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3487:. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. 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.
1719:
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3494:, 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.
2316:{\displaystyle {\begin{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}
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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
775:
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
492:
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
2562:, 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
2996:) 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
275:
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.
1127:
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.
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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
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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.
1612:
3153:, 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
2962:, 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
2034:
505:, 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:
78:, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Knowledge.
3117:
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3585:, 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
2731:
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
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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).
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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.
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3426:. 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
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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
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The subject of this article is basic in mathematics, and is treated in a lot of textbooks. Among them, Lay 2005, Meyer 2001, and Strang 2005 contain the material of this article.
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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
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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.
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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.
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1704:. 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.
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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
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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
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of the unknown functions. For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable.
3620:. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional
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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
2665:
427:
244:
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.
1932:
98:
Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.
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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.
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If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of
720:= 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such a transformation to an equation.
4289:: A web page for producing and downloading pdf or postscript plots of the solution sets to equations and inequations in two variables (
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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
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system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name
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The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called
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For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.
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are constants. An exponential
Diophantine equation is one for which exponents of the terms of the equation can be unknowns.
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to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A
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83:
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Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is
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are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values
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Content in this edit is translated from the existing French
Knowledge article at ]; see its history for attribution.
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is an equation in which the solutions for the variables are expressed as functions of some other variables, called
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3275:. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.
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2434:{\displaystyle {\begin{alignedat}{2}x&\,=\,&1\\y&\,=\,&-2\\z&\,=\,&-2\end{alignedat}}}
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in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and
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one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the
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of the unknown functions, where the function and its derivatives are evaluated at the same point, such as
1684:. To solve equations from either family, one uses algorithmic or geometric techniques that originate from
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The use of equations allows one to call on a large area of mathematics to solve geometric questions. The
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1865:. On the other hand, a polynomial equation may involve several variables, in which case it is called
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255:, who considered that nothing could be more equal than parallel straight lines with the same length.
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A large amount of research has been devoted to compute efficiently accurate approximations of the
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Equations often contain terms other than the unknowns. These other terms, which are assumed to be
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puts emphasis on qualitative analysis of systems described by differential equations, while many
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is the parameter. Together, these equations are called a parametric representation of the curve.
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is limited to between 0 and 45 degrees, one may use the above identity for the product to give:
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2872:, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.
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is an anonymous 2nd-century
Chinese book proposing a method of resolution for linear equations.
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1607:{\displaystyle \theta ={\frac {1}{2}}\arcsin \left({\frac {2}{3}}\right)\approx 20.9^{\circ }.}
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as the equation for the circle of radius of 2 around the origin. Hence, the equation with
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4283:: General Purpose plotter that can draw and animate 2D and 3D mathematical equations.
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the third page of the chapter "The rule of equation, commonly called
Algebers Rule."
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2981:
2029:{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}
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is an area where many identities exist; these are useful in manipulating or solving
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Weisstein, Eric W. "Parametric
Equations." From MathWorld--A Wolfram Web Resource.
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1012:, usually in several unknowns for which the common solutions are sought. Thus, a
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2077:) and of the common solutions of several multivariate polynomial equations (see
958:, ..., while coefficients (parameters) are denoted by letters at the beginning,
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to be between 0 and 45 degrees would restrict the solution to only one number.
997:. Such expressions of the solutions in terms of the parameters are also called
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252:
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4207:"A statement of equality between two expressions. Equations are of two types,
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is a univariate algebraic (polynomial) equation with integer coefficients and
55:
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One can use the same principle to specify the position of any point in three-
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2455:, a subject which is used in many parts of modern mathematics. Computational
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have been developed to determine solutions with a given degree of accuracy.
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of the curve and relations between the curves given by different equations.
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800:
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Raising both sides to the exponent of 2 (which means applying the function
3112:{\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}}
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revolutionized mathematics by providing the first systematic link between
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consisting of two expressions related with an equals sign is an equation.
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Usually, the unknowns are denoted by letters at the end of the alphabet,
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1091:{\displaystyle {\begin{aligned}3x+5y&=2\\5x+8y&=3\end{aligned}}}
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is restricted to be an integer, a difference equation is the same as a
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The above transformations are the basis of most elementary methods for
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in other languages may have subtly different meanings; for example, in
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to the source of your translation. A model attribution edit summary is
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equation is an equation in which both sides are polynomials (see also
2451:
In mathematics, the theory of linear systems is a fundamental part of
1648:. When there is only one variable, polynomial equations have the form
1621:, there are infinitely many solutions if there are no restrictions on
236:, and the values of the unknowns that satisfy the equality are called
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3359:. Modern algebraic geometry is based on more abstract techniques of
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Equations involving derivatives, integrals and finite differences:
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3386:. Examples of the most studied classes of algebraic varieties are:
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2992:. Using the Cartesian coordinate system, geometric shapes (such as
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Moreover, if the function is not defined at some values (such as 1/
509:
251:" symbol, which appears in every equation, was invented in 1557 by
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is a differential equation in which one or more of the terms is a
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3722:
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PDEs can be used to describe a wide variety of phenomena such as
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is a multivariate polynomial equation over the rational numbers.
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of the unknown functions, evaluated at multiple points, such as
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The invention of
Cartesian coordinates in the 17th century by
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or as the solution set of two linear equations with values in
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3034:
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4255:, Addison Wesley Publishing Co., fifth edition, 1979, p. 91.
