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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an equation is a formula that expresses the equality of two expressions, by connecting them with the
equals sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between tw ...
. The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...
an ''équation'' is defined as containing one or more variables, while in
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
, any
well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...
consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by an
equals sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between tw ...
("="). The expressions on the two 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. Assuming this does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides. The most common type of equation is a polynomial equation (commonly called also an ''algebraic equation'') in which the two sides are
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s. The sides of a polynomial equation contain one or more terms. For example, the equation : Ax^2 +Bx + C - y = 0 has left-hand side Ax^2 +Bx + C - y , which has four terms, and right-hand side 0 , consisting of just one term. The names of the variables suggest that and are unknowns, and that , , and are
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s, but this is normally fixed by the context (in some contexts, may be a parameter, or , , and may be ordinary variables). An equation is analogous to a scale into which weights are placed. When equal weights of something (e.g., grain) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. More generally, an equation remains in balance if the same operation is performed on its both sides. In Cartesian geometry, equations are used to describe geometric figures. As the equations that are considered, such as implicit equations or parametric equations, have infinitely many solutions, the objective is now different: instead of giving the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry, an important area of mathematics.
Algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
studies two main families of equations: polynomial equations and, among them, the special case of linear equations. When there is only one variable, polynomial equations have the form ''P''(''x'') = 0, where ''P'' is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
, and linear equations have the form ''ax'' + ''b'' = 0, where ''a'' and ''b'' are parameters. To solve equations from either family, one uses algorithmic or geometric techniques that originate from
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
or
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions. Differential equations are equations that involve one or more functions and their derivatives. They are ''solved'' by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics. The " =" symbol, which appears in every equation, was invented in 1557 by
Robert Recorde Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and las ...
, who considered that nothing could be more equal than parallel straight lines with the same length.Recorde, Robert, ''The Whetstone of Witte'' ... (London, England: Kyngstone, 1557)
the third page of the chapter "The rule of equation, commonly called Algebers Rule."
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Introduction


Analogous illustration

An equation is analogous to a weighing scale, balance, or seesaw. Each side of the equation corresponds to one side of the balance. Different quantities can be placed on each side: if the weights on the two sides are equal, the scale balances, and in analogy, the equality that represents the balance is also balanced (if not, then the lack of balance corresponds to an inequality represented by an
inequation In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific in ...
). In the illustration, ''x'', ''y'' and ''z'' are all different quantities (in this case real numbers) represented as circular weights, and each of ''x'', ''y'', and ''z'' has a different weight. Addition corresponds to adding weight, while subtraction corresponds to removing weight from what is already there. When equality holds, the total weight on each side is the same.


Parameters and unknowns

Equations often contain terms other than the unknowns. These other terms, which are assumed to be ''known'', are usually called ''constants'', ''coefficients'' or ''parameters''. An example of an equation involving ''x'' and ''y'' as unknowns and the parameter ''R'' is : x^2 +y^2 = R^2 . When ''R ''is chosen to have the value of 2 (''R ''= 2), this equation would be recognized in Cartesian coordinates as the equation for the circle of radius of 2 around the origin. Hence, the equation with ''R'' unspecified is the general equation for the circle. Usually, the unknowns are denoted by letters at the end of the alphabet, ''x'', ''y'', ''z'', ''w'', ..., while coefficients (parameters) are denoted by letters at the beginning, ''a'', ''b'', ''c'', ''d'', ... . For example, the general
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
is usually written ''ax''2 + ''bx'' + ''c'' = 0. The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called
solving the equation Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solutio ...
. Such expressions of the solutions in terms of the parameters are also called ''solutions''. A
system of equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single ...
is a set of ''simultaneous equations'', usually in several unknowns for which the common solutions are sought. Thus, a ''solution to the system'' is a set of values for each of the unknowns, which together form a solution to each equation in the system. For example, the system :\begin 3x+5y&=2\\ 5x+8y&=3 \end has the unique solution ''x'' = −1, ''y'' = 1.


Identities

An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable. In algebra, an example of an identity is the difference of two squares: :x^2 - y^2 = (x+y)(x-y) which is true for all ''x'' and ''y''.
Trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
is an area where many identities exist; these are useful in manipulating or solving
trigonometric equation In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
s. Two of many that involve the sine and cosine functions are: :\sin^2(\theta)+\cos^2(\theta) = 1 and :\sin(2\theta)=2\sin(\theta) \cos(\theta) which are both true for all values of ''θ''. For example, to solve for the value of ''θ'' that satisfies the equation: :3\sin(\theta) \cos(\theta)= 1\,, where ''θ'' is limited to between 0 and 45 degrees, one may use the above identity for the product to give: :\frac\sin(2 \theta) = 1\,, yielding the following solution for ''θ:'' :\theta = \frac \arcsin\left(\frac\right) \approx 20.9^\circ. Since the sine function is a
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
, there are infinitely many solutions if there are no restrictions on ''θ''. In this example, restricting ''θ'' to be between 0 and 45 degrees would restrict the solution to only one number.


