A formula is a claim; an expression is just a thing
Scientists pack relations into formulas, mathematical or chemical, so quantities talk to each other in symbols. Expressions name objects; formulas assert equalities or inequalities about them, the way sentences differ from noun phrases. Chemists use the idea too: a single line such as C2H6O records how many atoms of each element make up ethanol.
In science a formula is a compact, symbolic way of stating information, as in mathematical or chemical formulas; informally the word covers any general relationship between quantities. Its plural is either formulas, the ordinary English form, or formulae, a Latin form kept alive by scientific usage.
In mathematics a formula usually links one expression to another through an equation or inequality, and the weightiest formulas are theorems. Finding the volume of a sphere takes real work, through integral calculus or the older method of exhaustion, but once V = (4/3)πr³ is established, anyone can get the volume from the radius alone. Writing single letters instead of words lets mathematicians manipulate such results quickly. Some formulas, like Newton's second law, cover a huge range of physical situations; others, like a sine curve fitted to the tides in one bay, are built for a single problem.
The difference from an expression is grammatical. An expression names a mathematical object and normally has no equals or less-than sign; a formula makes a statement about objects, much as a noun phrase differs from a whole sentence. Computer algebra blurs the line by treating formulas as expressions that evaluate to true or false depending on the values given to their variables, and mathematical logic speaks of well-formed formulas assembled from a language's symbols by fixed rules.
Chemistry uses formulas to record the proportions of atoms in a compound on one line: H2O says each water molecule holds two hydrogen atoms and one oxygen, while ethanol's empirical formula is C2H6O. Boron carbide resists whole numbers altogether, with a ratio that varies from above 4 to beyond 6.5. In computing, a formula spells out a calculation, such as converting Fahrenheit to Celsius, and a spreadsheet cell holding A1+A2 simply stores its own result. Formulas in science nearly always require a choice of units, as in Boltzmann's entropy formula, whose constant k equals 1.380649 × 10⁻²³ joules per kelvin.
Source: Formula
