# equation

**equation** An expression that asserts the equality of two terms. To be precise, an equation has the following form. Let Σ be a signature and let *t*_{1}(*X*_{1},…, *X _{n}*) and

*t*

_{2}(

*X*

_{1},…,

*X*) be two terms over Σ involving the variables

_{n}*X*

_{1},…,

*X*. Then

_{n}*t*

_{1}(

*X*

_{1},…,

*X*) =

_{n}*t*

_{2}(

*X*

_{1},…,

*X*)

_{n}is an equation.

Equations are a natural means of expressing possible relationships between the functions in a signature. In fact, the equations can be used to specify or define the functions uniquely using initial algebra semantics (see equational specification).

Most systems in science and engineering are described mathematically using equations. Two stages are involved: a mathematical model of the system is made using sets and functions; some functions are known and others are to be found. Equations are postulated to define the unknown functions in terms of one another and the known functions. Research has shown that the same process is possible for computing systems. Indeed, theoretically it is known that any computing system, or any physical system that can be faithfully modeled using digital computation, can be characterized by small sets of equations. See also computable algebra.

# equation

e·qua·tion / iˈkwāzhən/ •
n. 1. Math. a statement that the values of two mathematical expressions are equal (indicated by the sign =).2. the process of equating one thing with another: *the equation of science with objectivity.* ∎ (the equation) a situation or problem in which several factors must be taken into account:

*money also came into the equation.*3. Chem. a symbolic representation of the changes that occur in a chemical reaction, expressed in terms of the formulae of the molecules or other species involved.

# equation

**equation** Mathematical statement of variables, equal to some subset of all possible variables. The equation x^{2} = 8 − 2x is true only for certain values (solutions) of x (x = 2 and x = −4). This type of equation is contrasted with an identity, such as (x+2)^{2} = x^{2} + 4x + 4, which is true for all values of *x*. Equations are said to be linear, quadratic, cubic, quartic, etc., according to whether their degree (the highest power of the variable) is 1, 2, 3, 4, etc. See also simultaneous equations

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