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Addition In N

Addition in N

Addition of natural numbers is the most basic arithmetic operation. In its simplest form, addition combines two numbers (terms, summands), the augend and addend, into a single number, the sum.

Notation and terms

The operation of addition, commonly written as the infix operator "+", is a function + : N × N → N. For natural numbers a, b, and c, we write :

a + b = c.\,

Here, a is the augend, b is the addend, and c is the sum.

Definition

We let S(a) denote the successor of a as defined in the Peano postulates. Addition is defined inductively by fixing the augend. In other words, we let a be any arbitrary, but fixed natural number, and we then make the following definitions:
- a + 0 = a [A1]
- a + S(b) = S(a + b) [A2] By the recursion theorem, this defines a unique function "a +" : N → N. In words, it says that adding zero to a gives back a, and that applying the successor function to the addend has the effect of applying the successor function to the sum. Since a was an arbitrary natural number, we can "put together" all these functions into a single binary operation N × N → N.

Properties

The following are three immediate and important properties of addition which can be deduced from the definition.
- Associativity: for all natural numbers a, b, and c, we have :

(a + b) + c = a + (b + c);\,

(proof)
- Commutativity: for all natural numbers a and b, we have :

a + b = b + a;\,

(proof)
- Identity element: for all natural numbers a, we have :

a + 0 = 0 + a = a.\,

(proof) Together, these three properties show that the set of natural numbers N under addition is a commutative monoid. Category:Elementary arithmetic ja:加法

Infix

Infix has meanings in linguistics, mathematics and computer science, and chemistry.

Linguistics

An infix is an affix inserted inside another morpheme. This is not uncommon in Semitic languages, in which roots are composed of three or occasionally four consonants and are conjugated by changing the vowels and sometimes inserting consonants between them. Several infixes are heard in colloquial English:
- Expletive infixation, a form of tmesis seen in profanity such as Massafuckingchusetts and absobloominlutely.
- Meaningless epenthetic sounds, such as the -iz- or -izn- of hip-hop slang (e.g., hizouse for house; shiznit for shit) or in several language games.
- The -ma- infix, whose distribution has been documented by linguist Alan C. Yu. The -ma- infix can imply that the person speaking is unintelligent, as with the words sophistimacated, saxomaphone, and edumacation.

Mathematics and computer science

In the syntax of notations used in mathematics and computer science, infix is used to describe an operator such as the usual addition sign + that is taken to bind to the variables immediately preceding and following it. See operator for more on the placement of operators.
- prefix: Polish notation
- postfix: reverse Polish notation
- infix: infix notation

Chemistry

In chemistry, infixes are used to describe molecular structure in IUPAC nomenclature. Chemical nomenclature includes the minuscule infixes -pe-, signifying complete hydrogenation (from piperidine); and -et- (from ethyl), signifying the ethyl radical C₂H₅. Thus, from picoline, we can derive pipecoline and from lutidine, we can derive lupetidine; from phenidine, we can derive phenetidine and from xanthoxylin, we can derive xanthoxyletin.

Function (mathematics)

In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics and every quantitative science. The terms function, mapping, map and transformation are usually used synonymously. The term operation is frequently used for binary functions; functions whose domain is a set of functions, or a vector space, are often called operators (see also operator (programming)).

Intuitive introduction

Essentially, a function is a "rule" or procedure that assigns an "output" value to each given "input" value. The following are examples of functions:
- In a group of people, each person has a favorite colour—from the set of red, orange, yellow, green, cyan, blue, indigo, or violet. Here, the input is the person, and the output is one of the 8 colours. The favorite colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Note that more than one person may be associated with a given colour (e.g., John and Kim may both like red), but one person cannot have more or less than one favorite color.
- A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second story, and 4 seconds to fall from the 8th story. Here, the input is the story, and the output is the number of seconds. The relevant function describes the relationship between the time it takes the stone to reach the ground and the story. (See acceleration) The "rule" defining a function can be specified by a formula, a relationship, or simply a table listing the outputs against inputs. The most important feature of a function is that it is consistent, or deterministic, always producing the same output from a given input. In this way, a function may be thought of as a mechanism or "machine" (a "black box") consistently converting a given valid input into its unique associated output. In certain technical contexts, the input is often called the argument of the function, and the output the value of the function. A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula. Consider for example :