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4242:(éd.), Van Nostrand, 1968, 3 ed. 1st ed. 1948, p. 131.
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The fundamental objects of study in algebraic geometry are
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zero or one. An example of linear
Diophantine equation is
1383:{\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}
87:
to this template: there are already 1,433 articles in the
291:. The sides of a polynomial equation contain one or more
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1306:{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}
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to both sides of the equation) changes the equation to
248:
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Equations can be classified according to the types of
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is a system of three equations in the three variables
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http://mathworld.wolfram.com/ParametricEquations.html
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is an equation where the unknowns are required to be
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since it makes all three equations valid. The word "
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unspecified is the general equation for the circle.
533:
to transform one side of the equation. For example,
71:
3523:or ODE is an equation containing a function of one
2459:for finding the solutions are an important part of
240:of the equation. There are two kinds of equations:
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3810:
3644:and quantities involved. Important types include:
3508:
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2723:
2691:
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2315:
2073:solutions of a univariate algebraic equation (see
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1915:
1818:
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1722:The solutions –1 and 2 of the polynomial equation
1700:. The techniques used are different and come from
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1525:{\displaystyle {\frac {3}{2}}\sin(2\theta )=1\,,}
526:both sides of an equation by a non-zero quantity.
153:The first use of an equals sign, equivalent to 14
4302:
2954:. As the equations that are considered, such as
2520:The blue and red line is the set of all points (
1460:{\displaystyle 3\sin(\theta )\cos(\theta )=1\,,}
429:, consisting of just one term. The names of the
4020:List of scientific equations named after people
3929:is an equation where the unknown is a function
803:) represented as circular weights, and each of
3818:. Differential equations are subdivided into
2084:
2054:). This can be done for all such equations of
501:Two equations or two systems of equations are
113:accompanying your translation by providing an
62:Click for important translation instructions.
54:expand this article with text translated from
4221:
4101:... (London, England: Jhon Kyngstone, 1557),
3624:, partial differential equations often model
3297:is a number that is a solution of a non-zero
3041:of the points of the curve as functions of a
1131:In algebra, an example of an identity is the
727:, as well as some less elementary ones, like
3845:is a functional equation involving both the
3244:, or more general object, and ask about the
409:, which has four terms, and right-hand side
4093:
4091:
3192:in two or more unknowns for which only the
931:= 2), this equation would be recognized in
799:are all different quantities (in this case
3161:parameter is used, for surfaces dimension
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2283:
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4215:(or usually simply "equations")". «
3753:is an equation in which the unknowns are
3630:stochastic partial differential equations
3445:
2770:
2741:
2418:
2414:
2394:
2390:
2373:
2369:
2094:The Nine Chapters on the Mathematical Art
1696:where the coefficients and solutions are
1518:
1453:
739:
4251:Thomas, George B., and Finney, Ross L.,
4088:
3455:
3378:, which are geometric manifestations of
3367:, with the language and the problems of
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2515:
2088:
1717:
1640:studies two main families of equations:
743:
148:
27:Mathematical formula expressing equality
3833:is a functional equation involving the
3822:for functions of a single variable and
3018:
2548:point, (2,3), satisfies both equations.
1707:
1400:For example, to solve for the value of
760:are real numbers, analogous to weights.
14:
4303:
4164:
3464:, which arises when solving a certain
3438:. More advanced questions involve the
3319:that are not algebraic are said to be
2886:
1393:which are both true for all values of
1204:{\displaystyle x^{2}-y^{2}=(x+y)(x-y)}
279:The most common type of equation is a
4054:As such an equation can be rewritten
3635:
3337:
2495:), a helpful technique when making a
1869:(multiple variables, x, y, z, etc.).
1664:, and linear equations have the form
1644:and, among them, the special case of
214:is defined as containing one or more
4112:
4110:
3933:that occurs in the equation through
3628:. PDEs find their generalisation in
3531:" is used in contrast with the term
3355:, classically studying solutions of
3279:Algebraic and transcendental numbers
2966:, an important area of mathematics.
2511:
1535:yielding the following solution for
849:An example of an equation involving
36:
4188:
3826:for functions of multiple variables
3767:is a functional equation involving
3660:). These are further classified by
3305:coefficients (or equivalently — by
748:Illustration of a simple equation;
157:+ 15 = 71 in modern notation. From
24:
2926:are the coordinates of the center
2868:. This point of view, outlined by
927:is chosen to have the value of 2 (
913:{\displaystyle x^{2}+y^{2}=R^{2}.}
271:("="). The expressions on the two
25:
4327:
4274:
4107:
3864:is a function equation involving
3122:are parametric equations for the
2950:, equations are used to describe
2849:{\displaystyle x^{2}+y^{2}=z^{2}}
2786:{\displaystyle \mathbb {R} ^{3}.}
974:, ... . For example, the general
4118:"Equation - Math Open Reference"
3981:stochastic differential equation
3858:functional differential equation
3172:
2754:{\displaystyle \mathbb {R} ^{2}}
2503:of a relatively complex system.
