Variable (mathematics)
In
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 any quadratic equation by substituting the numeric values of the coefficients of that equation for the variables that represent them in the quadratic formula. 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.
History
In ancient works such as Euclid's Elements, single letters refer to geometric points and shapes. In the 7th century, 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".[2]
At the end of the 16th century, François Viète introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple replacement. Viète's convention was to use consonants for known values, and vowels for unknowns.[3]
In 1637, René Descartes "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c".[4] Contrarily to Viète's convention, Descartes' is still commonly in use. The history of the letter x in math was discussed in an 1887 Scientific American article.[5]
Starting in the 1660s,
In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalized enough to deal with apparent paradoxes such as a nowhere differentiable continuous function. To solve this problem, Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion of limit by a formal definition. The older notion of limit was "when the variable x varies and tends toward a, then f(x) tends toward L", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula
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).
Notation
Variables are generally denoted by a single letter, most often from the
For example, a general quadratic function is conventionally written as , 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 , 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.[8]
Specific branches and applications of mathematics have specific
Specific kinds of variables
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
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
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
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
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.[9]
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.[10]
Examples
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 set of values) while n is a parameter (it does not vary within the formula).
In the theory of
Example: the ideal gas law
Consider the equation describing the ideal gas law,
One could rearrange this equation to obtain as a function of the other variables,
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
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
Moduli spaces
Considering constants and variables can lead to the concept of moduli spaces. For illustration, consider the equation for a parabola,
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.
Conventional variable names
- a, b, c, d (sometimes extended to e, f) for parameters or coefficients
- a0, a1, a2, ... for situations where distinct letters are inconvenient
- ai or ui for the i-th term of a sequence or the i-th coefficient of a series
- e for Euler's number
- f, g, h for functions (as in )
- i for the imaginary unit
- i, j, k (sometimes l or h) for varying unit vectors
- l and w for the length and width of a figure
- l also for a line, or in number theory for a prime number not equal to p
- n (with m as a second choice) for a fixed integer, such as a count of objects or the degree of an equation
- p for a prime number or a probability
- q for a prime power or a quotient
- r for a radius, a remainder or a correlation coefficient
- t for time
- x, y, z for the three Cartesian coordinates of a point in Euclidean geometry or the corresponding axes
- z for a complex number, or in statistics a normal random variable
- α, β, γ, θ, φ for angle measures
- ε (with δ as a second choice) for an arbitrarily small positive number
- λ for an eigenvalue
- Σ (capital sigma) for a sum, or σ (lowercase sigma) in statistics for the standard deviation[11]
- μ for a mean
See also
- Lambda calculus
- Observable variable
- Physical constant
- Propositional variable
References
- ^ Stover & Weisstein.
- ^ Tabak 2014, p. 40.
- ^ Fraleigh 1989, p. 276.
- ^ Sorell 2000, p. 19.
- ^ Scientific American. Munn & Company. September 3, 1887. p. 148.
- ^ Edwards Art. 4
- ^ Hosch 2010, p. 71.
- ^ Foerster 2006, p. 18.
- ^ Edwards Art. 5
- ^ Edwards Art. 6
- ^ Weisstein, Eric W. "Sum". mathworld.wolfram.com. Retrieved February 14, 2022.
Bibliography
- Edwards, Joseph (1892). An Elementary Treatise on the Differential Calculus (2nd ed.). London: MacMillan and Co.
- Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications (classics ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-13-165711-3.
- Fraleigh, John B. (1989). A First Course in Abstract Algebra (4th ed.). United States: Addison-Wesley. ISBN 978-0-201-52821-3.
- Hosch, William L., ed. (2010). The Britannica Guide to Algebra and Trigonometry. Britannica Educational Publishing. ISBN 978-1-61530-219-2.
- JSTOR 685170.
- ISBN 978-3-929979-53-4.
- JSTOR 985250.
- ISBN 978-0-19-285409-4.
- Stover, Christopher; Weisstein, Eric W. "Variable". In Weisstein, Eric W. (ed.). Wolfram MathWorld. Wolfram Research. Retrieved November 22, 2021.
- Tabak, John (2014). Algebra: Sets, Symbols, and the Language of Thought. Infobase Publishing. ISBN 978-0-8160-6875-3.