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Encyclopedia > Equation

An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). Equations are written with an equal sign, as in Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Proposition is a term used in logic to describe the content of assertions. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... See also the disambiguation page title equality. ...

2 + 3 = 5.

Equations are often used to state the equality of two expressions containing one or more variables. For example, given any value of x, it is always true that An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ...

xx = 0.

The two equations above are examples of identities: equations that are true regardless of the values of any variables that appear within them. The following equation is not an identity: In mathematics, the term identity has several important uses: identity can refer to an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ...

x + 1 = 2.

The above equation is false for an infinite number of values of x, and true for only one; the unique root of the equation, x = 1. Therefore, if the equation is known to be true, it carries information about the value of x. In general, the values of the variables for which the equation is true are called solutions. To solve an equation means to find its solutions. In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, equation solving is the problem of finding what values (numbers, functions, sets, etc. ...


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,

(x + 1)2 = x2 + 2x + 1

is an identity, while

(x + 1)2 = 2x2 + x + 1

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.


Letters from the beginning of the alphabet like a, b, c, ... are often considered constants in the context of the discussion at hand, while letters from end of the alphabet, like x, y, z, are usually considered variables. In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ...


Properties

If an equation in algebra is known to be true, the following operations may be used to produce another true equation: 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. ...

  1. Any quantity can be added to both sides.
  2. Any quantity can be subtracted from both sides.
  3. Any quantity can be multiplied to both sides.
  4. Any nonzero quantity can divide both sides.
  5. Generally, any function can be applied to both sides. (However, caution must be exercised to ensure that one does not encounter extraneous solutions.)

The algebraic properties (1-4) imply that equality is a congruence relation for a field; in fact, it is essentially the only one. 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... Partial plot of a function f. ... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...


The most well known system of numbers which allows all of these operations is the real numbers, which is an example of a field. However, if the equation were based on the natural numbers 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 are an example of an integral domain which does not allow all divisions as, again, whole numbers are needed. However, subtraction is allowed, and is the inverse operator in that system. Please refer to Real vs. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...


If a function that is not injective 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, not an equivalence, 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. Some generalized products, such as a dot product, are never injective. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... In logical calculus of mathematics, 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). ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...


See also

In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. ... The feasible regions of linear programming are defined by a set of inequalities. ... A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... An indeterminate equation is an equation for which there is an infinite set of solutions – for example, 2x = y. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ... This is a list of equations, by Wikipedia page. ... In mathematics, the theory of equations comprises a major part of traditional algebra. ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

External links

  • Mathematical equation plotter: Plots 2D mathematical equations, computes integrals, and finds solutions online.
  • Equation plotter: A web page that can plot general equations, not just functions.
  • WZGrapher: A Windows freeware program that plots Cartesian and polar equations, with both integration and differentiation solvers and graphing capabilities.
  • Equation Wizard: Automatic algebraic equation solver
  • EqWorld — contains information on solutions to many different classes of mathematical equations.
  • EquationSolver: A webpage that can solve single equations and linear equation systems.

 
 

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