real number - 360文档中心

2.4Rational and Irrational Numbers有理数与无理数

Rational numbers
fractions Integers Whole numbers
Irrational numbers
Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.
Rational and Irrational Numbers Essential Question
How do I distinguish between rational and irrational numbers?
The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers. Real Numbers
C. 4 0 not a real number
Summary:
• 1.Can you classify real number ,rational number, irrational number , • Whole number, natural number ? • 2.Do you have any question?
B. –12.75 –12.75 is a terminating decimal. rational, real C. 16 2 16 4 = =2 2 2
whole, integer, rational, real

实数实数 real number(English)

实数real numberreal numbers 实数 integers 整数 Fraction 分数decimals 小数 irrational Numbers 无理数 rational numbers 有理数positive number 正数 negative number 负数 opposite number 相反数count backwards 倒数Teaching aims1．了解无理数与实数的意义；Understanding of irrational Numbers and the notion of real Numbers2．了解有理数的运算法则(?)在实数范围内仍然适用；Understand the rational range of algorithms3．能利用化简对实数进行简单的四则运算；simple arithmetic of the real number by simplification4．了解实数的意义，能对实数按要求进行分类；The notion and the classification of the real numbers5．掌握有理数的运算法则在实数范围内仍然适用；Grasp the rational range of algorithms in real numbers6．能利用实数的性质熟练地进行四则运算；Master the arithmetic of the rational numbers well.7．注意：（1）无理数应满足：①是小数；②是无限小数；③不循环；Irrational Numbers should be: (1) a decimal number; (2) infinite decimals; (3)no cycles（2）无理数不是都带根号的数（例如π就是无理数），反之，带根号的数也不一定都是无理数（例如4，327就是有理数）．Not all irrational Numbers take square root (such as π is irrational ）,On the other hand, radicals are not necessarily irrational Numbers （for example4，327Is a rational number ）（1）按实数的定义分类：According to the definition of real Numbers ⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎨⎧⎭⎬⎫⎩⎨⎧⎪⎪⎪⎩⎪⎪⎪⎨⎧⎭⎬⎫⎩⎨⎧⎪⎩⎪⎨⎧decimal repeating -non infinite number irrational fraction negative fraction Positive fraction integer negative integer positive integer number rational 无限不循环小数负无理数正无理数无理数数有限小数或无限循环小负分数正分数分数负整数零正整数整数有理数实数zero（2）按实数的正负分类：⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎨⎧⎪⎩⎪⎨⎧⎩⎨⎧⎪⎩⎪⎨⎧⎩⎨⎧负无理数负分数负整数负有理数负实数数）（既不是正数也不是负零正无理数正分数正整数正有理数正实数实数number fractional negative integer negative number real negative negativenor positive Neither number arithmetic zero第1讲实数的有关概念【知识要点】1．实数的性质（1）实数范围内仍然适用在有理数范围内定义的一些概念（如倒数，相反数）；Real number range applies within the scope of the rational definition of some concepts, suchas countdown, opposite)（2）两实数的大小关系：正数大于0，0大于负数；两个正实数，绝对值大的实数大；两个负实数，绝对值大的实数反而小；Relations between the two the size of the real Numbers: a positive number greater than zero, 0 is greater than the negative; Two positive real number, the absolute value of real number; Two negative real number, the absolute value of real Numbers instead of small（3）在实数范围内，加、减、乘、除（除数不为零）、乘方五种运算是畅通无阻的，但是开方运算要注意，正实数和零总能进行开方运算，而负实数只能开奇次方，不能开偶次方；Within the scope of the real Numbers, addition, subtraction, multiplication, and division (divisor is not zero), power five kinds of arithmetic is unobstructed, but root operation should pay attention to, are real Numbers and zero always root operation, while the negative real number can only be open power, can't open my power（4）有理数范围内的运算律和运算顺序在实数范围内仍然相同．Rational number operations within the scope of law and order of operations within the scopeof the real number is still the same2．实数与数轴的关系Real relationships with a number line每一个实数都可以用数轴上的一个点表示；反之，数轴上每一个点都表示一个实数，即数轴上的点与实数是一一对应关系．Every real number can be represented by a point on the number axis; On the other hand,every point on a number line represents a real, namely the points on the number line is one-to-one correspondence relationship with real Numbers3．实数的分类The classification of real Numbers⎪⎪⎪⎩⎪⎪⎪⎨⎧⎭⎬⎫⎩⎨⎧⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧无限不循环小数负无理数正无理数无理数数有限小数或无限循环小负有理数零正有理数有理数实数4．实数的大小比较The size of the real number两实数的大小关系如下：正实数都大于0，负实数都小于0，正数大于一切负数；两个正实数，绝对值大的实数较大；两个负实数，绝对值大的实数反而小．Relations between the two the size of the real number is as follows: positive number isgreater than zero, negative real Numbers are less than zero, the positive is greater than all thenegative; Two positive real number, the absolute value of real number; Two negative real number,the absolute value of real Numbers instead of small实数和数轴上的点一一对应，在数轴上表示的两个实数，右边的数总大于左边的数．Real Numbers and at a point on the number line one to one correspondence, said on a numberline of two real Numbers, the right of the number of total is greater than the number on the left.【典型例题】A typical example例1若a 为实数,下列代数式中,一定是负数的是( )If a is real number, the following algebraic expression, must be a negative number isA. －a 2B. －( a +1)2C.－2a D.－(a -+1)分析：本题主要考查负数和非负数的概念,同时涉及考查字母表示数这个知识点.Subjectmainly examines the concept of the negative and the negative, involving check letters for thisknowledge at the same time.由于a 为实数, Because it is the set of real Numbers. a 2、( a +1)2、2a 均为非负数Both as a negative number ，∴－a 2≤0，－( a +1)2≤0，－2a ≤0．而0既不是正数也不是负数，是介于正数与负数之间的中性数．Zero is neither positive nor negative,neutral number is between positive and negative.因此，A 、B 、C 不一定是负数．又依据绝对值的概念及性质知Therefore, A, B, C are not necessarily negative. And based on the concept andnature of the absolute value －(a -+1)﹤0．So the choice is D.例2 实数a 在数轴上的位置如图所示,Real Numbers a position on a number line, as shown in the figure化简:2)2(1-+-a a =分析：这里考查了数形结合的数学思想,要去掉绝对值符号,必须清楚绝对值符号内的数是正还是负.由数轴可知:1﹤a ﹤2,于是,22)2(,112a a a a a -=-=--=- 所以, 2)2(1-+-a a =a －1+2－a =1.例3 如图所示,数轴上A 、B 两点分别表示实数1，5，点B 关于点A 的对称点为C ，则点C 所表示的实数为（）As shown in the figure on the number line A and Btwo points respectively according to the set of real Numbers 1，5，Point symmetry about pointA pointB to C, and the pointC said the real number for which one （） A.5－2 B. 2－5 C. 5－3 D.3－5分析：这道题也考查了数形结合的数学思想,同时又考查了对称的性质．B 、C 两点关于点A 对称，因而B 、C 两点到点A 的距离是相同的，点B 到点A 的距离是5－1，所以点C 到点A 的距离也是5－1，设点C 到点O 的距离为a ，所以a +1=5－1，即a =5－2．又因为点C 所表示的实数为负数，所以点C 所表示的实数为2－5．例4 已知a 、b 是有理数，且满足3-b +（a －2）2=0，则a b 的值为多少 A and b is a rational number is known, and satisfaction 3-b +（a －2）2=0，in that way ，a b has a value of .分析：因为（a －2）2+3-b =0，所以a －2=0，b －3=0。

2.4整数、有理数与实数数学专业英语

2.4 整数、有理数与实数1.单词，词组，短语翻译(1)整数integer，有理数rational number，无理数irrational number，实数real number，负数negative number, 相反数the negative，实直线real line，实轴real axis，尺度scale 在……的左边/右边to the left/right of(2)和sum，差difference，积product，商quotient，幂power，不等式inequality(3) 公理axiom，序公理the order axiom，域公理the field axiom(4)有序的ordered,完整的entirely complete,欧式的Euclidean,适当的appropriate,显著的distinguished,清晰的illuminating(5)可推导的公式deducible formula，可互相交换的运算interchangeable operation，采用一组公理to adopt a set of axioms，对应于某个对象corresponding to an object / to correspond to a particular object，用数学归纳法证明proof by induction，对检验结论的有效性很有用a very worthwhile aid to test the validity of conclusions，把这两个集加以区别to distinguish these two sets2. 汉译英（1）严格说，这样描述整数是不完整的，因为我们并没有说明“以此类推”或“反复加1” 的含义是什么。

