Posted in Algebra

What is a variable (in mathematics)?

A variable is often talked about in mathematics class as  a symbol that stands for numbers. As Prof Zalman Usiskin reminds us, a variable need not only stand for numbers. In Geometry it could stand for a point in Logic, a proposition, in Analysis, a function, in Linear Algebra, a matrix or a vector and even an operation in Higher mathematics. 

Here’s Usiskin on variables:

Many students think all variables are letters that stand for numbers. Yet the values a variable takes are not always numbers, even in high school mathematics. In geometry, variables often represent points, as seen by the use of the variables A, B, and C when we write “if AB = BC, then ΔABC is isosceles.” In logic, the variables p and q often stand for propositions; in analysis, the variable f often stands for a function; in linear algebra, the variable A may stand for a matrix or the variable v for a vector; and in higher algebra the variable * may represent an operation. The last of these demonstrates that variables need not be represented by letters.

Students also tend to believe that a variable is always a letter. This view is supported by many educators, for
3 + x = 7 and 3 + ? = 7
are usually considered algebra, whereas
3 + ___ = 7 and 3 + ? = 7
are not, even though the blank and the question mark are, in this context of desiring a solution to an equation, logically equivalent to the x and the !.

In summary, variables have many possible definitions, referents, and symbols. Trying to fit the idea of variable into a single conception oversimplifies the idea and in turn distorts the purposes of algebra.

Prof Usiskin also classified the various ways in which equation is used in algebra:  (1) a formula, (2) an equation (or open sentence) to solve, (3) an identity, (4) a property, and (5) an equation of a function (not to be solved). These different names reflect different uses to which the idea of variable is put.

See also: Variables, Constants and Parameters