An equation is a mathematical Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions statement A proposition is a sentence expressing something true or false. In philosophy, particularly in logic, a proposition is identified ontologically as an idea, concept, or abstraction whose token instances are patterns of symbols, marks, sounds, or strings of words. Propositions are considered to be syntactic entities and also truthbearers, in symbols The contour integral can also frequently be found with a subscript capital letter C, ∮C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮S, is used to denote that the integration is over a closed surface, that two things are exactly the same (or equivalent). Equations are written with an equal sign The equal sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Welshman Robert Recorde. The equals sign is placed between the things stated to be exactly the same, as in an equation. It is the Unicode and ASCII character 003D, as in
The equations above are examples of an equality The identity relation is the paradigmatic example of the more general concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and transitive. The relation of equality is also antisymmetric. These four properties uniquely determine the equality relation on any set S and render equality the only relation: a proposition A proposition is a sentence expressing something true or false. In philosophy, particularly in logic, a proposition is identified ontologically as an idea, concept, or abstraction whose token instances are patterns of symbols, marks, sounds, or strings of words. Propositions are considered to be syntactic entities and also truthbearers which states that two constants A mathematical constant is a number, usually a real number, that arises naturally in mathematics. Unlike physical constants, mathematical constants are defined independently of physical measurement are equal. Equalities may be true or false.
Equations are often used to state the equality of two expressions In mathematics, the word expression is a term for any well-formed combination of mathematical symbols[citation needed]. An algebraic expression is only a phrase, not a whole sentence, so it cannot contain an equality sign . For example, containing one or more variables A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use. The concepts of constants and variables are fundamental to all modern mathematics, science, engineering, and computer programming. In the reals In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an we can say, for example, that for any given value of x it is true that
The equation above is an example of an identity The concepts of "additive identity" and "multiplicative identity" are central to the Peano axioms. The number 0 is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all , that is, an equation that is true The word truth has a variety of meanings, from honesty, good faith, and sincerity in general, to agreement with fact or reality in particular. The term has no single definition about which a majority of professional philosophers and scholars agree, and various theories and views of truth continue to be debated. There are differing claims on such regardless of the values of any variables that appear in it. The following equation is not an identity:
It is false for an infinite number of values of x, and true for only two, the roots or solutions of the equation, x = 0 and x = 1. Therefore, if the equation is known to be true, it carries information about the value of x. To solve an equation In mathematics, equation solving is the problem of finding what values fulfill a condition stated as an equality (an equation). Usually, this condition involves expressions with variables (or unknowns), which are to be substituted by values in order for the equality to hold. More precisely, an equation involves some free variables means to find its solutions.
Many authors reserve the term equation for an equality which is not an identity. The distinction between the two concepts can be subtle; for example,
is an identity, while
is an equation, whose roots are x = 0 and x = 1. Whether a statement is meant to be an identity or an equation, carrying information about its variables can usually be determined from its context; or by making a distinction between the equality sign ( = ) for a statement not true except perhaps in particular situations, and the equivalence symbol () for statements know to be true without further specification.
Letters from the beginning of the alphabet like a, b, c... often denote constants In mathematics, a coefficient is a constant multiplicative factor of a specific object. For example, in the expression 9x2, the coefficient of x2 is 9 in the context of the discussion at hand, while letters from end of the alphabet, like x, y, z..., are usually reserved for the variables A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use. The concepts of constants and variables are fundamental to all modern mathematics, science, engineering, and computer programming, a convention initiated by Descartes René Descartes , (31 March 1596 – 11 February 1650), also known as Renatus Cartesius (Latinized form), was a French philosopher, mathematician, physicist, and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the "Father of Modern Philosophy", and much of subsequent Western philosophy is a response to.
Properties
If an equation in algebra Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. While in arithmetic only numbers and their arithmetical operations occur, in algebra one also uses symbols (such as x and y, or a and b) to denote numbers. These are is known to be true, the following operations may be used to produce another true equation:
- Any quantity can be added Addition is the mathematical process of combining quantities. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples, since 3 + 2 = 5. Besides counts of fruit, addition can also represent to both sides.
- Any quantity can be subtracted Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with. Subtraction is denoted by a minus sign in infix notation from both sides.
- Any quantity can be multiplied Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic to both sides.
- Any nonzero quantity can divide In mathematics, especially in elementary arithmetic, division is the arithmetic operation that is the inverse of multiplication. Division can be described as repeated subtraction whereas multiplication is repeated addition both sides.
