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 that asserts the equality of two expressions In mathematics, an expression is a finite combination of symbols that are well-formed according to the rules applicable in the context at hand. Symbols can designate values , variables, operations, relations, or can constitute punctuation or other syntactic entities. The use of expressions can range from simple arithmetic operations like.[1] Equations consist of the expressions that have to be equal on opposite sides of an equal sign The equality 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

One use of equations is in mathematical identities 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 , 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

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

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,

is an identity, while

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 () for an identity.

Letters from the beginning of the alphabet like a, b, c... often denote constants which makes the function-argument status of x clear, and thereby implicitly the constant status of a, b and c. In this example those constants are actually coefficients of the polynomial, and since c occurs in a term that does not involve x, it is called the constant term of the polynomial and can be thought of as the coefficient of x0 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 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; adjectival form: "Cartesian"), 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.

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. Elementary is known to be true, the following operations may be used to produce another true equation:

  1. Any quantity can be added Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . 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. Therefore, 3 + 2 = 5. Besides counts of fruit, to both sides.
  2. 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.
  3. 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 (the others being addition, subtraction and division) to both sides.
  4. Any nonzero quantity can divide In mathematics, especially in elementary arithmetic, division is the arithmetic operation that is the inverse of multiplication both sides.
  5. Generally, any function The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain 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 abstract algebra, a congruence relation is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation 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 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, which is an example of a field. However, if the equation were based on the natural numbers In mathematics, natural numbers are the ordinary counting numbers 1, 2, 3, ... . Since the development of set theory by Georg Cantor, it has become customary to view such numbers as a set. 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 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 formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as a subset of the real numbers, 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, are an example of an integral domain In abstract algebra, an integral domain is a commutative ring with 1 ≠ 0 that has no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity 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 preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain. If in addition all of the elements in the codomain are in fact mapped to by 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 material implication, is a binary truth function → such that the compound sentence p→q is logically equivalent to the negative compound: not both p and not q. A material conditional compound itself is often simply called a conditional. By definition of "→", the compound p→q is false if and, 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 Zero, written 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, 0 is used as a placeholder in place value systems. In the English language, 0 may be called zero, oh,. Some generalized products In 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 is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and adding up those products. The name is derived from the centered dot "·" that is often used to designate this operation; the alternative name scalar, are never injective.

See also

References

  1. ^ "Equation". Dictionary.com. Dictionary.com, LLC. http://dictionary.reference.com/browse/equation. Retrieved 2009-11-24.
  2. ^ a b Nahin, Paul J. (2006). Dr. Euler's fabulous formula: cures many mathematical ills. Princeton: Princeton University Press. p. 3. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0-691-11822-1.

External links

Categories: Elementary 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 | Equations

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