Expression (mathematics)

The use of expressions ranges from the simple:

${\displaystyle 3+8}$

${\displaystyle 8x-5}$   (linear polynomial)

${\displaystyle 7{{x}^{2}}+4x-10}$   (quadratic polynomial)

${\displaystyle {\frac {x-1}{{{x}^{2}}+12}}}$   (rational fraction)

to the complex:

${\displaystyle f(a)+\sum _{k=1}^{n}\left.{\frac {1}{k!}}{\frac {d^{k}}{dt^{k}}}\right|_{t=0}f(u(t))+\int _{0}^{1}{\frac {(1-t)^{n}}{n!}}{\frac {d^{n+1}}{dt^{n+1}}}f(u(t))\,dt.}$

Many mathematical expressions include variables. Any variable can be classified as being either a free variable or a bound variable.

For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the value of the expression may be undefined. Thus an expression represents a function whose inputs are the values assigned to the free variables and whose output is the resulting value of the expression.

For example, the expression

${\displaystyle x/y}$

evaluated for x = 10, y = 5, will give 2; but it is undefined for y = 0.

The evaluation of an expression is dependent on the definition of the mathematical operators and on the system of values that is its context.

Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function. Example:

The expression

${\displaystyle \sum _{n=1}^{3}(2nx)}$

has free variable x, bound variable n, constants 1, 2, and 3, two occurrences of an implicit multiplication operator, and a summation operator. The expression is equivalent to the simpler expression 12x. The value for x = 3 is 36.

• Redden, John (2011). "Elementary Algebra". Flat World Knowledge. Archived from the original on 2014-11-15. Retrieved 2012-03-18.