2348:to the system above is given by
1123:List of trigonometric identities
41:
4258:
3820:ordinary differential equations
3583:ordinary differential equations
3535:, which may be with respect to
3527:and its derivatives. The term "
3509:Ordinary differential equations
3384:systems of polynomial equations
3313:coefficients). Numbers such as
2463:, and play a prominent role in
2042:Some polynomial equations with
1625:. In this example, restricting
1232:. Two of many that involve the
348:{\displaystyle Ax^{2}+Bx+C-y=0}
4253:Calculus and Analytic Geometry
4245:
4201:
4182:
4158:
4134:
4068:
4048:
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3824:partial differential equations
3792:
3786:
3658:system of polynomial equations
3555:Partial differential equations
3521:ordinary differential equation
3515:Ordinary differential equation
2485:system of non-linear equations
2079:System of polynomial equations
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857:as unknowns and the parameter
764:An equation is analogous to a
624:
618:
263:An equation is written as two
258:
191:, by connecting them with the
123:You may also add the template
13:
1:
4081:
3843:integro-differential equation
3757:rather than simple quantities
3567:partial differential equation
3561:Partial differential equation
3533:partial differential equation
1617:Since the sine function is a
1404:that satisfies the equation:
1112:
496:
402:{\displaystyle Ax^{2}+Bx+C-y}
295:. For example, the equation
287:) in which the two sides are
4168:, Solomon; Watt, Stephen M.
3916:{\displaystyle f'(x)=f(x-2)}
3334:numbers are transcendental.
3188:A Diophantine equation is a
2608:{\displaystyle ax+by+cz+d=0}
1916:{\displaystyle x^{5}-3x+1=0}
1857:). An algebraic equation is
556:. For example, the equation
489:may be ordinary variables).
7:
3992:
3862:delay differential equation
3811:{\displaystyle f'(x)=x^{2}}
3731:is an equation involving a
3479:equation that relates some
2506:
2085:Systems of linear equations
2075:Root finding of polynomials
2046:have a solution that is an
1767:is an equation of the form
734:
716:, which is not defined for
95:will aid in categorization.
10:
4332:
3961:), for some whole integer
3746:appearing in the equations
3581:. (This is in contrast to
3558:
3539:one independent variable.
3512:
3449:
3341:
3282:
3181:
3022:
2551:
2102:system of linear equations
1841:with coefficients in some
1711:
1632:
1116:
823:
643:{\displaystyle f(s)=s^{2}}
125:{{Translated|fr|Équation}}
70:Machine translation, like
29:
4222:
3257:Hellenistic mathematician
2799:is the intersection of a
1733:are the points where the
1133:difference of two squares
283:(commonly called also an
56:the corresponding article
32:Equation (disambiguation)
4195:Encyclopædia Universalis
4191:"Équation, mathématique"
4142:"Equations and Formulas"
4041:
3626:multidimensional systems
3418:and quartic curves like
3137:has been generalized to
2544:=4, respectively. Their
2491:by a linear system (see
2461:numerical linear algebra
1861:if it involves only one
1101:has the unique solution
826:Expression (mathematics)
445:are unknowns, and that
3733:transcendental function
3729:transcendental equation
3575:multivariable functions
1692:. Algebra also studies
1656:) = 0, where
1230:trigonometric equations
820:Parameters and unknowns
676:{\displaystyle x^{2}=1}
471:may be a parameter, or
134:For more guidance, see
4223:Mathematics Dictionary
4170:"What is an Equation?"
4099:The Whetstone of Witte
4010:Indeterminate equation
3917:
3812:
3573:that contains unknown
3468:
3446:Differential equations
3388:plane algebraic curves
3113:
2943:
2850:
2787:
2755:
2725:
2693:
2661:
2641:
2609:
2549:
2435:
2317:
2112:involving one or more
2097:
2030:
1917:
1820:
1790:
1756:
1672: = 0, where
1608:
1526:
1461:
1384:
1307:
1214:which is true for all
1205:
1119:Identity (mathematics)
1092:
1014:solution to the system
1010:simultaneous equations
914:
761:
740:Analogous illustration
706:
677:
644:
602:
576:
423:
403:
349:
168:
160:The Whetstone of Witte
4213:conditional equations
3918:
3813:
3765:differential equation
3571:differential equation
3473:differential equation
3466:differential equation
3459:
3452:Differential equation
3307:clearing denominators
3301:in one variable with
3289:Transcendental number
3178:Diophantine equations
3114:
3004:satisfy the equation
2894:
2851:
2788:
2756:
2726:
2724:{\displaystyle a,b,c}
2694:
2692:{\displaystyle x,y,z}
2667:are real numbers and
2662:
2642:
2640:{\displaystyle a,b,c}
2610:
2519:
2436:
2318:
2108:) is a collection of
2092:
2044:rational coefficients
2031:
1918:
1821:
1791:
1721:
1694:Diophantine equations
1690:mathematical analysis
1609:
1527:
1462:
1385:
1308:
1206:
1093:
933:Cartesian coordinates
915:
834:, are usually called
787:In the illustration,
747:
707:
705:{\displaystyle x=-1.}
678:
645:
603:
577:
424:
404:
350:
152:
136:Knowledge:Translation
107:copyright attribution
3969:of the equation. If
3872:
3775:
3719:Diophantine equation
3545:elementary functions
3525:independent variable
3357:polynomial equations
3259:of the 3rd century,
3184:Diophantine equation
3056:
3019:Parametric equations
2960:parametric equations
2862:Cartesian coordinate
2807:
2765:
2736:
2703:
2671:
2651:
2619:
2566:
2355:
2123:
2060:Abel–Ruffini theorem
2052:solved algebraically
2048:algebraic expression
1933:
1879:
1804:
1774:
1708:Polynomial equations
1642:polynomial equations
1545:
1481:
1411:
1323:
1247:
1142:
1023:
995:solving the equation
868:
729:Gaussian elimination
687:
654:
612:
601:{\displaystyle x=1.}
586:
560:
554:extraneous solutions
413:
362:
302:
181:mathematical formula
30:For other uses, see
4122:www.mathopenref.com
4030:Theory of equations
3975:recurrence relation
3927:difference equation
3751:functional equation
3740:parametric equation
3579:partial derivatives
3376:algebraic varieties
3365:commutative algebra
3299:polynomial equation
3190:polynomial equation
3169:parameters, etc.).
