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From Wikipedia

Operation (mathematics)

The general operation as explained on this page should not be confused with the more specificoperators on vector spaces. For a notion in elementary mathematics, see arithmetic operation.

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation.

Operations can involve mathematical objects other than numbers. The logical valuestrue and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted. Rotations can be combined using the function composition operation, performing the first rotation and then the second. Operations on sets include the binary operations unionandintersectionand the unary operation ofcomplementation. Operations onfunctions include composition and convolution.

Operations may not be defined for every possible value. For example, in the real numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined from a set called its domain. The set which contains the values produced is called thecodomain, but the set of actual values attained by the operation is itsrange. For example, in the real numbers, the squaring operation only produces nonnegative numbers; the codomain is the set of real numbers but the range is the nonnegative numbers.

Operations can involve dissimilar objects. A vector can be multiplied by a scalar to form another vector. And the inner product operation on two vectors produces a scalar. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.

The values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs.

An operation is like an operator, but the point of view is different. For instance, one often speaks of "the operation of addition" or "addition operation" when focusing on the operands and result, but one says "addition operator" (rarely "operator of addition") when focusing on the process, or from the more abstract viewpoint, the function +: S×S → S.

General definition

An operationω is a function of the form ω : V→ Y, where V⊂ X1× … × Xk. The sets Xk are called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k (the number of arguments) is called the type or arityof the operation. Thus aunary operation has arity one, and a binary operation has arity two. An operation of arity zero, called a nullary operation, is simply an element of the codomain Y. An operation of arity k is called a k-ary operation. Thus a k-ary operation is a (k+1)-ary relation that is functional on its first k domains.

The above describes what is usually called a finitary operation, referring to the finite number of arguments (the value k). There are obvious extensions where the arity is taken to be an infinite ordinal or cardinal, or even an arbitrary set indexing the arguments.

Often, use of the term operation implies that the domain of the function is a power of the codomain (i.e. the Cartesian product of one or more copies of the codomain), although this is by no means universal, as in the example of multiplying a vector by a scalar.

Thus, since k can be 1, in the most general sense given here, operation is synonymous with function, map and mapping, that is, a relation, for which each element of the domain (input set) is associated with exactly one element of the codomain (set of possible outputs).



From Encyclopedia

Mathematics, Definition of

Over the centuries, people have thought of mathematics, and have defined it, in many different ways. Mathematics is constantly developing, and yet the mathematics of 2,000 years ago in Greece and of 4,000 years ago in Babylonia would look familiar to a student of the twenty-first century. Mathematics, says the mathematician Asgar Aaboe, is characterized by its permanence and its universality and by its independence of time and cultural setting. Try to think, for a moment, of another field of knowledge that is thus characterized. "In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to the old structure," noted Hermann Henkel in 1884. The mathematician and philosopher Bertrand Russell said that math is "the subject in which we never know what we are talking about nor whether what we are saying is true." Mathematics, in its purest form, is a system that is complete in itself, without worrying about whether it is useful or true. Mathematical truth is not based on experience but on inner consistency within the system. Yet, at the same time, mathematics has many important practical applications in every facet of life, including computers, space exploration, engineering, physics, and economics and commerce. In fact, mathematics and its applications have, throughout history, been inextricably intertwined. For example, mathematicians knew about binary arithmetic , using only the digits 0 and 1, for years before this knowledge became practical in computers to describe switches that are either off (0) or on (1). Gamblers playing games of chance led to the development of the laws of probability . This knowledge in turn led to our ability to predict behaviors of large populations by sampling . The desire to explain the patterns in 100 years of weather data led, in part, to the development of mathematical chaos theory . Therefore, mathematics develops as it is needed as a language to describe the real world, and the language of mathematics in turn leads to practical developments in the real world. Another way to think of mathematics is as a game. When players decide to join in a game—say a game of cards, a board game, or a baseball game—they agree to play by the rules. It may not be "fair" or "true" in the real world that a player is "out" if someone touches the player with a ball before the player's foot touches the base, but within the game of baseball, that is the rule, and everyone agrees to abide by it. One of the rules of the game of mathematics is that a particular problem must have the same answer every time. So, if Bill says that 3 divided by 2 is 1½, and Maria says that 3 divided by 2 is 1.5, then mathematics asks if these two different-looking answers really represent the same number (as they do). The form of the answers may differ, but the value of the two answers must be identical if both answers are correct. Another rule of the game of mathematics is consistency. If a new rule is introduced, it must not contradict or lead to different results from any of the rules that went before. These rules of the game explain why division by 0 must be undefined. For example, when checking division by multiplication it is clear that 10 divided by 2 is 5 because 2 × 5 is 10. Suppose 10/0 is defined as 0. Then 0 × 0 must be 10, and that contradicts the rule that 0 times anything is 0. One may believe that 0 divided by 0 is 5 because 0 × 5 is 0, but then 0 divided by 0 is 4, because 0 × 4 is also 0. There is another rule in the game of mathematics that says if 0 divided by 0 is 5 and 0 divided by 0 is 4, then 5 must be equal to 4—and that is a contradiction that no mathematician or student will accept. Mathematics depends on its own internal rules to test whether something is valid. This means that validity in mathematics does not depend on authority or opinion. A third-grade student and a college professor can disagree about an answer, and they can appeal to the rules of the game to decide who is correct. Whoever can prove the point, using the rules of the game, must be correct, regardless of age, experience, or authority. Mathematics is often called a language. Numbers and symbols are understood without the barrier of translation, and mathematics can be used to describe many aspects of today's world, from airline reservation systems to theories about the shape of space. Yet learning the vocabulary of mathematics is often a challenge and can be confusing. For example, mathematicians speak of the "bottom" of a fraction as the "denominator," which is a pretty frightening word to a beginner. But, like any language, mathematics vocabulary can be learned, just as Spanish speakers learn to say anaranjado, and English speakers learn to say "orange" for the same color. In Islands of Truth (1990), the mathematician Ivars Peterson says that "the understanding of mathematics requires hard, concentrated work. It combines the learning of a new language and the rigor of logical thinking, with little room for error." He goes on to say "I've also learned that mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve." see also Mathematics, New Trends in. Lucia McKay Aaboe, Asger. Episodes from the Early History of Mathematics. New York: Random House, 1964. Denholm, Richard A. Mathematics: Man's Key to Progress. Chicago: Franklin Publications, 1968. Flegg, Graham. Numbers, Their History and Meaning. New York: Barnes & Noble Books, 1983. Peterson, Ivars. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman and Company, 1990.


