linguistic:paraphrasing, information:extraction
In mathematics

In mathematics, a formula generally refers to an equation or inequality relating one mathematical expression to another, with the most important ones being mathematical theorems. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius:

Having obtained this result, the volume of any sphere can be computed as long as its radius is known. Here, notice that the volume V and the radius r are expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex. Mathematical formulas are often algebraic, analytical or in closed form.

In a general context, formulas often represent mathematical models of real world phenomena, and as such can be used to provide solutions (or approximate solutions) to real world problems, with some being more general than others. For example, the formula

is an expression of Newton's second law, and is applicable to a wide range of physical situations. Other formulas, such as the use of the equation of a sine curve to model the movement of the tides in a bay, may be created to solve a particular problem. In all cases, however, formulas form the basis for calculations.

Expressions are distinct from formulas in the sense that they don't usually contain relations like equality (=) or inequality (<). Expressions denote a mathematical object, where as formulas denote a statement about mathematical objects. This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, 8 x − 5 {\displaystyle 8x-5} is an expression, while 8 x − 5 ≥ 3 {\displaystyle 8x-5\geq 3} is a formula.

However, in some areas mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the variables occurring in the expressions. For example 8 x − 5 ≥ 3 {\displaystyle 8x-5\geq 3} takes the value false if x is given a value less than 1, and the value true otherwise. (See Boolean expression)