Plugging in is a colloquial mathematical term for the process of substitution

Substitution

Substitution may refer to:- Sciences :* Substitution , a syntactic transformation on strings of symbols of a formal language* Substitution of variables* Substitution cipher, a method of encryption...

, typically of a number or formula, in a formula

Formula

In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

. If the independent variable

Independent variable

The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...

s in the formula are visualised as certain required inputs, then the metaphor is of some black box

Black box

A black box is a device, object, or system whose inner workings are unknown; only the input, transfer, and output are known characteristics.The term black box can also refer to:-In science and technology:*Black box theory, a philosophical theory...

 picture in which a given input is a socket

Socket

Socket may refer to:In mechanics:* Socket wrench, a type of wrench that uses separate, removable sockets to fit different sizes of nuts and bolts...

, into which a plug

Plug

Plug may refer to:* Plug for the plughole in the bathtub, washbasin and sink* Plug , a short lived British comic that ran from 1977 until 1979, when it merged with The Beezer, which featured a character from The Bash Street Kids called Plug as it's main star.* Plug , a family of fishing lures*...

 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

Output

Output is the term denoting either an exit or changes which exit a system and which activate/modify a process. It is an abstract concept, used in the modeling, system design and system exploitation.-In control theory:...

 of one black box into the input of another.

Caveats

In algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

 the process of substituting a number for a variable, or a more general expression

Expression (mathematics)

In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...

, 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

Lambda calculus

In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...

.

A further difficulty is the possibility of obtaining an indeterminate form

Indeterminate form

In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution...

as a result.