Plugging in (algebra)

"Plugging in" is a colloquial mathematical term for the process of substitution, typically of a number or formula, in a formula. If the independent variables in the formula are visualised as certain required inputs, then the metaphor is of some black box picture in which a given input is a socket, into which a plug is then inserted. A formula may be substituted algebraically into another formula; this is seen, by extension of the metaphor, as a procedure of plugging an output of one black box into the input of another.

Caveats

In algebra the process of substituting a number for a variable, or a more general expression, is taught in passing. All instances of a given variable have to be replaced, simultaneously and by the same expression. This may require parentheses to be used. One practical hazard is the bound variable clash: letters for variables in the two formulae should be disjoint, or confusion can arise.

The formalization of the process of substitution is carried out as a basic step in developing lambda calculus.

A further difficulty is the possibility of obtaining an indeterminate form as a result.