 7 is substituted for x, the inequality becomes a true statement; if the numeral 2 is substituted, the inequality becomes a false statement. What are called bound variables do not mean anything in themselves but perform purely syntactical functions; when certain elementary precautions are observed, bound variables can be “renamed”—that is, they can be replaced by other variables. An example is the variable y in the expressions
in the interpretations of which y does not appear and can be replaced by any other variable. Thus, the first expression, which can be read “the sum of the integers from 5 through 25,” can be replaced by
The second expression, which might be read “all numbers have the property p,” can be replaced by ∀tP(t). Individual, proposition, predicate, function, number, and other types of variables are distinguished. They can be replaced, in accordance with special substitution rules, by symbols for, respectively, the objects from the domain under consideration (“terms”), specific statements, predicates, functions, numbers, and so on. Thus, a variable in a formula can be meaningfully understood as an empty place equipped with an indication that the place can be filled—the variable is like a container for a strictly defined piece of merchandise.
Free occurrences of variables in the expressions of meaningful scientific theories and in the formulas of logical and mathematical calculi correspond to the use of indefinite pronouns in ordinary speech and permit of different interpretations. One type of interpretation, which corresponds to the use of a variety of substitution procedures, is the predicate interpretation: the formula A (x1, … xn) of some calculus is interpreted as an n-place predicate. The same formula can also be interpreted as a proposition, or statement, specifically as the proposition ∀x1 … ∀xnA (x1, …, xn), which is a “closure” of the formula—this is the interpretation of universality, which is commonly used, for example, in formulating the axioms of various scientific theories. Finally, to free variables there can be ascribed values constant within some context—for example, the context of a conclusion from a given set of formulas. The variables are then called the parameters of the context, and we speak of the conditional interpretation. For example, the variable x in the expression cos x, taken in isolation, has a predicate interpretation. In the identity sin2x + cos2x = 1 it has the interpretation of universality. In the equation cos x = 1 (in the process of solving this equation, when the variable is called an “unknown”) it has the conditional interpretation.
Thus, at different levels of formalization the concept of the variable acts as a refinement of means commonly used in ordinary spoken languages—such as indefinite pronouns and indefinite articles—and of various methods of employing these means.
REFERENCES
Kleene, S. C. Vvedenie ν metamatematiku, subsecs. 31, 32, 45. Moscow, 1957. (Translated from English.)Church, A. Vvedenie ν matematicheskuiu logiku, vol. 1, subsecs. 02, 04, 06. Moscow, 1960. (Translated from English.)
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