3147:algebraic varieties
3135:parametric equation
3031:parametric equation
3025:Parametric equation
2887:Cartesian equations
2877:functional analysis
2501:computer simulation
1819:{\displaystyle P=Q}
1789:{\displaystyle P=0}
1765:polynomial equation
1714:Polynomial equation
1006:system of equations
978:is usually written
575:{\displaystyle x=1}
358:has left-hand side
281:polynomial equation
230:Solving an equation
224:well-formed formula
183:that expresses the
4311:Elementary algebra
4146:www.mathsisfun.com
4005:History of algebra
3985:stochastic process
3913:
3808:
3674:quadratic equation
3650:algebraic equation
3636:Types of equations
3543:represent them by
3469:
3436:points at infinity
3349:Algebraic geometry
3344:Algebraic geometry
3338:Algebraic geometry
3109:
3107:
2986:Euclidean geometry
2964:algebraic geometry
2956:implicit equations
2948:Cartesian geometry
2944:
2846:
2783:
2751:
2721:
2689:
2657:
2637:
2605:
2560:Euclidean geometry
2550:
2497:mathematical model
2431:
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2313:
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2275:
2098:
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1913:
1816:
1786:
1761:algebraic equation
1757:
1739:quadratic function
1604:
1522:
1457:
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1303:
1201:
1088:
1086:
976:quadratic equation
910:
780:represented by an
762:
702:
673:
640:
598:
572:
419:
399:
345:
285:algebraic equation
267:, connected by an
169:
115:interlanguage link
4189:Lachaud, Gilles.
4097:Recorde, Robert,
4015:List of equations
3831:integral equation
3622:dynamical systems
3618:quantum mechanics
3503:numerical methods
3499:dynamical systems
3462:strange attractor
3432:inflection points
3242:algebraic surface
2952:geometric figures
2866:analytic geometry
2660:{\displaystyle d}
2554:Analytic geometry
2512:Analytic geometry
2274:
2024:
1982:
1962:
1619:periodic function
1582:
1562:
1492:
1105: = −1,
582:has the solution
422:{\displaystyle 0}
147:
146:
63:
16:(Redirected from
4323:
4287:Equation plotter
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3712:for degree eight
3706:for degree seven
3692:quintic equation
3686:quartic equation
3682:for degree three
3492:pure mathematics
3390:, which include
3361:abstract algebra
3317:
3295:algebraic number
3285:Algebraic number
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2110:linear equations
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1847:rational numbers
1825:
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1792:
1787:
1755:cuts the x-axis.
1754:
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1646:linear equations
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1097:
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1087:
990: = 0.
919:
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725:equation solving
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167:of Wales (1557).
126:
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94:
93:|topic=
91:, and specifying
76:Google Translate
61:
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4240:Robert C. James
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3875:
3873:
3870:
3869:
3851:antiderivatives
3835:antiderivatives
3802:
3798:
3778:
3776:
3773:
3772:
3735:of its unknowns
3704:septic equation
3698:sextic equation
3694:for degree five
3688:for degree four
3668:linear equation
3638:
3606:electrodynamics
3563:
3557:
3517:
3511:
3454:
3448:
3428:singular points
3416:elliptic curves
3351:is a branch of
3346:
3340:
3315:
3291:
3283:Main articles:
3281:
3238:algebraic curve
3208:
3186:
3180:
3175:
3106:
3105:
3089:
3083:
3082:
3066:
3059:
3057:
3054:
3053:
3049:. For example,
3027:
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2823:
2814:
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2144:
2136:
2126:
2124:
2121:
2120:
2116:. For example,
2087:
2062:demonstrates.
2016:
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1855:complex numbers
1805:
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1775:
1772:
1771:
1759:In general, an
1741:
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4275:External links
4273:
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4035:Cancelling out
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3713:
3710:octic equation
3707:
3701:
3700:for degree six
3695:
3689:
3683:
3680:cubic equation
3677:
3676:for degree two
3671:
3670:for degree one
3637:
3634:
3602:electrostatics
3587:computer model
3559:Main article:
3556:
3553:
3513:Main article:
3510:
3507:
3450:Main article:
3447:
3444:
3342:Main article:
3339:
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3321:transcendental
3280:
3277:
3255:refers to the
3246:lattice points
3182:Main article:
3179:
3176:
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3171:
3133:The notion of
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3037:expresses the
3023:Main article:
3020:
3017:
2982:René Descartes
2942:is the radius.
