Imagine that you were writing an elementary book, for example for high school learners, and at the beginning you had a glossary where you wanted to write the definitions for common mathematical words (e.g. numerical expression, algebraic expression, variable, equation, etc.) to help the reader make sense of those words when they appeared throughout the text. In this setting, what would you write to explain/define what a mathematical formula is?
Imagine that you wanted to put an entry for mathematical formula in a table like in these screenshots, taken from the book Algebra and Trigonometry: Structure and Method, Book 2 which you can read here, if you have create an account.
I would like to avoid circular definitions.
The term formula seems to raise confusion with some frequency and this can be seen by the number of Math Stackexchange questions that appeared over the years:
After some investigation it seems that people use the term mathematical formula to mean (mainly) two things:
- Any mathematical sentence that uses symbols (e.g. equations, inequations/inequalities, identities). Which makes equations a kind of formula.
- A rule for expressing a subject in terms of some other variables (as stated in this comment). Or a procedure to compute some value of interest. Usually written as two expressions connected by an equals sign. Which makes formulas a kind of equation.
Mathworld seems to define a formula as the first. Many books use the term formula for the rules used to compute areas and volumes, and there is also the quadratic formula which provides the solution(s) to a quadratic equation. These are instances of the second definition. Less frequently, I've also seen people call formula to any string of mathematical symbols, making an expression a kind of formula. Another example is the book Introduction to Algebra that defines formula as a synonym for expression in page 23 (according to this comment).
EDIT After reading some of the comments I feel the need reword the first part of my original question to:
Imagine that you were writing a book that revisits elementary mathematics subjects/areas but from an advanced standpoint, for example for advanced high school learners or people preparing to an undergraduate mathematics heavy course. One example of such book might be Serge Lang's Basic Mathematics.
Now consider that in the beginning of this book you had a glossary where you wanted to write the definitions for common mathematical words (e.g. numerical expression, algebraic expression, variable, equation, etc.) to help the reader make sense of those words when they appeared throughout the text.
In this setting, what would you write to explain/define what a mathematical formula is?
I personally think that the answers given up to the point in time of this edit fit both the original question and this reword, but would prefer if new answers take this revision into consideration.
EDIT 2 I accepted @ryang's answer because it has all the information requested by the Question, but also found Charles Wells' work, which helped me interiorize stuff: