算术
维基百科，自由的百科全书
算術是數學最古老且最簡單的一個分支，幾乎被每個人使用著，從日常上簡單的算數到高深的科學及工商業計算都會用到。一般而言，算術這一詞指的是記錄數字某些運算基本性質的數學分支。常用的运算有加法、减法、乘法、除法，有时候，更复杂的运算如指数和平方根，也包括在算术运算的范畴内。算术运算要按照特定规则来进行。
自然数、整数、有理数（以分数的形式）和实数（以十进制指数的形式）的运算主要是在小学和中学的时候学习。用百分比形式进行运算也主要是在这个时候学习。然而，在成人中，很多人使用计算器，计算机或者算盘来进行数学计算。
專業數學家有時會使用高等算術來指數論，但這不應該和初等算術相搞混。另外，算術也是初等代數的重要部份之一。
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[编辑] 歷史
算術的史前史只有極小部份能有加法與減法等明確概念的人造物，最著名的一件是在非洲發明的伊珊郭骨頭，距今約有兩萬年的時間。
比較清楚的是，巴比倫尼亞在西元前1850年已有關於各方面初等算術的堅實知識，但歷史學家也只能依其算術成果來推斷其使用的方式(看巴比倫楔形泥版322(Plimpton322))。同樣地，乘法和單位分數的運用的可靠演算法也在古埃及的賴因德數學古本中被發現，其約在西元前1650年的時期。
西元前六世紀中葉，畢達哥拉斯學派的時代，算術已被視為四種計量或數學科學中的其中一種了。 These were carried over in mediæval universities as the Quadrivium which, together with the Trivium of grammar, rhetoric and dialectic, constituted the septem liberales artes (seven liberal arts).
Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Arabic numerals and decimal place notation for numbers. Although it is now considered elementary, its simplicity is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician Archimedes devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.
[编辑] 十進位算術
Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,...,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: for example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10-1) plus 6 hundredths (10-2). An essential part of this notation (and a major stumbling block in achieving it) was conceiving of 0 as a number comparable to the other basic digits.
Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations {...,102,10,1,10-1,...} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.
[编辑] 算術運算
算術運算指加法、減法、乘法和除法,但有時也包括較高級的運算(例如百分比、平方根、取冪和對數)。算術按運算次序進行， Any set of objects upon which all four operations of arithmetic can be performed (except division by zero, and wherein these four operations obey the usual laws, is called a field.
[编辑] 加法(+)
- 主條目：加法
加法是基本算術運算。In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum.
Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting.
Addition is commutative and associative so the order in which the terms are added does not matter. The identity element of addition (the additive identity) is 0, that is, adding zero to any number will yield that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself will yield the additive identity, 0. For example, the opposite of 7 is (-7), so 7 + (-7) = 0.
[编辑] 減法(−)
- 主條目：減法
減法是加法的相反。Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero.
Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is a − b = a + (−b). When written as a sum, all the properties of addition hold.
[编辑] 乘法(× 或 *)
- 主條目：乘法
Multiplication is in essence repeated addition, or the sum of a list of identical numbers. Multiplication finds the product of two numbers, the multiplier and the multiplicand, sometimes both just called factors.
Multiplication, as it is really repeated addition, is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverse is the reciprocal of any number, that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity, 1.
[编辑] 除法(÷ 或 /)
- 主條目：除法
除法是乘法的相反。 Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers and negative one). The quotient multiplied by the divisor always yields the dividend.
Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1⁄b. When written as a product, it will obey all the properties of multiplication.
[编辑] 例子
[编辑] 加法表
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[编辑] 乘法表
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[编辑] 數論
The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry. Number theory is also referred to as 'the higher arithmetic', as in the title of H. Davenport's book on the subject.
[编辑] 算術教育
國小時的數學通常專注在自然數、整數、有理數(分數和實數(使用十進位法)等算術的演算法。此一學習有時被稱為algorism。
這種演算法的困難性及無目的性的樣貌已讓教育學家們很長時間地去思考其課程內容，主張早期應該教導較中心且直覺的數學概念。在此一方向上的著名進展為1960年代至1970年代的新數學運動，它試圖以集合論中公理化(高等數學的主流)的精神來教導算術。
當能比人腦更有效地執行運算的電子計算機被發明後，有影響力的學校的教育家們開始聲稱標準算術演化法的機械化熟練已不再是必須的了。在他們的觀點，一年級的數學可以花更多在了解更高等的概念上，如數字被使用來哪裡和數字、數量和度量之間的關係等。但無論如何，許多的數學家依然認為手算的熟練會是學習代數和電腦科學的必要基礎。這一爭論主要集中在加州1990年代國小課程上頭，並且延續至今日。
在台灣，算術教育是要採建構式數學，亦或採台灣傳統的九九乘法表也有一段爭議的時間。但是，因為政府沒有清楚說明何為建構式數學，老師們又沒多少人懂建構式數學的精神，到現在，學校內幾乎沒有再聽到建構式數學的聲音了。