From Yahoo Answers

Question:so i have a problem that says... 3XaXaXaX5XyX2XaXaXaXa sorry if thats hard to read... the x's are the times sign. (not variables). so i got the answer of 3xa^7x5xYx2 is that right or wrong? am i supposed to add 3, 5, and 2 together? or leave them by themselves? or times them together? thanks for your help~

Answers:you can leave the times sign out, that way it's easier to read the answer is: 30ya^7 your answer is right, you just gotta simplify it by multiplying all the numbers

Question:what's the definition? for example i got for expanded form... a number written as the sum of it's parts.

Answers:Standard form is the way you write any number normally. Like writing five thousand and six in stand form would be 5006.

Question:I know standard form is Ax+By=C or something but I dont know the definition or how to find it. Can someone please tell me how to find it and the definition? Also can you please help me with this equation. I need to put the equation of the line with slope 1/2 through (1,-5) into standard form, or y=1/2x-11/2 or something

Answers:y = mx + b is slope-intercept form and you need m and b to write the equation of a line in your case y = -5, x = 1 and m = (1/2) plug them in and solve for b. - 5 = (1/2)(1) + b - 5 = (1/2) + b - 5.5 = b or (-11/2) Now that you have m and b... y = (1/2)x - (11/2) Now you have to change it to Standard Form, which just means moving stuff around. Multiply through by 2 (to get rid of the fractions) 2y = x - 11 Subtract x -x + 2y = -11 Multiply through by -1 (to get rid of the -1x) x - 2y = 11

Question:1- Power of a Power Property 2-Power of a Product Property 3-A Nonzero number to the zero power is 1. Give an example:____ 4-a^-n is the reciprocal of a^n : a^-n = 1/a^n "a" cannot = zero 5-Quotient of Powers Property 6-Power of a Quotient Property 7- Scientific Notation 8-Exponential Growth 9-Exponential Growth Thanks! =D

Answers:Dear,I searched the whole net..but couldn't find any as they are not proper mathematical terms. But I know what a Scientific Notation is. Scientific notation is a mathematical format used to write very large and very small numbers; this system avoids using a lot of zeros by using powers (exponents). Hope this helps!!

From Youtube

Math Definitions : What Is Short Word Form in Math? :In math, short word form is used to simplify a representation of large numbers. Discover the definition of short word form inmath with tips from a mathematics tutor in this free video on math lessons. Expert: Ken Au Bio: Ken Au is a math teacher and tutor for middle school through college levels. Au holds several international patents and has published numerous technical papers. Filmmaker: Mark Bullard

Calculus - Derivatives of Exponential Functions :Free Math Help at Brightstorm! www.brightstorm.com How to use the definition of the derivative to find a derivative formula for exponential functions.