"算術"的搜尋報告

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。

歷史
十进制计数法 在基数（前十个非负整数0，1，2，……，9）的基础上构建所有实数。一个十进制数由一个基数序列组成，每一位数字的命名取决于其相对于小数点的位置。例如：507.36表示5个100（10,...} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.

十进制计数法
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.

算術運算



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.

加法(+)



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.

減法(−)



乘法本质上是一组相同数字的重复累加或总和。乘法运算可得出乘数与被乘数（有时被通称为因数）的乘积。
乘法运算（由于其本质是重复累加）具有交换性和结合性；进而，它对加法和减法运算具有分配性。乘法单位为1，即，用1乘以任意数的结果仍为该数。并且，任意数字的乘法逆元素是其倒数，即，用一个数的倒数乘以该数，其结果为乘法单位：1。

乘法(× 或 ·)



除法是乘法的逆运算。除法运算得到两个数的商：被除数除以除数。任何被除数被零除是没有定义的。对于正数，如果被除数大于除数，其商大于1，否则商小于1（对于负数和-1有类似的规则）。商乘以乘数其结果总是被乘数。
除法运算不具有交换性和结合性。正如可以将减法视为加法，除法亦可被视作被除数和除数的倒数之间的乘法运算，即，a ÷ b = a × ⁄_b 。当被写为乘积形式，运算遵循乘法的所有特性。

乘法表
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.

數論
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。

算術教育

加法逆元
交換律
分配律
結合律
數線
有限體的算術
算術編碼
算術平均數