f(x)=x^

which for any number x, assigns to x the associated value the square of x. A straightforward generalization is to allow functions depending on several arguments. For instance, :

g(x,y) = xy

is a function which takes the input, two expressions x and y, and assigns to it its product (output), xy. It might seem that this is not really a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single pair (x, y), then we can interpret g as a function -- the argument (unified single input) is the ordered pair (x, y), and the function value (output) is xy. Such functions whose input consists of ordered pairs are called "binary" or "2-ary". In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on earth over time: this is a function which takes location and time as arguments and gives as output value the temperature at the indicated location at the indicated moment in time. We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics and the quantitative sciences.

History

As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, and these measurements are the basis of calculus. The word function was later used by Euler during the mid-18th century to describe an expression or formula involving various arguments, e.g. f(x) = sin(x) + x³. During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis). By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis have shown that these functions are in some sense "more common" than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian motion. Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using set theory, and they sought to define every mathematical object as a set. Dirichlet and Lobachevsky independently and almost simultaneously gave the modern "formal" definition of function (see formal definition below). In this definition, a function is a special case of a relation. In most cases of practical interest, however, the differences between the modern definition and Euler's definition are negligible. The notion of function as a rule for computing, rather than a special kind of relation, has been formalized in mathematical logic and theoretical computer science by means of several systems, including the lambda calculus, the theory of recursive functions and the Turing machine.

Formal definition

Formally a function f from a set X to a set Y, written f : X → Y, is an ordered triple (X, Y, G(f)), where G(f) is a subset of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in G(f). X is called the domain of f, Y is called the codomain of F, and G(f) is called the graph of f. For each "input value" x in the domain, the corresponding unique "output value" y in the codomain is denoted by f(x). Equivalently a function f can be defined as a relation between X and Y which satisfies: # f is total, or entire: for all x in X, there exists a y in Y such that x f y (x is f-related to y), i.e. for each input value, there is at least one output value in Y. # f is many-to-one, or functional: if x f y and x f z, then y = z. i.e., many input values can be related to one output value, but one input value cannot be related to many output values. A relation between X and Y that satisfies condition (1) is a multivalued function. Every function is a multivalued function, but not every multivalued function is a function. A relation between X and Y that satisfies condition (2) is a partial function. Every function is a partial function, but not every partial function is a function. In this encyclopedia, the term "function" will mean a relation satisfying both conditions (1) and (2), unless otherwise stated. Consider the following three examples:

image:notMap1.png

This relation is total but not many-to-one; the element 3 in X is related to two elements b and c in Y. Therefore, this is a multivalued function, but not a function.

image:notMap2.png

This relation is many-to-one but not total; the element 1 in X is not related to any element of Y. Therefore, this is a partial function, but not a function.

image:mathmap2.png

This relation is both total and many-to-one, and so it is a function from X to Y. Note that the emphasis is on "-to-one" as "many" may actually mean "one". The function can be given explicitly by specifying its graph G(f) = or as :

f(x)=\left\

Associative

:This article is about associativity in mathematics. For associativity in central processor unit memory cache architecture see CPU cache. In mathematics, associativity is a property that a binary operation can have. It means that the order of evaluation is immaterial if the operation appears more than once in an expression. Put another way, no parentheses are required for an associative operation. Consider for instance the equation :(5+2)+1 = 5+(2+1) Adding 5 and 2 gives 7, and adding 1 gives an end result of 8 for the left hand side. To evaluate the right hand side, we start with adding 2 and 1 giving 3, and then add 5 and 3 to get 8, again. So the equation holds true. In fact, it holds true for all real numbers, not just for 5, 2 and 1. We say that "addition of real numbers is an associative operation". Associative operations are abundant in mathematics, and in fact most algebraic structures explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; one common example would be the vector cross product.