Properties

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: * Adding or 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. * Multiplying or dividing both sides of an equation by a non-zero quantity. * Applying an
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an ...
to transform one side of the equation. For example, expanding a product or factoring a sum. * For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity. If some 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 In mathematics, an extraneous solution (or spurious solution) is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem. A missing solution is a solution that is a ...
s. For example, the equation x=1 has the solution x=1. Raising both sides to the exponent of 2 (which means applying the function f(s)=s^2 to both sides of the equation) changes the equation to x^2=1, which not only has the previous solution but also introduces the extraneous solution, x=-1. 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 above transformations are the basis of most elementary methods for equation solving, as well as some less elementary one, like Gaussian elimination.


Algebra


Polynomial equations

In general, an ''algebraic equation'' or polynomial equation is an equation of the form :P = 0, or :P = Q where ''P'' and ''Q'' are
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s with coefficients in some field (e.g.,
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
, real numbers, complex numbers). An algebraic equation is ''univariate'' if it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called ''multivariate'' (multiple variables, x, y, z, etc.). For example, :x^5-3x+1=0 is a univariate algebraic (polynomial) equation with integer coefficients and :y^4+\frac=\frac-xy^2+y^2-\frac is a multivariate polynomial equation over the rational numbers. Some polynomial equations with rational coefficients have a solution that is an
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
, with a finite number of operations involving just those coefficients (i.e., can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the Abel–Ruffini theorem demonstrates. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see
Root finding of polynomials In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex n ...
) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).


Systems of linear equations

A system of linear equations (or ''linear system'') is a collection of linear equations involving one or more variables. For example, :\begin 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac y &&\; - \;&& z &&\; = \;&& 0 & \end is a system of three equations in the three variables . A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A
solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solutio ...
to the system above is given by :\begin x &\,=\,& 1 \\ y &\,=\,& -2 \\ z &\,=\,& -2 \end since it makes all three equations valid. The word "''system''" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is a fundamental part of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, a subject which is used in many parts of modern mathematics. Computational
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s for finding the solutions are an important part of
numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematic ...
, and play a prominent role in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
,
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, and
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
. A system of non-linear equations can often be approximated by a linear system (see
linearization In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linea ...
), a helpful technique when making a
mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
or computer simulation of a relatively complex system.


Geometry


Analytic geometry

In Euclidean geometry, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional space can be expressed as the solution set of an equation of the form ax+by+cz+d=0, where a,b,c and d are real numbers and x,y,z are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values a,b,c are the coordinates of a vector perpendicular to the plane defined by the equation. A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values in \mathbb^2 or as the solution set of two linear equations with values in \mathbb^3. A
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
is the intersection of a cone with equation x^2+y^2=z^2 and a plane. In other words, in space, all conics are defined as the solution set of an equation of a plane and of the equation of a cone just given. This formalism allows one to determine the positions and the properties of the focuses of a conic. The use of equations allows one to call on a large area of mathematics to solve geometric questions. The Cartesian coordinate system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
. This point of view, outlined by Descartes, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians. Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
.


Cartesian equations

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
directed lines, that are marked using the same unit of length. One can use the same principle to specify the position of any point in three-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
by the use of three Cartesian coordinates, which are the signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). The invention of Cartesian coordinates in the 17th century by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
( Latinized name: ''Cartesius'') revolutionized mathematics by providing the first systematic link between Euclidean geometry and
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
. Using the Cartesian coordinate system, geometric shapes (such as curves) 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 .


Parametric equations

A parametric equation for a curve expresses the
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
of the points of the curve as functions of a variable, called a
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
.Weisstein, Eric W. "Parametric Equations." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParametricEquations.html For example, :\begin x&=\cos t\\ y&=\sin t \end are parametric equations for the unit circle, where ''t'' is the parameter. Together, these equations are called a parametric representation of the curve. The notion of ''parametric equation'' has been generalized to surfaces, manifolds and algebraic varieties of higher
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is ''one'' and ''one'' parameter is used, for surfaces dimension ''two'' and ''two'' parameters, etc.).


