equation

American Heritage Dictionary:

e·qua·tion

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(ĭ-kwā'zhən, -shən) pronunciation

The act or process of equating or of being equated.
The state of being equal.
Mathematics. A statement asserting the equality of two expressions, usually written as a linear array of symbols that are separated into left and right sides and joined by an equal sign.
Chemistry. A representation of a chemical reaction, usually written as a linear array in which the symbols and quantities of the reactants are separated from those of the products by an equal sign, an arrow, or a set of opposing arrows.
A complex of variable elements or factors: "The world was full of equations . . . there must be an answer for everything, if only you knew how to set forth the questions" (Anne Tyler).

equational e·qua'tion·al adj.
equationally e·qua'tion·al·ly adv.

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Home > Library > Miscellaneous > Britannica Concise Encyclopedia

Statement of equality between two expressions consisting of variables and/or numbers. In essence, equations are questions, and the development of mathematics has been driven by attempts to find answers to those questions in a systematic way. Equations vary in complexity from simple algebraic equations (involving only addition or multiplication) to differential equations, exponential equations (involving exponential expressions), and integral equations. They are used to express many of the laws of physics. system of equations.

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Computer Desktop Encyclopedia:

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An arithmetic expression that equates one set of conditions to another; for example, A = B + C. In a programming language, assignment statements take the form of an equation. The above example assigns the sum of B and C to the variable A.

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Roget's Thesaurus:

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noun

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same/different/compare

Columbia Encyclopedia:

equation

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equation, in mathematics, a statement, usually written in symbols, that states the equality of two quantities or algebraic expressions, e.g., x+3=5. The quantity x+3, to the left of the equals sign (=), is called the left-hand, or first, member of the equation, that to the right (5) the right-hand, or second, member. A numerical equation is one containing only numbers, e.g., 2+3=5. A literal equation is one that, like the first example, contains some letters (representing unknowns or variables). An identical equation is a literal equation that is true for every value of the variable, e.g., the equation (x+1)²=x²+2x+1. A conditional equation (usually referred to simply as an equation) is a literal equation that is not true for all values of the variable, e.g., only the value 2 for x makes true the equation x+3=5. To solve an equation is to find the value or values of the variable that satisfy it. Polynomial equations, containing more than one term, are classified according to the highest degree of the variable they contain. Thus the first example is a first degree (also called linear) equation. The equation ax²+bx+c=0 is a second degree, or quadratic, equation in the unknown x if the letters a, b, and c are assumed to represent constants. In algebra, methods are evolved for solving various types of equations. To be valid the solution must satisfy the equation. Whether it does can be ascertained by substituting the supposed solution for the variable in the equation. The simultaneous solution of two or more equations is a set of values of the variables that satisfies each of the equations. In order that a solution may exist, the number of equations (i.e., conditions) must generally be no greater than the number of variables. In chemistry an equation (see chemical equation) is used to represent a reaction.

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IN BRIEF: A mathematical statement showing that two quantities have the same value by putting a special sign (=) between them.

Love, and you shall be loved. All love is mathematically just, as much as the two sides of an algebraic equation. — Ralph Waldo Emerson (1803-1882).

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Dictionary of Cultural Literacy: Science:

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An expression of equality between two formulas in mathematics. The two formulas are written with an equal sign between them: 2 + 2 = 4 is an equation, as is E = mc².

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Quantities, Relationships, and Operations - equation: mathematical statement containing the equals (=) sign between two different but equal algebraic formulations

Rhymes:

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Wikipedia on Answers.com:

Equation

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This article is about equations in mathematics. For the chemistry term, see chemical equation.

The first use of an equals sign, equivalent to 14x+15=71 in modern notation. From The Whetstone of Witte by Robert Recorde (1557).

An equation is a mathematical statement that asserts the equality of two expressions.^[1] In modern notation, this is written by placing the expressions on either side of an equals sign (=), for example

$x + 3 = 5\,$

asserts that x+3 is equal to 5. The = symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more equal than parallel straight lines with the same length.