2888:
2885:
2881:linear algebra
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2803:with equation
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1073:
1070:
1067:
1064:
1061:
1058:
1057:
1054:
1051:
1048:
1046:
1044:
1041:
1038:
1035:
1032:
1029:
1028:
921:
920:
909:
904:
900:
896:
891:
887:
883:
878:
874:
821:
818:
768:, balance, or
766:weighing scale
741:
738:
736:
733:
701:
698:
695:
692:
672:
669:
664:
660:
637:
633:
629:
626:
623:
620:
617:
597:
594:
591:
571:
568:
565:
546:
545:
542:
527:
517:
498:
495:
418:
398:
395:
392:
389:
386:
383:
380:
375:
371:
367:
356:
355:
344:
341:
338:
335:
332:
329:
326:
323:
320:
315:
311:
307:
260:
257:
253:Robert Recorde
165:Robert Recorde
145:
144:
140:
139:
132:
121:
99:
96:
84:adding a topic
79:
68:
65:
51:
50:
49:
47:
40:
26:
9:
6:
4:
3:
2:
4328:
4317:
4314:
4312:
4309:
4308:
4306:
4296:
4292:
4288:
4285:
4282:
4279:
4278:
4267:
4261:
4254:
4248:
4241:
4235:
4230:
4226:
4218:
4214:
4210:
4204:
4196:
4192:
4185:
4171:
4167:
4161:
4147:
4143:
4137:
4123:
4119:
4113:
4111:
4104:
4100:
4094:
4092:
4087:
4071:
4062:
4058:
4051:
4047:
4036:
4033:
4031:
4028:
4026:
4023:
4021:
4018:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3997:
3986:
3982:
3978:
3976:
3972:
3968:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3936:
3932:
3928:
3924:
3907:
3904:
3901:
3895:
3892:
3886:
3879:
3876:
3867:
3863:
3859:
3855:
3852:
3848:
3844:
3840:
3836:
3832:
3828:
3825:
3821:
3803:
3799:
3795:
3789:
3782:
3779:
3770:
3766:
3762:
3761:
3759:
3756:
3752:
3748:
3745:
3741:
3737:
3734:
3730:
3726:
3724:
3720:
3716:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3665:
3663:
3659:
3655:
3651:
3647:
3646:
3645:
3643:
3633:
3631:
3627:
3623:
3619:
3615:
3611:
3607:
3603:
3599:
3595:
3590:
3588:
3584:
3580:
3576:
3572:
3568:
3562:
3552:
3550:
3546:
3540:
3538:
3534:
3530:
3526:
3522:
3516:
3506:
3504:
3500:
3495:
3493:
3488:
3486:
3482:
3478:
3474:
3467:
3463:
3458:
3453:
3443:
3441:
3437:
3433:
3429:
3425:
3424:Cassini ovals
3421:
3417:
3413:
3409:
3405:
3401:
3397:
3393:
3389:
3385:
3381:
3377:
3372:
3370:
3366:
3363:, especially
3362:
3358:
3354:
3350:
3345:
3335:
3333:
3329:
3326:
3322:
3318:
3312:
3308:
3304:
3300:
3296:
3290:
3286:
3276:
3274:
3270:
3266:
3262:
3258:
3254:
3249:
3247:
3243:
3239:
3234:
3232:
3228:
3224:
3219:
3215:
3211:
3206:
3202:
3198:
3195:
3191:
3185:
3173:Number theory
3170:
3168:
3164:
3160:
3156:
3152:
3148:
3144:
3140:
3136:
3131:
3129:
3125:
3102:
3099:
3096:
3093:
3091:
3086:
3079:
3076:
3073:
3070:
3068:
3063:
3052:
3051:
3050:
3048:
3044:
3040:
3036:
3032:
3026:
3016:
3012:
3008:
3003:
2999:
2995:
2991:
2987:
2983:
2978:
2975:
2972:
2967:
2965:
2961:
2957:
2953:
2949:
2941:
2935:
2931:
2925:
2921:
2916:
2912:
2908:
2904:
2900:
2893:
2884:
2882:
2878:
2873:
2871:
2867:
2863:
2858:
2841:
2837:
2833:
2828:
2824:
2820:
2815:
2811:
2802:
2798:
2797:conic section
2793:
2780:
2775:
2746:
2718:
2715:
2712:
2709:
2706:
2686:
2683:
2680:
2677:
2674:
2654:
2634:
2631:
2628:
2625:
2622:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2561:
2555:
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2518:
2504:
2502:
2498:
2494:
2493:linearization
2490:
2487:can often be
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2449:
2447:
2424:
2421:
2415:
2412:
2407:
2400:
2397:
2391:
2388:
2383:
2376:
2370:
2367:
2362:
2351:
2350:
2349:
2347:
2343:
2338:
2334:
2330:
2305:
2298:
2291:
2284:
2277:
2271:
2268:
2258:
2251:
2248:
2240:
2237:
2230:
2223:
2220:
2213:
2206:
2203:
2196:
2189:
2186:
2178:
2171:
2164:
2157:
2150:
2147:
2140:
2133:
2130:
2119:
2118:
2117:
2115:
2111:
2107:
2106:linear system
2103:
2095:
2091:
2082:
2080:
2076:
2072:
2068:
2063:
2061:
2057:
2053:
2049:
2045:
2040:
2021:
2018:
2013:
2008:
2004:
2000:
1995:
1991:
1987:
1984:
1979:
1974:
1970:
1964:
1959:
1955:
1952:
1946:
1941:
1937:
1929:
1928:
1927:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1887:
1883:
1875:
1874:
1873:
1872:For example,
1870:
1868:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1813:
1810:
1807:
1800:
1799:
1783:
1780:
1777:
1770:
1769:
1768:
1766:
1762:
1752:
1748:
1744:
1740:
1736:
1730:
1726:
1720:
1715:
1705:
1703:
1702:number theory
1699:
1695:
1691:
1687:
1683:
1679:
1675:
1671:
1668: +
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1630:
1628:
1624:
1620:
1601:
1596:
1592:
1588:
1584:
1579:
1576:
1571:
1567:
1564:
1559:
1556:
1551:
1548:
1541:
1540:
1539:
1538:
1519:
1515:
1512:
1506:
1503:
1497:
1494:
1489:
1486:
1477:
1476:
1475:
1473:
1454:
1450:
1447:
1441:
1435:
1432:
1426:
1420:
1417:
1414:
1407:
1406:
1405:
1403:
1398:
1396:
1374:
1368:
1365:
1359:
1353:
1350:
1347:
1344:
1338:
1335:
1329:
1326:
1319:
1318:
1317:
1300:
1297:
1291:
1285:
1280:
1276:
1272:
1266:
1260:
1255:
1251:
1243:
1242:
1241:
1239:
1235:
1231:
1227:
1223:
1221:
1217:
1195:
1192:
1189:
1180:
1177:
1174:
1168:
1163:
1159:
1155:
1150:
1146:
1138:
1137:
1136:
1134:
1129:
1124:
1120:
1110:
1108:
1104:
1081:
1078:
1076:
1071:
1068:
1065:
1062:
1059:
1052:
1049:
1047:
1042:
1039:
1036:
1033:
1030:
1019:
1018:
1017:
1015:
1011:
1007:
1002:
1000:
996:
991:
989:
986: +
985:
982: +
981:
977:
973:
969:
965:
961:
957:
953:
949:
945:
940:
938:
934:
930:
926:
907:
902:
898:
894:
889:
885:
881:
876:
872:
864:
863:
862:
860:
856:
852:
847:
845:
841:
837:
833:
827:
817:
814:
810:
806:
802:
798:
794:
790:
785:
783:
779:
773:
771:
767:
759:
755:
751:
746:
732:
730:
726:
721:
719:
715:
699:
696:
693:
690:
670:
667:
662:
658:
635:
631:
627:
621:
615:
595:
592:
589:
569:
566:
563:
555:
551:
543:
540:
537:a product or
536:
532:
528:
525:
521:
518:
515:
511:
508:
507:
506:
504:
494:
490:
487:
481:
475:
466:
461:
455:
449:
443:
437:
433:suggest that
432:
416:
396:
393:
390:
387:
384:
381:
378:
373:
369:
365:
342:
339:
336:
333:
330:
327:
324:
321:
318:
313:
309:
305:
298:
297:
296:
294:
290:
286:
282:
277:
274:
270:
266:
256:
254:
250:
245:
243:
239:
235:
231:
227:
225:
221:
217:
213:
209:
205:
201:
194:
190:
186:
182:
178:
174:
166:
162:
161:
156:
151:
137:
133:
130:
122:
116:
112:
108:
104:
100:
97:
90:
89:main category
86:
85:
80:
77:
73:
69:
67:
66:
59:
57:
52:You can help
48:
39:
38:
33:
19:
4294:
4290:
4260:
4252:
4247:
4220:
4219: », in
4216:
4212:
4208:
4203:
4197:(in French).
4194:
4184:
4173:. Retrieved
4160:
4149:. Retrieved
4145:
4136:
4125:. Retrieved
4121:
4098:
4070:
4060:
4056:
4050:
4025:Term (logic)
3970:
3966:
3962:
3958:
3954:
3950:
3946:
3942:
3938:
3934:
3930:
3639:
3591:
3564:
3541:
3536:
3528:
3518:
3496:
3489:
3477:mathematical
3470:
3412:cubic curves
3373:
3347:
3292:
3252:
3250:
3235:
3230:
3226:
3222:
3217:
3213:
3209:
3187:
3166:
3162:
3158:
3154:
3134:
3132:
3127:
3121:
3028:
3010:
3006:
3001:
2997:
2979:
2968:
2945:
2939:
2933:
2929:
2923:
2919:
2914:
2910:
2906:
2902:
2898:
2874:
2859:
2794:
2557:
2546:intersection
2541:
2537:
2533:
2529:
2528:) such that
2525:
2521:
2489:approximated
2450:
2445:
2443:
2341:
2336:
2332:
2328:
2325:
2105:
2099:
2064:
2041:
2038:
1925:
1871:
1867:multivariate
1866:
1858:
1851:real numbers
1834:
1830:
1828:
1760:
1758:
1750:
1746:
1742:
1728:
1724:
1677:
1673:
1669:
1665:
1657:
1653:
1649:
1636:
1626:
1622:
1616:
1536:
1534:
1471:
1469:
1401:
1399:
1394:
1392:
1315:
1226:Trigonometry
1224:
1219:
1215:
1213:
1130:
1126:
1106:
1102:
1100:
1013:
1009:
1008:is a set of
1003:
998:
992:
987:
983:
979:
971:
967:
963:
959:
955:
951:
947:
943:
941:
936:
928:
924:
922:
858:
854:
850:
848:
843:
840:coefficients
839:
835:
831:
829:
812:
808:
804:
801:real numbers
796:
792:
788:
786:
774:
763:
757:
753:
749:
722:
717:
713:
547:
529:Applying an
502:
500:
491:
485:
479:
473:
459:
453:
447:
441:
435:
357:
284:
278:
262:
246:
233:
228:
211:
199:
176:
170:
158:
154:
111:edit summary
102:
82:
53:
4232: [
4229:Glenn James
3965:called the
3866:derivatives
3847:derivatives
3769:derivatives
3569:(PDE) is a
3485:derivatives
3420:lemniscates
3353:mathematics
3253:Diophantine
3124:unit circle
3045:, called a
3039:coordinates
2971:dimensional
2469:engineering
1839:polynomials
520:Multiplying
514:subtracting
289:polynomials
269:equals sign
265:expressions
259:Description
218:, while in
198:. The word
193:equals sign
189:expressions
173:mathematics
4305:Categories
4209:identities
4175:2019-02-27
4151:2020-09-01
4127:2020-09-01
4082:References
3949:−1), ...,
3744:parameters
3654:polynomial
3642:operations
3614:elasticity
3610:fluid flow
3577:and their
3408:hyperbolas
3325:Almost all
3265:Alexandria
3261:Diophantus
3149:of higher
2457:algorithms
1859:univariate
1682:parameters
1662:polynomial
1113:Identities
844:parameters
824:See also:
782:inequation
778:inequality
503:equivalent
497:Properties
465:parameters
242:identities
4316:Equations
3905:−
3755:functions
3549:numerical
3537:more than
3483:with its
3400:parabolas
3380:solutions
3269:symbolism
3251:The word
3201:monomials
3197:solutions
3151:dimension
3143:manifolds
3100:
3077:
3047:parameter
2870:Descartes
2481:economics
2473:chemistry
2422:−
2398:−
2285:−
2249:−
2238:−
2197:−
2158:−
2114:variables
2014:−
1985:−
1893:−
1597:∘
1589:≈
1568:
1549:θ
1507:θ
1498:
1442:θ
1436:
1427:θ
1421:
1375:θ
1369:
1360:θ
1354:
1339:θ
1330:
1292:θ
1286:
1267:θ
1261:
1193:−
1156:−
999:solutions
836:constants
697:−
539:factoring
535:expanding
431:variables
394:−
334:−
238:solutions
216:variables
129:talk page
81:Consider
58:in French
18:Equations
4217:Equation
3993:See also
3880:′
3849:and the
3783:′
3723:integers
3529:ordinary
3481:function
3440:topology
3434:and the
3404:ellipses
3369:geometry
3303:rational
3139:surfaces
3126:, where
3043:variable
2615:, where
2536:=5 and -
2507:Geometry
2346:solution
2342:solution
1863:variable
1698:integers
735:Examples
550:function
548:If some
531:identity
524:dividing
234:unknowns
212:équation
204:cognates
202:and its
200:equation
185:equality
177:equation
105:provide
4281:Winplot
4000:Formula
3838:surface
3396:circles
3332:complex
3311:integer
3309:— with
3273:algebra
3248:on it.
3194:integer
2990:algebra
2465:physics
2071:complex
1845:(e.g.,
1737:of the
1731:+ 2 = 0
1638:Algebra
1633:Algebra
220:English
187:of two
127:to the
109:in the
4166:Marcus
3662:degree
3430:, the
3422:, and
3229:, and
3221:where
3205:degree
3033:for a
2994:curves
2918:where
2479:, and
2446:system
2056:degree
1829:where
1565:arcsin
1470:where
1238:cosine
811:, and
770:seesaw
541:a sum.
510:Adding
483:, and
457:, and
222:, any
208:French
4236:]
4042:Notes
3967:order
3616:, or
3594:sound
3475:is a
3414:like
3392:lines
3271:into
3035:curve
2974:space
2905:) + (
1843:field
1735:graph
1660:is a
923:When
832:known
293:terms
273:sides
247:The "
179:is a
175:, an
72:DeepL
4293:and
4211:and
3598:heat
3330:and
3328:real
3287:and
3165:and
3157:and
3145:and
3000:and
2988:and
2938:and
2922:and
2913:) =
2879:and
2801:cone
2647:and
2483:. A
2340:. A
2104:(or
2067:real
1837:are
1833:and
1796:, or
1680:are
1676:and
1593:20.9
1316:and
1236:and
1234:sine
1218:and
1121:and
853:and
795:and
463:are
439:and
103:must
101:You
4238:et
4063:= 0
3941:),
3860:of
3841:An
3829:An
3652:or
3648:An
3519:An
3490:In
3382:of
3293:An
3263:of
3203:of
3167:two
3163:two
3159:one
3155:one
3097:sin
3074:cos
3013:= 4
2958:or
2946:In
2558:In
2499:or
2081:).
2069:or
1763:or
1753:+ 2
1688:or
1495:sin
1433:cos
1418:sin
1366:cos
1351:sin
1327:sin
1277:cos
1252:sin
861:is
842:or
784:).
776:an
522:or
512:or
210:an
171:In
163:by
74:or
60:.
4307::
4297:).