Number theory


Diophantine equations

A Diophantine equation is a polynomial equation in two or more unknowns for which only the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
solutions are sought (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An example of linear Diophantine equation is where ''a'', ''b'', and ''c'' are constants. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve,
algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
, or more general object, and ask about the lattice points on it. The word ''Diophantine'' refers to the Hellenistic mathematician of the 3rd century, Diophantus of
Alexandria Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandri ...
, who made a study of such equations and was one of the first mathematicians to introduce symbolism into
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.


Algebraic and transcendental numbers

An algebraic number is a number that is a solution of a non-zero polynomial equation in one variable with rational coefficients (or equivalently — by clearing denominators — with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coefficients). Numbers such as that are not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.


Algebraic geometry

Algebraic geometry is a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, classically studying solutions of polynomial equations. Modern algebraic geometry is based on more abstract techniques of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, especially
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, with the language and the problems of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines,
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
s,
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
s, ellipses,
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s, cubic curves like
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
s and quartic curves like lemniscates, and Cassini ovals. 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 points, the
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s and the points at infinity. More advanced questions involve the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of the curve and relations between the curves given by different equations.


Differential equations

A
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
is a mathematical equation that relates some function with its
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
, and
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
. In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.


Ordinary differential equations

An
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
or ODE is an equation containing a function of one independent variable and its derivatives. The term "''ordinary''" is used in contrast with the term partial differential equation, which may be with respect to ''more than'' one independent variable. Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.


Partial differential equations

A partial differential equation (PDE) is a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena such as
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
,
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
, electrostatics, electrodynamics, fluid flow, elasticity, or
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. 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 In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables. Important problems such as factorization and stability ...
. PDEs find their generalisation in
stochastic partial differential equations Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They have ...
.


Types of equations

Equations can be classified according to the types of operations and quantities involved. Important types include: * An algebraic equation or
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by degree: ** linear equation for degree one **
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
for degree two ** cubic equation for degree three ** quartic equation for degree four ** quintic equation for degree five **
sextic equation In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More prec ...
for degree six **
septic equation In algebra, a septic equation is an equation of the form :ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\, where . A septic function is a function of the form :f(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\, where . In other words, it is a polynomial o ...
for degree seven **
octic equation In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus ...
for degree eight * A Diophantine equation is an equation where the unknowns are required to be
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s * A transcendental equation is an equation involving a
transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed ...
of its unknowns * A parametric equation is an equation in which the solutions for the variables are expressed as functions of some other variables, called
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s appearing in the equations * A functional equation is an equation in which the unknowns are functions rather than simple quantities * Equations involving derivatives, integrals and finite differences: ** A
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
is a functional equation involving
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of the unknown functions, where the function and its derivatives are evaluated at the same point, such as f'(x) = x^2. Differential equations are subdivided into
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s for functions of a single variable and partial differential equations for functions of multiple variables ** An integral equation is a functional equation involving the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
s of the unknown functions. For functions of one variable, such an equation differs from a differential equation primarily through a change of variable substituting the function by its derivative, however this is not the case when the integral is taken over an open surface ** An
integro-differential equation In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. General first order linear equations The general first-order, linear (only with respect to the term involving deriva ...
is a functional equation involving both the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s and the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
s of the unknown functions. For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable. ** A
functional differential equation A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values. Functiona ...
of
delay differential equation In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time ...
is a function equation involving
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of the unknown functions, evaluated at multiple points, such as f'(x) = f(x-2) ** A difference equation is an equation where the unknown is a function ''f'' that occurs in the equation through ''f''(''x''), ''f''(''x''−1), ..., ''f''(''x''−''k''), for some whole integer ''k'' called the ''order'' of the equation. If ''x'' is restricted to be an integer, a difference equation is the same as a recurrence relation ** A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process


See also

* Formula * History of algebra * Indeterminate equation * List of equations *
List of scientific equations named after people This is a list of scientific equations named after people (eponymous equations)."Reflections on the Natural History of Eponymy and Scientific Law", Donald deB. Beaver, ''Social Studies of Science'', volume 6, number 1 (February, 1976), pages 89–9 ...
* Term (logic) * Theory of equations *
Cancelling out Cancelling out is a mathematical process used for removing subexpressions from a mathematical expression, when this removal does not change the meaning or the value of the expression because the subexpressions have equal and opposing effects. ...


Notes


References


External links


Winplot
General Purpose plotter that can draw and animate 2D and 3D mathematical equations.
Equation plotter
A web page for producing and downloading pdf or postscript plots of the solution sets to equations and inequations in two variables (''x'' and ''y''). {{Authority control Elementary algebra