Strictly speaking,，this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”.（2）两个整数的和、差或积是一个整数，但是两个整数的商未必是一个整数。

number意思的用法总结

number的意思
n. 数字,数量,号码,编号
v. 标号,总计,把…算作
变形：过去式: numbered; 现在分词：numbering; 过去分词：numbered;
number用法
number可以用作名词
number的基本意思是“数字”,引申可作“编号,电话号码,房间号码”解,是可数名词。

常缩写为No.或no.。

number也可作“数目,数量”解,其后常与介词of连用,引申可表示语法上名词或动词的“数”或“数词”。

number的单数形式也可指“一群人,一帮人”。

number用作名词的用法例句
He picked a winning number on the first draw.他第一次便抽到一个中奖号码。

I slipped up and gave you the wrong phone number.我粗心大意给错了你电话号码。

The number of fish in coastal waters has decreased.沿海鱼的数量已减少了。

number可以用作动词
number用作动词意思是“数,算”,引申可作“编号”“加号码于…”“总共,共计”等解。

实数实数 real number(English)

实数real numberreal numbers 实数 integers 整数 Fraction 分数decimals 小数 irrational Numbers 无理数 rational numbers 有理数positive number 正数 negative number 负数 opposite number 相反数count backwards 倒数Teaching aims1．了解无理数与实数的意义；Understanding of irrational Numbers and the notion of real Numbers2．了解有理数的运算法则(?)在实数范围内仍然适用；Understand the rational range of algorithms3．能利用化简对实数进行简单的四则运算；simple arithmetic of the real number by simplification4．了解实数的意义，能对实数按要求进行分类；The notion and the classification of the real numbers5．掌握有理数的运算法则在实数范围内仍然适用；Grasp the rational range of algorithms in real numbers6．能利用实数的性质熟练地进行四则运算；Master the arithmetic of the rational numbers well.7．注意：（1）无理数应满足：①是小数；②是无限小数；③不循环；Irrational Numbers should be: (1) a decimal number; (2) infinite decimals; (3)no cycles（2）无理数不是都带根号的数（例如π就是无理数），反之，带根号的数也不一定都是无理数（例如4，327就是有理数）．Not all irrational Numbers take square root (such as π is irrational ）,On the other hand, radicals are not necessarily irrational Numbers （for example4，327Is a rational number ）（1）按实数的定义分类：According to the definition of real Numbers ⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎨⎧⎭⎬⎫⎩⎨⎧⎪⎪⎪⎩⎪⎪⎪⎨⎧⎭⎬⎫⎩⎨⎧⎪⎩⎪⎨⎧decimal repeating -non infinite number irrational fraction negative fraction Positive fraction integer negative integer positive integer number rational 无限不循环小数负无理数正无理数无理数数有限小数或无限循环小负分数正分数分数负整数零正整数整数有理数实数zero（2）按实数的正负分类：⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎨⎧⎪⎩⎪⎨⎧⎩⎨⎧⎪⎩⎪⎨⎧⎩⎨⎧负无理数负分数负整数负有理数负实数数）（既不是正数也不是负零正无理数正分数正整数正有理数正实数实数number fractional negative integer negative number real negative negativenor positive Neither number arithmetic zero第1讲实数的有关概念【知识要点】1．实数的性质（1）实数范围内仍然适用在有理数范围内定义的一些概念（如倒数，相反数）；Real number range applies within the scope of the rational definition of some concepts, suchas countdown, opposite)（2）两实数的大小关系：正数大于0，0大于负数；两个正实数，绝对值大的实数大；两个负实数，绝对值大的实数反而小；Relations between the two the size of the real Numbers: a positive number greater than zero, 0 is greater than the negative; Two positive real number, the absolute value of real number; Two negative real number, the absolute value of real Numbers instead of small（3）在实数范围内，加、减、乘、除（除数不为零）、乘方五种运算是畅通无阻的，但是开方运算要注意，正实数和零总能进行开方运算，而负实数只能开奇次方，不能开偶次方；Within the scope of the real Numbers, addition, subtraction, multiplication, and division (divisor is not zero), power five kinds of arithmetic is unobstructed, but root operation should pay attention to, are real Numbers and zero always root operation, while the negative real number can only be open power, can't open my power（4）有理数范围内的运算律和运算顺序在实数范围内仍然相同．