4234:de
4227:,
4193:.
4144:.
4120:.
4109:^
4090:^
4059:–
3979:A
3925:A
3856:A
3763:A
3749:A
3738:A
3727:A
3717:A
3664::
3632:.
3612:,
3608:,
3604:,
3600:,
3596:,
3589:.
3565:A
3471:A
3460:A
3410:,
3406:,
3402:,
3398:,
3394:,
3371:.
3323:.
3240:,
3225:,
3216:=
3214:by
3212:+
3210:ax
3141:,
3029:A
3015:.
3009:+
2932:,
2909:−
2901:−
2883:.
2795:A
2540:+2
2475:,
2471:,
2467:,
2335:,
2331:,
2100:A
1853:,
1849:,
1749:–
1745:=
1727:–
1666:ax
1537:θ:
1397:.
1222:.
1135::
1004:A
1001:.
984:bx
980:ax
970:,
966:,
962:,
954:,
950:,
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925:R
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838:,
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772:.
756:,
752:,
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700:1.
596:1.
477:,
451:,
4295:y
4291:x
4178:.
4154:.
4130:.
4061:Q
4057:P
3971:x
3963:k
3959:k
3957:−
3955:x
3953:(
3951:f
3947:x
3945:(
3943:f
3939:x
3937:(
3935:f
3931:f
3911:)
3908:2
3902:x
3899:(
3896:f
3893:=
3890:)
3887:x
3884:(
3877:f
3804:2
3800:x
3796:=
3793:)
3790:x
3787:(
3780:f
3316:π
3231:c
3227:b
3223:a
3218:c
3128:t
3103:t
3094:=
3087:y
3080:t
3071:=
3064:x
3011:y
3007:x
3002:y
2998:x
2940:r
2936:)
2934:b
2930:a
2928:(
2924:b
2920:a
2915:r
2911:b
2907:y
2903:a
2899:x
2897:(
2842:2
2838:z
2834:=
2829:2
2825:y
2821:+
2816:2
2812:x
2781:.
2776:3
2771:R
2747:2
2742:R
2719:c
2716:,
2713:b
2710:,
2707:a
2687:z
2684:,
2681:y
2678:,
2675:x
2655:d
2635:c
2632:,
2629:b
2626:,
2623:a
2603:0
2600:=
2597:d
2594:+
2591:z
2588:c
2585:+
2582:y
2579:b
2576:+
2573:x
2570:a
2542:y
2538:x
2534:y
2532:+
2530:x
2526:y
2524:,
2522:x
2425:2
2416:=
2408:z
2401:2
2392:=
2384:y
2377:1
2371:=
2363:x
2337:z
2333:y
2329:x
2306:0
2299:=
2292:z
2278:y
2272:2
2269:1
2259:+
2252:x
2241:2
2231:=
2224:z
2221:4
2214:+
2207:y
2204:2
2190:x
2187:2
2179:1
2172:=
2165:z
2151:y
2148:2
2141:+
2134:x
2131:3
2022:7
2019:1
2009:2
2005:y
2001:+
1996:2
1992:y
1988:x
1980:3
1975:3
1971:x
1965:=
1960:2
1956:y
1953:x
1947:+
1942:4
1938:y
1911:0
1908:=
1905:1
1902:+
1899:x
1896:3
1888:5
1884:x
1835:Q
1831:P
1814:Q
1811:=
1808:P
1784:0
1781:=
1778:P
1751:x
1747:x
1743:y
1729:x
1725:x
1678:b
1674:a
1670:b
1658:P
1654:x
1652:(
1650:P
1627:θ
1623:θ
1602:.
1585:)
1580:3
1577:2
1572:(
1560:2
1557:1
1552:=
1520:,
1516:1
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1504:2
1501:(
1490:2
1487:3
1472:θ
1455:,
1451:1
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1445:)
1439:(
1430:)
1424:(
1415:3
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1395:θ
1378:)
1372:(
1363:)
1357:(
1348:2
1345:=
1342:)
1336:2
1333:(
1301:1
1298:=
1295:)
1289:(
1281:2
1273:+
1270:)
1264:(
1256:2
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1196:y
1190:x
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1184:)
1181:y
1178:+
1175:x
1172:(
1169:=
1164:2
1160:y
1151:2
1147:x
1107:y
1103:x
1082:3
1079:=
1072:y
1069:8
1066:+
1063:x
1060:5
1053:2
1050:=
1043:y
1040:5
1037:+
1034:x
1031:3
988:c
972:d
968:c
964:b
960:a
956:w
952:z
948:y
944:x
937:R
908:.
903:2
899:R
895:=
890:2
886:y
882:+
877:2
873:x
859:R
855:y
851:x
813:z
809:y
805:x
797:z
793:y
789:x
758:z
754:y
750:x
718:x
714:x
694:=
691:x
671:1
668:=
663:2
659:x
636:2
632:s
628:=
625:)
622:s
619:(
616:f
593:=
590:x
570:1
567:=
564:x
486:C
480:B
474:A
469:y
460:C
454:B
448:A
442:y
436:x
417:0
397:y
391:C
388:+
385:x
382:B
379:+
374:2
370:x
366:A
343:0
340:=
337:y
331:C
328:+
325:x
322:B
319:+
314:2
310:x
306:A
249:=
196:=
155:x
138:.
131:.
34:.
20:)
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