Променљива (математика)

У математици, променљива је услован наслов за скуп значења. Такође, променљива је број представљен словом који се добија када се од приказаног резултата бројевног израза одузме резултат свих бројева без променљиве. Свака променљива може постојати само у контексту, јер свака променљива је сама по себи асоцирана са датим скупом значења, изван којег она ништа не значи. Променљиве су инструменти логике који чине основицу савремене математике; оне су тамо, можда, најважнији прибор апстракције. Појам променљива је постао део математичког језика током развоја аналитичке геометрије. In particular, a variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.

Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the quadratic formula solves every quadratic equation by substituting the numeric values of the coefficients of the given equation for the variables that represent them. In mathematical logic, a variable is either a symbol representing an unspecified term of the theory (a meta-variable), or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation.

Нотација

Variables are generally denoted by a single letter, most often from the Latin alphabet and less often from the Greek, which may be lowercase or capitalized. The letter may be followed by a subscript: a number (as in x₂), another variable (x_i), a word or abbreviation of a word (x_total) or a mathematical expression (x_{2i + 1}). Under the influence of computer science, some variable names in pure mathematics consist of several letters and digits. Following René Descartes (1596–1650), letters at the beginning of the alphabet such as (a, b, c) are commonly used for known values and parameters, and letters at the end of the alphabet such as (x, y, z) are commonly used for unknowns and variables of functions. In printed mathematics, the norm is to set variables and constants in an italic typeface.

For example, a general quadratic function is conventionally written as $ax^{2}+bx+c\,$, where a, b and c are parameters (also called constants, because they are constant functions), while x is the variable of the function. A more explicit way to denote this function is $x\mapsto ax^{2}+bx+c\,$, which clarifies the function-argument status of x and the constant status of a, b and c. Since c occurs in a term that is a constant function of x, it is called the constant term.

Specific branches and applications of mathematics have specific naming conventions for variables. Variables with similar roles or meanings are often assigned consecutive letters or the same letter with different subscripts. For example, the three axes in 3D coordinate space are conventionally called x, y, and z. In physics, the names of variables are largely determined by the physical quantity they describe, but various naming conventions exist. A convention often followed in probability and statistics is to use X, Y, Z for the names of random variables, keeping x, y, z for variables representing corresponding better-defined values.

Специфичне врсте варијабли

It is common for variables to play different roles in the same mathematical formula, and names or qualifiers have been introduced to distinguish them. For example, the general cubic equation

$ax^{3}+bx^{2}+cx+d=0,$

is interpreted as having five variables: four, a, b, c, d, which are taken to be given numbers and the fifth variable, x, is understood to be an unknown number. To distinguish them, the variable x is called an unknown, and the other variables are called parameters or coefficients, or sometimes constants, although this last terminology is incorrect for an equation, and should be reserved for the function defined by the left-hand side of this equation.

In the context of functions, the term variable refers commonly to the arguments of the functions. This is typically the case in sentences like "function of a real variable", "x is the variable of the function f: x ↦ f(x)", "f is a function of the variable x" (meaning that the argument of the function is referred to by the variable x).

In the same context, variables that are independent of x define constant functions and are therefore called constant. For example, a constant of integration is an arbitrary constant function that is added to a particular antiderivative to obtain the other antiderivatives. Because the strong relationship between polynomials and polynomial function, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.

This use of "constant" as an abbreviation of "constant function" must be distinguished from the normal meaning of the word in mathematics. A constant, or mathematical constant is a well and unambiguously defined number or other mathematical object, as, for example, the numbers 0, 1, π and the identity element of a group. Since a variable may represent any mathematical object, a letter that represents a constant is often called a variable. This is, in particular, the case of e and π, even when they represents Euler's number and 3.14159...

Other specific names for variables are:

• An unknown is a variable in an equation which has to be solved for.
• An indeterminate is a symbol, commonly called variable, that appears in a polynomial or a formal power series. Formally speaking, an indeterminate is not a variable, but a constant in the polynomial ring or the ring of formal power series. However, because of the strong relationship between polynomials or power series and the functions that they define, many authors consider indeterminates as a special kind of variables.
• A parameter is a quantity (usually a number) which is a part of the input of a problem, and remains constant during the whole solution of this problem. For example, in mechanics the mass and the size of a solid body are parameters for the study of its movement. In computer science, parameter has a different meaning and denotes an argument of a function.
• Free variables and bound variables
• A random variable is a kind of variable that is used in probability theory and its applications.

All these denominations of variables are of semantic nature, and the way of computing with them (syntax) is the same for all.

Dependent and independent variables

Главни чланак: Dependent and independent variables

In calculus and its application to physics and other sciences, it is rather common to consider a variable, say y, whose possible values depend on the value of another variable, say x. In mathematical terms, the dependent variable y represents the value of a function of x. To simplify formulas, it is often useful to use the same symbol for the dependent variable y and the function mapping x onto y. For example, the state of a physical system depends on measurable quantities such as the pressure, the temperature, the spatial position, ..., and all these quantities vary when the system evolves, that is, they are function of the time. In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.

Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other) variables. An independent variable is a variable that is not dependent.

The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic. For example, in the notation f(x, y, z), the three variables may be all independent and the notation represents a function of three variables. On the other hand, if y and z depend on x (are dependent variables) then the notation represents a function of the single independent variable x.

Examples

If one defines a function f from the real numbers to the real numbers by

$f(x)=x^{2}+\sin(x+4)$

then x is a variable standing for the argument of the function being defined, which can be any real number.

In the identity

$\sum _{i=1}^{n}i={\frac {n^{2}+n}{2}}$

the variable i is a summation variable which designates in turn each of the integers 1, 2, ..., n (it is also called index because its variation is over a discrete set of values) while n is a parameter (it does not vary within the formula).

In the theory of polynomials, a polynomial of degree 2 is generally denoted as ax² + bx + c, where a, b and c are called coefficients (they are assumed to be fixed, i.e., parameters of the problem considered) while x is called a variable. When studying this polynomial for its polynomial function this x stands for the function argument. When studying the polynomial as an object in itself, x is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.

Литература

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Спољашње везе

• Khan Academy: Conceptual videos and worked examples
• Khan Academy: Origins of Algebra, free online micro lectures
• Algebrarules.com: An open source resource for learning the fundamentals of Algebra
• Polyvalued logic and Quantity Relation Logic
• forall x: an introduction to formal logic, a free textbook by P. D. Magnus.
• A Problem Course in Mathematical Logic, a free textbook by Stefan Bilaniuk.