Contents

1 Knowns and unknowns
2 Types of equations
3 Identities
4 Properties
5 See also
6 References
7 External links

Knowns and unknowns

Types of equations

Equations can be classified according to the types of operations and quantities involved. Important types include:

An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree.
A linear equation is an algebraic equation of degree one.
A polynomial equation is an equation in which a polynomial is set equal to another polynomial.
A transcendental equation is an equation involving a transcendental function of one of its variables.
A functional equation is an equation in which the unknowns are functions rather than simple quantities.
A differential equation is an equation involving derivatives.
An integral equation is an equation involving integrals.
A Diophantine equation is an equation where the unknowns are required to be integers.
A quadratic equation

Identities

One use of equations is in mathematical identities, assertions that are true independent of the values of any variables contained within them. For example, for any given value of x it is true that

$x (x-1) = x^2-x\,.$

However, equations can also be correct for only certain values of the variables.^[2] In this case, they can be solved to find the values that satisfy the equality. For example, consider the following.

$x^2-x = 0\,.$

The equation is true only for two values of x, the solutions of the equation. In this case, the solutions are $x=0$ and $x=1$ .

Many mathematicians^[2] reserve the term equation exclusively for the second type, to signify an equality which is not an identity. The distinction between the two concepts can be subtle; for example,

$(x + 1)^2 = x^2 + 2x + 1\,$

is an identity, while

$(x + 1)^2 = 2x^2 + x + 1\,$

is an equation with solutions $x=0$ and $x=1$ . Whether a statement is meant to be an identity or an equation can usually be determined from its context. In some cases, a distinction is made between the equality sign ( $=$ ) for an equation and the equivalence symbol ( $\equiv$ ) for an identity.

Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand, while letters from the end of the alphabet, like ...x, y, z, are usually reserved for the variables, a convention initiated by Descartes.

Properties

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

Any real number can be added to both sides.
Any real number can be subtracted from both sides.
Any real number can be multiplied to both sides.
Any non-zero real number can divide both sides.
Some functions can be applied to both sides. Caution must be exercised to ensure that the operation does not cause missing or extraneous solutions. For example, the equation $yx=x$ has 2 sets of solutions: $y=1$ (with any x) and $x=0$ (with any y). Raising both sides to the exponent of 2 (which means, applying the function $f(s)=s^2$ to both sides of the equation) changes our equation into $(xy)^2=x^2$ , which not only has all the previous solutions but also introduces a new set of extraneous solutions, with $y=-1$ and x being any number.

The algebraic properties (1-4) imply that equality is a congruence relation for a field; 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, 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.

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.

More information at Equation solving.

References

^ "Equation". Dictionary.com. Dictionary.com, LLC. http://dictionary.reference.com/browse/equation. Retrieved 2009-11-24.
^ ^a ^b Nahin, Paul J. (2006). Dr. Euler's fabulous formula: cures many mathematical ills. Princeton: Princeton University Press. p. 3. ISBN 0-691-11822-1.

External links

Winplot: General Purpose 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.

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Translations:

Equation

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Dansk (Danish)
n. - lighed, udligning, ligning, reaktionsligning

Nederlands (Dutch)
vergelijking, het gelijkmaken, factor/ samenspel van factoren, associatie, correctie van een onnauwkeurigheid

Français (French)
n. - assimilation, égalisation, (Chim, Math) équation, (Astron) équation

Deutsch (German)
n. - Gleichung

Ελληνική (Greek)
n. - (μαθημ.) εξίσωση

Italiano (Italian)
equazione

Português (Portuguese)
n. - equação (f)

Русский (Russian)
выравнивание, уравнение

Español (Spanish)
n. - ecuación

Svenska (Swedish)
n. - ekvation, utjämning, jämställande

中文（简体）(Chinese (Simplified))
相等, 平衡, 等式

中文（繁體）(Chinese (Traditional))
n. - 相等, 平衡, 等式

한국어 (Korean)
n. - 평균, 평형상태, 방정식

日本語 (Japanese)
n. - 等しくすること, 等しい状態, 平衡状態, 等しいとみなすこと, 誤差, 等式, 方程式, 反応式, 均分

العربيه (Arabic)
‏(الاسم) معادله, توازن, تسويه‏

עברית (Hebrew)
n. - ‮משוואה, השוואה‬

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Dictionary of Cultural Literacy: Science The New Dictionary of Cultural Literacy, Third Edition Edited by E.D. Hirsch, Jr., Joseph F. Kett, and James Trefil. Copyright © 2002 by Houghton Mifflin Company. Published by Houghton Mifflin. All rights reserved. Read

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