- Generally, any function In mathematics a function is a relation between a given set of elements and another set of elements (the codomain), which associates each element in the domain with exactly one element in the codomain. The elements so related can be any kind of thing (words, objects, qualities) but are typically mathematical quantities, such as real numbers can be applied to both sides. (However, caution must be exercised to ensure that one does not encounter extraneous solutions In mathematics, an extraneous solution represents 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 original problem. A missing solution is a solution that was a valid solution to the original problem, but disappeared during the process of solving the problem. Both are.)
The algebraic properties (1-4) imply that equality is a congruence relation In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation for a field In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic; in fact, it is essentially the only one.
The most well known system of numbers which allows all of these operations is the real numbers In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in, which is an example of a field In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic. However, if the equation were based on the natural numbers In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century for example, some of these operations (like division and subtraction) may not be valid as negative numbers and non-whole numbers are not allowed. The integers The integers are natural numbers including 0 (0, 1, 2, 3, ...) and their negatives (0, −1, −2, −3, ...). They are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 65, 7, and −756 are integers; 1.6 and 1½ are not integers. In other terms, integers are an example of an integral domain In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain which does not allow all divisions as, again, whole numbers are needed. However, subtraction is allowed, and is the inverse operator In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A (or from B to A to B) returns each element of the initial set to itself. Thus, if an input x into the function ƒ produces an output y, then inputting y into the in that system.
If a function that is not injective In mathematics, an injective function is a function that associates distinct arguments with distinct values; in other words, every unique argument produces a unique result. It is not necessary that all elements in codomain must be mapped is applied to both sides of a true equation, then the resulting equation will still be true, but it may be less useful. Formally, one has an implication The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function from truth-values to truth-values. In predicate logic, it can be viewed as a subset relation between the extension of predicates. In, not an equivalence In logic and mathematics, the logical biconditional is a logical operator connecting two statements to assert "p if and only if q", where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). The operator is denoted using a doubleheaded arrow (↔), an equality sign (=), an equivalence sign (≡), or EQV. It is, so the solution set may get larger. The functions implied in properties (1), (2), and (4) are always injective, as is (3) if we do not multiply by zero 0 is both a number and the numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems. In the English language, zero may also be called oh, null, nil,. Some generalized products In the a mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied the product, such as a dot product In mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the orthonormal Euclidean space. It contrasts with the cross product which produces a vector result, are never injective.
See also
- Cubic equation where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function
- Differential equation A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics and other disciplines
- Diophantine equation In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. 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 or more
- Formula editor Content for formula editors can be provided manually using a markup language,eg TeX or MathML, via a point-and-click GUI, or as computer generated results from symbolic computations such as Mathematica
- Functional equation In mathematics or its applications, a functional equation is any equation that specifies a function in implicit form . Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional equations they satisfy. The term
- Indeterminate equation An indeterminate equation, in mathematics, is an equation for which there is an infinite set of solutions; for example, 2x = y is a simple indeterminate equation. Indeterminate equations cannot be directly solved from the given information. For example, the equations
- Inequality In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not
- Inequation In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. This relation is written with a crossed-out equal sign, like
- Integral equation In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Maxwell's equations
- Linear equation A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable
- List of equations This is a list of equations, by Wikipedia page. See also list of equations in classical mechanics, list of relativistic equations, equation solving, theory of equations
- Quadratic equation In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is
- Quartic equation where a is nonzero; or in other words, a polynomial of degree of four. Such a function is sometimes called a biquadratic function, but the latter term can occasionally also refer to a quadratic function of a square, having the form
- Quintic equation
- Parametric equation
- Polynomial equation
- Scientific equations named after people
- Simultaneous equation
- Theory of equations
External links
- Winplot: General Purpouse plotter which can draw and animate 2D and 3D mathematical equations.
- Mathematical equation plotter: Plots 2D mathematical equations, computes integrals, and finds solutions online.
- 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).
- EqWorld — contains information on solutions to many different classes of mathematical equations.
- EquationSolver: A webpage that can solve single equations and linear equation systems.
Categories: Elementary algebra | Equations
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Google Docs added the . equation. editor previously available in Knol. It's a basic LaTeX editor that's not very easy to use if you aren't familiar with LaTeX or.
Q. I'm not sure of the process to find the Cartesian equation. My book has examples of how to solve it but none are similar to this question. If you could show me the steps that'd be great! Find the Cartesian equation: rcos(theta)=6 Thanks!
Asked by Jackie - Wed Jul 15 14:51:07 2009 - - 1 Answers - 0 Comments
A. x = 6
Answered by Melissa Baker - Wed Jul 15 14:56:15 2009
