Variable (mathematics)

Not to be confused with Variable (computer science).

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol, typically a letter, that holds a place for constants, often numbers.[1][2][3][4][5][6] One say colloquially that the variable represents or denotes the object, and that the object is the value of the variable.

Originally, the term "variable" was used primarily for the argument of a function, in which case its value can vary in 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 y in

A variable may represent a 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 π generally represents the number π, but has also been used to denote a projection. Similarly the letter e 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.[7]

Variables are often used for representing matrices, functions, their arguments, sets and their elements, vectors, spaces, etc.[8]

In mathematical logic, a variable is either a symbol representing an unspecified constant of the theory, or a variable which is being quantified over.[9][10][11]

  1. ^ Sobolev, S.K. (originator). Individual variable. Springer. ISBN 1402006098. Retrieved September 5, 2024. {{cite book}}: |website= ignored (help)
  2. ^ Beckenbach, Edwin F (1982). College algebra (5th ed.). Wadsworth. ISBN 0-534-01007-5.
  3. ^ Landin, Joseph (1989). An Introduction to Algebraic Structures. New York: Dover Publications. p. 204. ISBN 0-486-65940-2.
  4. ^ Ely, Robert; Adams, Anne E. (February 22, 2012). "Unknown, placeholder, or variable: what is x?". Mathematics Education Research Journal. 24 (1): 19–38. Bibcode:2012MEdRJ..24...19E. doi:10.1007/s13394-011-0029-9 – via Springer Science+Business Media.
  5. ^ Oxford English Dictionary, s.v. “variable (n.), sense 1.a,” March 2024. "Mathematics and Physics. A quantity or force which, throughout a mathematical calculation or investigation, is assumed to vary or be capable of varying in value."
  6. ^ Collins English Dictionary. Variable, (noun) mathematics a. an expression that can be assigned any of a set of values b. a symbol, esp x, y, or z, representing an unspecified member of a class of objects
  7. ^ "ISO 80000-2:2019" (PDF). Quantities and units, Part 2: Mathematics. International Organization for Standardization. Archived from the original on September 15, 2019. Retrieved September 15, 2019.
  8. ^ Stover & Weisstein.
  9. ^ van Dalen, Dirk (2008). "Logic and Structure" (PDF). Springer-Verlag (4th ed.): 57. doi:10.1007/978-3-540-85108-0. ISBN 978-3-540-20879-2.
  10. ^ Feys, Robert; Fitch, Frederic Brenton (1969). Dictionary of symbols of mathematical logic. Amsterdam: North-Holland Pub. Co. LCCN 67030883.
  11. ^ Shapiro, Stewart; Kouri Kissel, Teresa (2024), "Classical Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved September 1, 2024