Definition

Formally, a binary operation

-

on a set S is called associative if it satisfies the associative law: :

(x - y) - z=x - (y - z)\qquad\mboxx,y,z\in S.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of

-

operations. Thus, when

-

is associative, the evaluation order can therefore be left unspecified without causing ambiguity, by omitting the parentheses and writing simply: :

x - y - z.

Examples

Some examples of associative operations include the following.
- In arithmetic, addition and multiplication of real numbers are associative; i.e., ::

\left.
    \begin
     (x+y)+z=x+(y+z)=x+y+z\quad
    \\
     (x\,y)z=x(y\,z)=x\,y\,z\qquad\qquad\qquad\quad\ \ \,
    \end
   \right\{for all {R{matrix{gcd{gcd{gcd{gcd{gcd{lcm{lcm{lcm{lcm{lcm{matrix

Commutative

:For other meanings of commutation, see commutation (disambiguation).

Mathematical meaning

In mathematics, especially abstract algebra, a binary operation

\times

on a set S is commutative if :

x\times y = y\times x

for all x and y in S. Otherwise, the operation is noncommutative. Additionally, if :

x\times y = y\times x

for a particular pair of elements x and y, then x and y are said to commute. Every element commutes with itself and, in a group, every element commutes with the identity, with its own inverse, and with its powers. The most well-known examples of commutative binary operations are addition and multiplication of real numbers; for example:
- 4 + 5 = 5 + 4 (since both expressions evaluate to 9)
- 2 × 3 = 3 × 2 (since both expressions evaluate to 6) Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets. Among the noncommutative binary operations are subtraction (a − b), division (a/b), exponentiation (a^b), function composition (f o g), tetration (a↑↑b), matrix multiplication, and quaternion multiplication. An abelian group is a group whose group operation is commutative. A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.) In a field both addition and multiplication are commutative.

Neurophysiological meaning

In neurophysiology, commutative has much the same meaning as in algebra. Physiologist Douglas A. Tweed and coworkers consider whether certain neural circuits in the brain exhibit noncommutativity and state: :In non-commutative algebra, order makes a difference to multiplication, so that

a\times b\neq b\times a

. This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuits that steer the eyes, head and limbs, and sensory circuits that handle spatial information. This idea is controversial: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain, but none that clearly rules out all commutative models. (Douglas A. Tweed and others, Nature 399, 261 - 263; 20 May 1999). Tweed goes on to demonstrate non-commutative computation in the vestibulo-ocular reflex by showing that subjects rotated in darkness can hold their gaze points stable in space---correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system.

Identity element

:For other uses, see identity (disambiguation). In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related concepts. The term identity element is often shortened to identity when there is no possibility of confusion; we do so in this article. Let (S,
- ) be a set S with a binary operation
- on it (known as a magma. Then an element e of S is called a left identity if e
- a = a for all a in S, and a right identity if a
- e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

Examples

As the last example shows, it is possible for (S,
- ) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identity and r is a right identity then l = l
- r = r. In particular, there can never be more than one two-sided identity.

Category:Elementary arithmetic

Elementary arithmetic encompases topics from arithmetic that are frequently taught at the primary or secondary school level. Category:Elementary mathematics Category:Arithmetic

庄子

庄子（约前369年—前286年），名周，战国时代宋国蒙（今安徽蒙城人，另说河南商丘）人。著名思想家、哲学家、文学家，是道家学派的代表人物，老子思想的继承和发展者。后世将他与老子并称为“老庄”。他也被称为蒙吏、孟庄和孟叟。

生平

庄子曾做过漆园（今安徽蒙城县）小吏，生活很穷。据《庄子·外篇·秋水》记载，楚威王曾派人邀请庄周管理楚国政事。庄子以宁为泥里嬉戏的龟而不为庙堂之龟为由，拒绝了楚威王的邀请。他一生淡泊名利，主张修身养性，清静无为，一直过着深居简出的隐居生活。和惠施交好。对于庄子的行为，有些人认为这是真正的逍遥，也有人认为是愤世嫉俗的表现，清代胡文英在《庄子独见》持此观点，他说：“人只知三闾之哀怨，而不知漆园之哀怨有甚于三闾也。盖三闾之哀怨在一国，而漆园之哀在天下；三闾之哀怨在一时，而漆园之哀怨在万世。”（三闾为屈原）

