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算数

(2009-04-06 23:39:57)
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杂谈

Decimal arithmetic

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).

十进制算术
十进制符号构造所有实数的基本数字,第一个10非负整数0,1,2 ,..., 9 。十进制数字组成的序列,这些基本的数字,与“面值”的每个数字取决于其立场对小数点:例如, 507.36是指5数百名( 102 ) ,加上0数万( 101 ) ,加上7单位( 100 ) ,加上十分之三( 10月1日) ,另加百分之六( 10月2日) 。的一个重要组成部分这个符号(和一个主要的绊脚石实现它)是构想零一些可比的其他基本数字。

An essential part of this notation (and a major stumbling block in achieving it) was conceiving of zero 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.

算法包括所有规则的执行算术运算使用十进制数字系统的代表在书面使用数字符号有10 0到9的价值观相结合,用一个价值系统(定位符号) ,其中每个符号的10倍重量1至其权利。

 

这符号允许添加任意数量的增加数字在每个地方,这是完成了10 × 10此外就座。 (甲总和超过9位数字,必须有它的10位数字进行到下一个地方左倾。 )一个可以作出类似的算法乘以任意号码,因为一套面值{..., 10 ² , 10,1,10 - 1 ,...}是封闭乘法。减法和司所取得的类似,但更复杂的算法。

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.

[edit] Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject.

[编辑]算术运算
传统的算术运算的加,减,乘除,但更先进的业务(如操纵的百分比,平方根,指数,以及对数函数)有时也包含在这个问题。

算术根据执行的命令行动。任何一套物体赖以所有四个业务的算术就可以执行(除司零) ,其中这四个行动服从一般的法律,被称为领域。
Arithmetic is performed according to an
order of operations. 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.

[edit] Addition (+)

Main article: Addition
 

Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum of the numbers.[编辑]加成( + )
主要文章:增加
此外,这是基本的操作算法。最简单的形式,除了结合了两个号码, addends或条款,到一个单一的号码,金额的数目。

新增两个以上的号码可以被视为重复此外,这个程序被称为总和,包括如何购买无穷多个号码在无穷级数;一再增加了排名第一的是最基本的形式计数

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.

Addition can be given geometrically as follows:

If a and b are the lengths of two sticks, then if we place the sticks one after the other, the length of the stick thus formed will be a + b.

[edit] Subtraction (−)

Main article: Subtraction
 

Subtraction is essentially the opposite of addition. 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.[编辑]减法( - )
主要文章:减法
减法基本上是相反的另外。减法发现之间的区别两个号码,减数减去减数。如果减数大于减数,差异将是积极的;如果减数小于减数,差额将负面的影响;如果他们是平等的,差额将为零。

减法既不是交换,也没有关联。出于这个原因,往往是有帮助看看减为增加减数和对面的减数,这是一个-乙= 1 + ( - b )项。当书面作为总之,所有的属性除了举行。

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.

[edit] Multiplication (×, ·, or *)

Main article: Multiplication
 

Multiplication is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product.

[编辑]乘法( × ,或* )
主要文章:乘法
乘法是第二个基本操作的算法。乘法还结合了两个数字到一个单一的号码,该产品。这两个数字被称为原始的乘数和被乘数,有时都只是所谓的因素。

The two original numbers are called the multiplier and the multiplicand, sometimes both simply called factors.

 

Multiplication is best viewed as a scaling operation. If the real numbers are imagined as lying in a line, multiplication by a number, say x, greater than 1 is the same as stretching everything away from zero uniformly, in such a way that the number 1 itself is stretched to where x was.

乘法是最好看作是一个缩放操作。如果真正的数字是想象作为躺在线,乘法的一个数字,说十,大于1是一样的伸展一切远离零均匀,以这样一种方式的数量1本身延伸到其中x是。同样,乘以若干不到1可以想像作为压缩实现零。 (同样,在这样一种方式1去被乘数。 )
Similarly, multiplying by a number less than 1 can be imagined as squeezing towards zero. (Again, in such a way that 1 goes to the multiplicand.)

乘法是最好看作是一个缩放操作。如果真正的数字是想象作为躺在线,乘法的一个数字,说十,大于1是一样的伸展一切远离零均匀,以这样一种方式的数量1本身延伸到其中x是。同样,乘以若干不到1可以想像作为压缩实现零。 (同样,在这样一种方式1去被乘数。 )

 

Multiplication 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.乘法的交换和关联;进一步它分配了加减。倍增的身份是1 ,也就是说,任何数字乘以1将产生同样的号码。此外,乘法逆是互惠的任何数量,也就是乘以倒数任何数目的数量本身将产生倍增的身份。

[edit] Division (÷ or /)

 

Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. [编辑]司( ÷或/ )
主要文章:司(数学)
司基本上是相反的乘法。司认为,智商的两个数字,该股息除以除数。任何股息除以零未定义。

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). The quotient multiplied by the divisor always yields the dividend.

 

为正数,如果股息大于除数,商将大于1 ,否则这将是不到一年(也适用类似的规则为负数) 。商乘以除数总是股息收益率。

司既不是交换,也没有关联。因为它是帮助看看减法作为此外,它有助于看看司乘红利次的倒数除数,这是一个÷ b = 1 × 1 /湾当书面作为一个产品,它会遵守所有的性能倍增。

 

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 × 1b. When written as a product, it will obey all the properties of multiplication.

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