Equation (mathematics)

Equation is kind of mathematical problem that is formulated in the following way: some equality is given, and at least one side of this equality is dependent on some variable (or variables); it is suggested to find values of variables, at which value of equality is true. Equation may have form

(1) [math]\displaystyle{ 
F(x)=0
 }[/math]

or

(2) [math]\displaystyle{ F(x)=G(x) }[/math]

where [math]\displaystyle{ F }[/math] and [math]\displaystyle{ G }[/math] are known functions and [math]\displaystyle{ x }[/math] is unknown variable.

An equation may have solution, id est, value of variable ([math]\displaystyle{ x }[/math] in the example above, although variable can be denoted by any letter, including letters from exotic alphabets), at which the equality has value true.

Usulally, it is supposed, that the variable belongs to some set [math]\displaystyle{ \mathrm A }[/math]; for example, it may be specified that [math]\displaystyle{ x }[/math] is real.

The equation may have no solution, may have one solution and may have many solutions, dependently on the set [math]\displaystyle{ \mathrm A }[/math].

Equations are common in language used in science; especially in mathematics and physics: Newton equation, equation of oscillator, Schrödinger equation. The termin equation is used also in chemistry, indicating conservation of atomic or isotopic content at the chemical reactions.

Examples

For function [math]\displaystyle{ F(x)=1+x }[/math], the equation (1) has no solutions among natural numbers, but has solution [math]\displaystyle{ x=-1 }[/math] in the set of integer numbers.

Equation [math]\displaystyle{ x^2=2 }[/math] has no solutions among rational numbers, has one solution ([math]\displaystyle{ x=\sqrt{2} }[/math]) among positive real numbers, and has two soluitons ([math]\displaystyle{ x=\sqrt{2} }[/math] and [math]\displaystyle{ x=-\sqrt{2} }[/math]) among real numbers.

Inverse function

Function [math]\displaystyle{ F }[/math] in equation (1) ; or functions [math]\displaystyle{ F }[/math] and [math]\displaystyle{ G }[/math] in equation (2) may also depend on some parameter(s). In this case, the solution(s) [math]\displaystyle{ x }[/math] also may depend on parameter(s). Indicating the function, the parameter, say, [math]\displaystyle{ b }[/math], can be specified as a second argument, writing [math]\displaystyle{ F(x,b) }[/math], [math]\displaystyle{ G(x,b) }[/math] or as subscript, writing [math]\displaystyle{ F_b(x) }[/math], [math]\displaystyle{ G_b(x) }[/math].

In relatively simple case, function [math]\displaystyle{ F }[/math] depends only on the unknown variable, and [math]\displaystyle{ G }[/math] depends on the parameter; for example,

(3) [math]\displaystyle{ F(x)=b }[/math] .

In this case, the solution [math]\displaystyle{ x }[/math] is considered as inverse function of [math]\displaystyle{ b }[/math], which can be written as

(4) [math]\displaystyle{ x=F^{-1}(b) }[/math].

Dependently on function [math]\displaystyle{ F }[/math], range of values of [math]\displaystyle{ b }[/math] and set [math]\displaystyle{ \mathbf A }[/math], there may exist no inverse function, one inverse function or several inverse functions.

Graphical solution of equations

File:ExampleEquationLog01.png

FIg.1. Example of graphic solution of equation [math]\displaystyle{ x=\log_b(x) }[/math] for [math]\displaystyle{ b=\sqrt{2} }[/math] (two solutions, [math]\displaystyle{ x=2 }[/math] and [math]\displaystyle{ x=4 }[/math]), [math]\displaystyle{ b=\exp(1/\rm e) }[/math] (one solution [math]\displaystyle{ x=\rm e }[/math]), and [math]\displaystyle{ b=2 }[/math] (no real solutions).

Solving equations, it may worth to begin with graphic solution of the equation, which allows the quick and dirty estimates. One plots both functions, [math]\displaystyle{ F }[/math] and [math]\displaystyle{ G }[/math] at the same graphic, and watch the point (s) of the intersection of the curves. In figire 1, functions [math]\displaystyle{ y=F(x)=x }[/math] is plotted with black line, and function [math]\displaystyle{ y=G(x)=\log_{\sqrt{2}}(x) }[/math] is plotted with red curve. The intersections with black curve indicate values of [math]\displaystyle{ x }[/math] which are solutions.

At the same figure, the cases [math]\displaystyle{ G(x)=\log_{\exp(1/\rm e)}(x) }[/math] (only one solution, [math]\displaystyle{ x=\rm e }[/math]) and [math]\displaystyle{ G(x)=\log_2(x) }[/math] (no solutions among real numbers) are shown with green and blue curves.

System of equations

In the equations (1) or (2), [math]\displaystyle{ x }[/math] may denote several numbers at once, [math]\displaystyle{ x=\{x_0,x_2,.. x_{n-1}\} }[/math]; and functions [math]\displaystyle{ F }[/math] and [math]\displaystyle{ G }[/math] may return values from multidimentional space [math]\displaystyle{ \{F_0,F_1,F_2,..F_{m-1} \} }[/math]. In this case, one says that there is system of equations. For example, there is well developed theory of systems of linear equations, while unknown variables [math]\displaystyle{ x }[/math] are real or complex numbers.

Differential equations and integral equations

In particular, [math]\displaystyle{ x }[/math] may denote an element of a Hilbert space, for example, set of functions of one or several variables; and function [math]\displaystyle{ F }[/math] may be expressed in terms of derivatives or integrals of [math]\displaystyle{ x }[/math] with respect to these variables. In these cases, the equation is called differential equation of integral equation.

Operator equations

Equations can be used for objects of any origen, as soon, as the operation of equality is defined. In particular, in Quantum mechanics, the Heisenberg equation deals with non-commuting objects (operators).