Variable (mathematics)

In mathematics, a variable (from Latin ) is a symbol, typically a letter, that refers to an unspecified mathematical object. One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a set, such as the set of real numbers. The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables and and require that the value of the square of is twice the square of , which in algebraic notation can be written . A definitive proof that this relationship is impossible to satisfy when and are restricted to nonzero integers isn't obvious, but it has been known since ancient times and has had a big influence on mathematics ever since. Originally, the term variable was used primarily for the argument of a function, in which case its value could be thought of as varying within the domain of the function. This is the motivation for the choice of the term. Also, variables are used for denoting values of functions, such as the symbol in the equation , where is the argument and denotes the function itself.…

In mathematics, a variable (from Latin ) is a symbol, typically a letter, that refers to an unspecified mathematical object. One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a set, such as the set of real numbers. The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables and and require that the value of the square of is twice the square of , which in algebraic notation can be written . A definitive proof that this relationship is impossible to satisfy when and are restricted to nonzero integers isn't obvious, but it has been known since ancient times and has had a big influence on mathematics ever since. Originally, the term variable was used primarily for the argument of a function, in which case its value could be thought of as varying within the domain of the function. This is the motivation for the choice of the term. Also, variables are used for denoting values of functions, such as the symbol in the equation , where is the argument and denotes the function itself. A variable may represent an unspecified number that remains fixed during the resolution of a problem; in which case, it is often called a parameter. A variable may denote an unknown number that has to be determined; in which case, it is called an unknown; for example, in the quadratic equation , the variables , , are parameters, and is the unknown. Sometimes the same symbol can be used to denote both a variable and a constant, that is a well defined mathematical object. For example, the Greek letter generally represents the number , but has also been used to denote a projection. Similarly, the letter often denotes Euler's number, but has been used to denote an unassigned coefficient for quartic function and higher degree polynomials. Even the symbol has been used to denote an identity element of an arbitrary field. These two notions are used almost identically, therefore one usually must be told whether a given symbol denotes a variable or a constant. Variables are often used for representing matrices, functions, their arguments, sets and their elements, vectors, spaces, etc. In mathematical logic, a variable is a symbol that either represents an unspecified constant of the theory, or is being quantified over. x^3 - 2x^2 + 10x -1, would be written in Diophantus's syncopated notation as: \Kappa^{\upsilon} \overline{\alpha} \; \zeta \overline{\iota} \;\, \pitchfork \;\, \Delta^{\upsilon} \overline{\beta} \; \Mu \overline{\alpha} \,\; In the 7th century BC, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuṭasiddhānta. One section of this book is called "Equations of Several Colours". Greek and other ancient mathematical advances, were often trapped in long periods of stagnation, and so there were few revolutions in notation, but this began to change by the early modern period. (\forall \epsilon >0) (\exists \eta >0) (\forall x) \;|x-a| in which none of the five variables is considered as varying. This static formulation led to the modern notion of variable, which is simply a symbol representing a mathematical object that either is unknown, or may be replaced by any element of a given set (e.g., the set of real numbers). ax^3+bx^2+cx+d=0 is interpreted as having five variables: four, , which are taken to be given 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 refers commonly to the arguments of the functions. This is typically the case in sentences like "function of a real variable", " is the variable of the function ", " is a function of the variable " (meaning that the argument of the function is referred to by the variable ). In the same context, variables that are independent 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 of the strong relationship between polynomials and polynomial functions, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates. 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. 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 is a summation variable which designates in turn each of the integers (it is also called index because its variation is over a discrete set of values) while is a parameter (it does not vary within the formula). In the theory of polynomials, a polynomial of degree 2 is generally denoted as , where , and are called coefficients (they are assumed to be fixed, i.e., parameters of the problem considered) while is called a variable. When studying this polynomial for its polynomial function this stands for the function argument. When studying the polynomial as an object in itself, is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status. PV = Nk_\text{B}T. This equation would generally be interpreted to have four variables, and one constant. The constant is , the Boltzmann constant. One of the variables, , the number of particles, is a positive integer (and therefore a discrete variable), while the other three, , and , for pressure, volume and temperature, are continuous variables. One could rearrange this equation to obtain as a function of the other variables, P(V, N, T) = \frac{Nk_\text{B}T}{V}. Then , as a function of the other variables, is the dependent variable, while its arguments, , and , are independent variables. One could approach this function more formally and think about its domain and range: in function notation, here is a function P: \mathbb{R}_{>0} \times \mathbb{N} \times \mathbb{R}_{>0} \rightarrow \mathbb{R} . However, in an experiment, in order to determine the dependence of pressure on a single one of the independent variables, it is necessary to fix all but one of the variables, say . This gives a function P(T) = \frac{Nk_\text{B}T}{V}, where now and are also regarded as constants. Mathematically, this constitutes a partial application of the earlier function . This illustrates how independent variables and constants are largely dependent on the point of view taken. One could even regard as a variable to obtain a function P(V, N, T, k_\text{B}) = \frac{Nk_\text{B}T}{V}. y = ax^2 + bx + c, where , , , and are all considered to be real. The set of points in the 2D plane satisfying this equation trace out the graph of a parabola. Here, , and are regarded as constants, which specify the parabola, while and are variables. Then instead regarding , and as variables, we observe that each set of 3-tuples corresponds to a different parabola. That is, they specify coordinates on the 'space of parabolas': this is known as a moduli space of parabolas.

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