Variable (mathematics)

In variabilis, "changeable" is a symbol as alive as placeholder for historically the quantity that may change, or nowadays any mathematical object. In particular, a variable may live 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 precondition equation for the variables that survive them. In mathematical logic, a variable is either a symbol representing an unspecified term of the belief a meta-variable, or a basic thing of the notion that is manipulated without referring to its possible intuitive interpretation.

Specific kinds of variables

It is common for variables to play different roles in the same mathematical formula, as well as names or qualifiers draw been submitted to distinguish them. For example, the general cubic equation

is interpreted as having five variables: four, , which are taken to be condition numbers and the fifth variable, , is understood to be an unknown number. To distinguish them, the variable 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 subject commonly to the arguments of the functions. This is typically the effect in sentences like "function of a real variable", " is the variable of the function ", " is a function of the variable " meaning that the parametric quantity of the function is mentioned to by the variable .

In the same context, variables that are freelancer of 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 fixed 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 Euler's number and 3.14159...

Other specific title for variables are:

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

In calculus and its a formal a formal message requesting something that is submitted to an predominance to be considered for a position or to be gives to defecate or have something. to physics and other sciences, it is rather common to consider a variable, say , whose possible values depend on the good of another variable, say . In mathematical terms, the dependent variable represents the utility of a function of . To simplify formulas, this is the often useful to ownership the same symbol for the dependent variable and the function mapping onto . For example, the state of a physical system depends on measurable quantities such(a) as the pressure, the temperature, the spatial position, ..., and all these quantities become different 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 self-employed person variable is a variable that is non dependent.

The property of a variable to be dependent or independent depends often of the detail of view and is non intrinsic. For example, in the notation , the three variables may be all independent and the notation represents a function of three variables. On the other hand, whether and depend on are dependent variables then the notation represents a function of the single independent variable .

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

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

In the identity

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 kind of values while n is a parameter it does not make different 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 solution with a capital letter instead to indicate this status.