Rational number operations within the scope of law and order of operations within the scopeof the real number is still the same2．实数与数轴的关系Real relationships with a number line每一个实数都可以用数轴上的一个点表示；反之，数轴上每一个点都表示一个实数，即数轴上的点与实数是一一对应关系．Every real number can be represented by a point on the number axis; On the other hand,every point on a number line represents a real, namely the points on the number line is one-to-one correspondence relationship with real Numbers3．实数的分类The classification of real Numbers⎪⎪⎪⎩⎪⎪⎪⎨⎧⎭⎬⎫⎩⎨⎧⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧无限不循环小数负无理数正无理数无理数数有限小数或无限循环小负有理数零正有理数有理数实数4．实数的大小比较The size of the real number两实数的大小关系如下：正实数都大于0，负实数都小于0，正数大于一切负数；两个正实数，绝对值大的实数较大；两个负实数，绝对值大的实数反而小．Relations between the two the size of the real number is as follows: positive number isgreater than zero, negative real Numbers are less than zero, the positive is greater than all thenegative; Two positive real number, the absolute value of real number; Two negative real number,the absolute value of real Numbers instead of small实数和数轴上的点一一对应，在数轴上表示的两个实数，右边的数总大于左边的数．Real Numbers and at a point on the number line one to one correspondence, said on a numberline of two real Numbers, the right of the number of total is greater than the number on the left.【典型例题】A typical example例1若a 为实数,下列代数式中,一定是负数的是( )If a is real number, the following algebraic expression, must be a negative number isA. －a 2B. －( a +1)2C.－2a D.－(a -+1)分析：本题主要考查负数和非负数的概念,同时涉及考查字母表示数这个知识点.Subjectmainly examines the concept of the negative and the negative, involving check letters for thisknowledge at the same time.由于a 为实数, Because it is the set of real Numbers. a 2、( a +1)2、2a 均为非负数Both as a negative number ，∴－a 2≤0，－( a +1)2≤0，－2a ≤0．而0既不是正数也不是负数，是介于正数与负数之间的中性数．Zero is neither positive nor negative,neutral number is between positive and negative.因此，A 、B 、C 不一定是负数．又依据绝对值的概念及性质知Therefore, A, B, C are not necessarily negative. And based on the concept andnature of the absolute value －(a -+1)﹤0．So the choice is D.例2 实数a 在数轴上的位置如图所示,Real Numbers a position on a number line, as shown in the figure化简:2)2(1-+-a a =分析：这里考查了数形结合的数学思想,要去掉绝对值符号,必须清楚绝对值符号内的数是正还是负.由数轴可知:1﹤a ﹤2,于是,22)2(,112a a a a a -=-=--=- 所以, 2)2(1-+-a a =a －1+2－a =1.例3 如图所示,数轴上A 、B 两点分别表示实数1，5，点B 关于点A 的对称点为C ，则点C 所表示的实数为（）As shown in the figure on the number line A and Btwo points respectively according to the set of real Numbers 1，5，Point symmetry about pointA pointB to C, and the pointC said the real number for which one （） A.5－2 B. 2－5 C. 5－3 D.3－5分析：这道题也考查了数形结合的数学思想,同时又考查了对称的性质．B 、C 两点关于点A 对称，因而B 、C 两点到点A 的距离是相同的，点B 到点A 的距离是5－1，所以点C 到点A 的距离也是5－1，设点C 到点O 的距离为a ，所以a +1=5－1，即a =5－2．又因为点C 所表示的实数为负数，所以点C 所表示的实数为2－5．例4 已知a 、b 是有理数，且满足3-b +（a －2）2=0，则a b 的值为多少 A and b is a rational number is known, and satisfaction 3-b +（a －2）2=0，in that way ，a b has a value of .分析：因为（a －2）2+3-b =0，所以a －2=0，b －3=0。

