Equation: Difference between revisions

{{Short description|Mathematical formula expressing equality}}

In [[mathematics]], an '''equation''' is a [[mathematical formula]] that expresses the [[equality (mathematics)|equality]] of two [[Expression (mathematics)|expressions]], by connecting them with the [[equals sign]] {{char|1==}}.<ref name=":1">{{Cite web|title=Equation - Math Open Reference|url=https://www.mathopenref.com/equation.html|access-date=2020-09-01|website=www.mathopenref.com}}</ref><ref>{{Cite web|title=Equations and Formulas|url=https://www.mathsisfun.com/algebra/equation-formula.html|access-date=2020-09-01|website=www.mathsisfun.com}}</ref> The word ''equation'' and its [[cognate]]s in other languages may have subtly different meanings; for example, in [[French language|French]] an ''équation'' is defined as containing one or more [[variable (mathematics)|variables]], while in [[English language|English]], any [[well-formed formula]] consisting of two expressions related with an equals sign is an equation.<ref>{{cite web |last1=[[Solomon Marcus|Marcus]] |first1=Solomon |last2=Watt |first2=Stephen M. |title=What is an Equation? |url=https://www.academia.edu/3287674 |access-date=2019-02-27 }}</ref>

The "[[=]]" symbol, which appears in every equation, was invented in 1557 by [[Robert Recorde]], who considered that nothing could be more equal than parallel straight lines with the same length.<ref name="Whetstone" />

An equation is written as two [[expression (mathematics)|expressions]], connected by an [[equals sign]] ("=").<ref name=":1" /> The expressions on the two [[Sides of an equation|sides]] of the equals sign are called the "left-hand side" and "right-hand side" of the equation. Very often the right-hand side of an equation is assumed to be zero. This does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides.

Two equations or two systems of equations are ''equivalent'', if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to:

* [[addition|Adding]] or [[subtraction|subtracting]] the same quantity to both sides of an equation. This shows that every equation is equivalent to an equation in which the right-hand side is zero.

If some [[function (mathematics)|function]] is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called [[extraneous solution]]s. For example, the equation <math>x=1</math> has the solution <math>x=1.</math> Raising both sides to the exponent of 2 (which means applying the function <math>f(s)=s^2</math> to both sides of the equation) changes the equation to <math>x^2=1</math>, which not only has the previous solution but also introduces the extraneous solution, <math>x=-1.</math> Moreover, if the function is not defined at some values (such as 1/''x'', which is not defined for ''x'' = 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such a transformation to an equation.

The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called [[Equation solving|solving the equation]]. Such expressions of the solutions in terms of the parameters are also called ''solutions''.

:<math>\begin{align}

3x+5y&=2\\

5x+8y&=3

\end{align}

has the unique solution ''x''&nbsp;=&nbsp;−1, ''y''&nbsp;=&nbsp;1.

and

which are both true for all values of ''θ''.

For example, to solve for the value of ''θ'' that satisfies the equation:

where ''θ'' is limited to between 0 and 45 degrees, one may use the above identity for the product to give:

In general, an ''algebraic equation'' or [[polynomial equation]] is an equation of the form

:<math>P = Q</math> {{efn|As such an equation can be rewritten {{math|1=''P'' – ''Q'' = 0}}, many authors do not consider this case explicitly.}}

The invention of Cartesian coordinates in the 17th century by [[René Descartes]] revolutionized mathematics by providing the first systematic link between [[Euclidean geometry]] and [[algebra]]. Using the Cartesian coordinate system, geometric shapes (such as [[curve]]s) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates ''x'' and ''y'' satisfy the equation {{nowrap|1=''x''² + ''y''² = 4}}.

{{main|Parametric equation}}

:<math>\begin{align}

x&=\cos t\\

An [[algebraic number]] is a number that is a solution of a non-zero [[polynomial equation]] in one variable with [[rational number|rational]] coefficients (or equivalently — by [[clearing denominators]] — with [[integer]] coefficients). Numbers such as [[Pi|{{pi}}]] that are not algebraic are said to be [[transcendental number|transcendental]]. [[Almost all]] [[real number|real]] and [[complex number|complex]] numbers are transcendental.

{{main|Algebraic geometry}}

[[Algebraic geometry]] is a branch of [[mathematics]], classically studying solutions of [[polynomial equations]]. Modern algebraic geometry is based on more abstract techniques of [[abstract algebra]], especially [[commutative algebra]], with the language and the problems of [[geometry]].

The fundamental objects of study in algebraic geometry are [[algebraic variety|algebraic varieties]], which are geometric manifestations of [[solution set|solutions]] of [[systems of polynomial equations]]. Examples of the most studied classes of algebraic varieties are: [[plane algebraic curve]]s, which include [[line (geometry)|lines]], [[circle]]s, [[parabola]]s, [[ellipse]]s, [[hyperbola]]s, [[cubic curve]]s like [[elliptic curve]]s and quartic curves like [[lemniscate of Bernoulli|lemniscates]], and [[Cassini oval]]s. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the [[singular point of a curve|singular points]], the [[inflection point]]s and the [[point at infinity|points at infinity]]. More advanced questions involve the [[topology]] of the curve and relations between the curves given by different equations.

{{main|Differential equation}}

[[File:Attracteur étrange de Lorenz.png|thumb|A [[strange attractor]], which arises when solving a certain [[differential equation]]]]

PDEs can be used to describe a wide variety of phenomena such as [[sound]], [[heat]], [[electrostatics]], [[electrodynamics]], [[fluid flow]], [[Elasticity (physics)|elasticity]], or [[quantum mechanics]]. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional [[dynamical systems]], partial differential equations often model [[multidimensional systems]]. PDEs find their generalisation in [[stochastic partial differential equations]].

<!--Linked from [[Simultaneous equations]]-->

Equations can be classified according to the types of [[Operation (mathematics)|operations]] and quantities involved. Important types include:

** [[sextic equation]] for degree six

* A [[transcendental equation]] is an equation involving a [[transcendental function]] of its unknowns

** A [[functional differential equation]] of [[delay differential equation]] is a function equation involving [[derivative]]s of the unknown functions, evaluated at multiple points, such as <math>f'(x) = f(x-2)</math>

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* [[Formula]]

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