The specialized use of the term variable should not be the first one given in an algebra text or dictionary, mathematical or not. We can talk about numbers and quantities, and among them identify those which are changing or varying, and those which are constant, known, unknown, given, confidential and so on. Here a number and quantity which may vary, or take many values in the circumstances of interest, is called a variable. We can talk about variables without using the shorthand notation, that is, letters and symbols, employed in algebra. Examples follow below.
Formulas use shorthand notation, symbols or letters, to represent numbers and quantities.
This suggests that when a symbol or letter is the shorthand notation for a number or quantity which may vary, we may also call that symbol or letter a variable. But this symbol free notion clarifies and refines the concept of a variable in mathematics. We can talk about numbers and quantities. The words or adjectives used here may be used in mathematics after arithmetic. There is more to mathematics than just doing arithmetic.