有理数和无理数统称实数(real

绝对值：正数的绝对值是它本身，负数的绝对值是它的相反数， 0的绝对值是0，
a ∣a∣= 0
－a
(a＞0) (a＝0) ( a＜0)
例3观察下图，然后再回答问题
1.如图，OA=OB，数轴上A点对应的数是什么？它介于哪两个整数之间？ 2.如果将所有有理数都标到数轴上，那么数轴被填满了吗？
课堂练习
小结 :
1、实数概念 2、实数的分类 3、在实数范围内，相反数、倒数和绝对值的意义仍
然与有理数范围内的意义相同。 4、实数与数轴上的点一一对应
课后作业习题2.8 1、2、3、
姚志盛
实数的概念：
有理数和无理数统称实数（real number）,即实数可分为有理数和无理数
填完以后大家有什么发现？
实数也可按正负进行分类正有理数
正实数正无理数实数零
负实数负有理数负无理无理数
在实数范围内的几个概念。
相反数：a与-a互为相反数，o的相反数是o。倒数：若a≠o，则a与互为倒数。

Oracle中Numberdecimal（numeric）、float和real数据类型的区别

Oracle中Numberdecimal（numeric）、float和real数据类型的区别在Oracle中Number类型可以⽤来存储0，正负定点或者浮点数，可表⽰的数据范围在1.0 * 10(-130) —— 9.9...9 * 10(125) {38个9后边带88个0}的数字，当Oracle中的数学表达式的值>=1.0*10(126)时，Oracle就会报错。

Number的数据声明如下：表⽰作⽤说明Number(p, s) 声明⼀个定点数 p(precision)为精度，s(scale)表⽰⼩数点右边的数字个数，精度最⼤值为38，scale的取值范围为-84到127 Number(p) 声明⼀个整数相当于Number(p, 0)Number 声明⼀个浮点数其精度为38，要注意的是scale的值没有应⽤，也就是说scale的指不能简单的理解为0，或者其他的数。

定点数的精度(p)和刻度(s)遵循以下规则：当⼀个数的整数部分的长度 > p-s 时，Oracle就会报错当⼀个数的⼩数部分的长度 > s 时，Oracle就会舍⼊。

当s(scale)为负数时，Oracle就对⼩数点左边的s个数字进⾏舍⼊。

当s > p 时, p表⽰⼩数点后第s位向左最多可以有多少位数字，如果⼤于p则Oracle报错，⼩数点后s位向右的数字被舍⼊decimal(numeric ) 同义，⽤于精确存储数值float 和 real 不能精确存储数值decimal 数据类型最多可存储 38 个数字，所有数字都能够放到⼩数点的右边。

decimal 数据类型存储了⼀个准确（精确）的数字表达法；不存储值的近似值。

定义 decimal 的列、变量和参数的两种特性如下：p ⼩数点左边和右边数字之和，不包括⼩数点。

如 123.45,则 p=5，s=2。

指定精度或对象能够控制的数字个数。

s指定可放到⼩数点右边的⼩数位数或数字个数。

1.1 The Real Number System

5
Example 1 – Classifying Real Numbers
Which of the numbers in the set are (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers? Solution a. Natural numbers: {5} b. Integers: {–7, –1, 0, 5}
11
The Real Number Line and Order
12
Example 3 – Ordering Real Numbers
Place the correct inequality symbol (< or >) between the real numbers. a. –4 0 b. –3 –5 c. d.
1.1 The Real Number System
What You Will Learn
Understand the set of real numbers and the
subsets of real numbers.
Use the real number line to order real
on the real number line,
than , and write
<
.
14
Example 3 – Ordering Real Numbers
cont’d
d. Because lies to the right of on the real number line, you can say that is greater than , and write > .