《庄子》一书和其哲学思想

庄子和其弟子的思想主要表现在《庄子》一书中。《庄子》现存33篇，分内篇、外篇、杂篇。传统上一般认为，《庄子》一书内篇为庄子所著，外篇和杂篇由庄子及其弟子以及后来学者所著。但也有人认为外篇和杂篇的大部分篇目仍为庄子本人所著，只是著述时期不同，后人托名的作品虽然也有，但是并不多。内篇最集中表现庄子哲学思想的有《齐物论》、《逍遥游》、《[養生主》等。庄子的哲学思想大体可归纳为以道为实体的本体论、“万物齐一”的相对主义认识论，并由此引发出其独有的主观唯心主义倾向和相对主义诡辩倾向。以此为基础，在深刻揭露“窃国者为诸侯”的政治现实的同时，庄子与老子一脉相承，主张一种无为而治、小国寡民的社会理想。同时，庄子崇尚一种“天人合一”，提倡“天地与我并生，万物与我为一”的精神境界，并且认为，人生的最高境界是逍遥自得，是精神的自由，而非名利。这种本于自然的人性论与伦理观，为后世的中国知识分子提供了另一种生存方式和价值观念的可能性。庄子的哲学提倡破除肉身我与认知我,追求超然物外的审美态度,于事于物不着痕迹。

《庄子》的其文学贡献

庄子同时也采用了一种不同于其他诸子的论说风格的方式阐述自己的思想，就是寓言的方式。这种方式让庄子的思想像水一般，不会惧怕后人的肢解。同时让他的观点不会被历史湮没。不同的时期拜读，会得更新的意义。庄周梦蝶、混沌开窍、庖丁解牛……都是其出色的寓言。庄子的文字，堪称中国文学史上的一宛奇葩，将先秦散文推向了一个新的高峰。相对老子而言，庄子的思想倾于对艺术及自由的追求。从庄周梦蝶、子非鱼安知鱼之乐等事情可见。

对后世的影响

虽然不像儒家一样被奉为经典，但庄子却对后世的中国有着重大的影响。与儒家对官方意识形态的塑造不同，庄子对后世的影响主要在非官方的意识形态。许多民间的行为方式、格言据考证均源出《庄子》：
- 江湖一词是对后世中国非官方社会的概括性名词，源出《庄子·内篇·逍遥游第一》
- “盗亦有道”，后世非官方社会的重要行为准则，源出《庄子·外篇·胠箧第十》

道教中的庄子

请参看

- 道家
- 道教
- 老子

外部链接

- [http://www.clas.ufl.edu/users/gthursby/taoism/cz-list.htm 庄子]
- [http://www.chinacattle.com/culture/zhuangzi/zh0001.htm 蒙城庄子]
- [http://jiezhuang.myrice.com/index.html 庄子的玄机]
- [http://www.hongdao.org/dxygl/dxygl11.html 新道家的奠基之作--沈善增《还吾庄子》的卓特成就简评] Category:中国哲学家 Category:中国思想家 Category:中国作家 Category:道家 category:春秋戰國人物 Z Z ja:荘子 ko:장자

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Addition in N

Notation and terms

Definition

Properties

Infix

Linguistics

Mathematics and computer science

Chemistry

See also

Function (mathematics)

Intuitive introduction

History

Formal definition

Associative

Definition

Examples

Commutative

Mathematical meaning

Neurophysiological meaning

See also

Identity element

Examples

See also

Category:Elementary arithmetic

庄子

生平

《庄子》一书和其哲学思想

《庄子》的其文学贡献

对后世的影响

道教中的庄子

请参看

外部链接

Ilgumos nustatymas