新人教版七年级下册数学第六章实数知识点总结及阶梯练习

新人教版七年级下册数学第六章实数知识点总结及阶梯练习第六章实数考点一、实数的概念及分类1、实数的分类2、无理数在理解无理数时，要抓住“无限不循环”这一点，归纳起来有四类3(1)开方开不尽的数，如等;7,2π(2)有特定意义的数，如圆周率π，或化简后含有π的数，如+8等;3(3)有特定结构的数，如0.1010010001„等;o(4)某些三角函数，如sin60等(这类在初三会出现)0,,16判断一个数是否是无理数，不能只看形式，要看运算结果，如是有理数，而不是无理数。3、有理数与无理数的区别(1)有理数指的是有限小数和无限循环小数，而无理数则是无限不循环小数;(2)所有的有理数都能写成分数的形式(整数可以看成是分母为1的分数)，而无理数则不能写成分数形式。考点二、平方根、算术平方根、立方根1、概念、定义(1)如果一个正数x的平方等于a，即，那么这个正数x叫做a的算术平方根。(2)如果一个数的平方等于a，那么这个数就叫做a的平方根(或二次方跟)。如果，那么x叫做a的平方根。(3)如果一个数的立方等于a，那么这个数就叫做a的立方根(或a的三次方根)。如果，那么x叫做a的立方根。2、运算名称(1)求一个正数a的平方根的运算，叫做开平方。平方与开平方互为逆运算。(2)求一个数的立方根的运算，叫做开立方。开立方和立方互为逆运算。3、运算符号(1)正数a的算术平方根，记作“”。a(2)a(a?0)的平方根的符号表达为。thedrawings,firsttobecomefamiliarwiththedrawings,mainlyaboutthefollowingaspects:1)pagesofdrawingsanddiagrams,maps;2)acomprehensiveunderstandingofdrawing;3)findingofdesign-drivendimensions(),donotresizeandadjustthesize;4)questionedthedrawings;5)developburiesaconstructionprogrammeandtechnicalclarification;6)clearcornersandspecialtreatment;7)controlledconstructiondrawingstoverifyconstructionschemeanddesign.(2)attheconstructionsitetofindsubjectplantedarea:variousengineering,firstonthescenetofindplantedarea,itmustfirstunderstandthecurtainwallinstallationsection,andsomeprojectsareallcurtainwall,theentireprojectisaregion-wide,onlypartialwalls,thezoneislocal.(3)identifytheaxispositioning:positioningthedrawingsshowninaxiscomparedwiththeactualconstructionsitetofindoutexactlywheretheaxispositioning,axispositioningfunctionare:1)tohelpdeterminetheoriginofcurtainwalls;2)acceptanceoftheinstallationisaccurate;3)adjusterrors,anddeterminetheerrorrange.(4)findanchorpoints:accordingtositetofindtheexactlocationoftheaxis,accordingtothedrawingsprovidedbythecontentdeterminethepositioningpoint;numberofanchorpointsshallnotbelessthantwopoints.Repeatedmeasurementsdeterminethelocationpointsmustensurethatthepositioningiscorrect.(5)takelevel(level):level(minorworksavailableinhorizontalpipe),thetwoanchorpointsdeterminethehorizontalposition.Levelaccordingtouse(3)一个数a的立方根，用表示，其中a是被开方数，3是根指数。4、运算公式4、开方规律小结a,a(1)若a?0，则a的平方根是，a的算术平方根;正数的平方根有两个，它们互为相反数，其中正的那个叫它的算术平方根;0的平方根和算术平方根都是0;负数没有平方根。实数都有立方根，一个数的立方根有且只有一个，并且它的符号与被开方数的符号相同。正数的立方根是正数，负数的立方根是负数，0的立方根是0。(2)若a<0，则a没有平方根和算术平方根;若a为任意实数，则a的立方根是。(3)正数的两个平方根互为相反数，两个互为相反数的实数的立方根也互为相反数。考点三、实数的性质有理数的一些概念，如倒数、相反数、绝对值等，在实数范围内仍然不变。1、相反数(1)实数a的相反数是-a;实数与它的相反数是一对数(只有符号不同的两个数叫做互为相反数，零的相反数是零)(2)从数轴上看，互为相反数的两个数所对应的点关于原点对称，如果a与b互为相反数，则有a+b=0，a=-b，反之亦成立。2、绝对值(1)要正确的理解绝对值的几何意义，它表示的是数轴上的点到数轴原点的距离，数轴分为正负两半，那么不管怎样总有两个数字相等的正负两个数到原点的距离相等。|a|?0。(2)若|a|=a，则a?0;若|a|=-a，则a?0，零的绝对值是它本身。a(a,0),,,a(a,0),(3)3、倒数(1)如果a与b互为倒数，则有ab=1，反之亦成立。实数a的倒数是1/a(a?0)(2)倒数等于本身的数是1和-1。零没有倒数。考点四、实数的三个非负性及性质1、在实数范围内，正数和零统称为非负数。2、非负数有三种形式(1)任何一个实数a的绝对值是非负数，即|a|?0;(2)任何一个实数a的平方是非负数，即?0;thedrawings,firsttobecomefamiliarwiththedrawings,mainlyaboutthefollowingaspects:1)pagesofdrawingsanddiagrams,maps;2)acomprehensiveunderstandingofdrawing;3)findingofdesign-drivendimensions(),donotresizeandadjustthesize;4)questionedthedrawings;5)developburiesaconstructionprogrammeandtechnicalclarification;6)clearcornersandspecialtreatment;7)controlledconstructiondrawingstoverifyconstructionschemeanddesign.(2)attheconstructionsitetofindsubjectplantedarea:variousengineering,firstonthescenetofindplantedarea,itmustfirstunderstandthecurtainwallinstallationsection,andsomeprojectsareallcurtainwall,theentireprojectisaregion-wide,onlypartialwalls,thezoneislocal.(3)identifytheaxispositioning:positioningthedrawingsshowninaxiscomparedwiththeactualconstructionsitetofindoutexactlywheretheaxispositioning,axispositioningfunctionare:1)tohelpdeterminetheoriginofcurtainwalls;2)acceptanceoftheinstallationisaccurate;3)adjusterrors,anddeterminetheerrorrange.(4)findanchorpoints:accordingtositetofindtheexactlocationoftheaxis,accordingtothedrawingsprovidedbythecontentdeterminethepositioningpoint;numberofanchorpointsshallnotbelessthantwopoints.Repeatedmeasurementsdeterminethelocationpointsmustensurethatthepositioningiscorrect.(5)takelevel(level):level(minorworksavailableinhorizontalpipe),thetwoanchorpointsdeterminethehorizontalposition.Levelaccordingtouse(3)任何非负数的算术平方根是非负数，即()。3、非负数具有以下性质(1)非负数有最小值零;(2)非负数之和仍是非负数;(3)几个非负数之和等于0，则每个非负数都等于0.考点五、实数大小的比较实数的大小比较的法则跟有理数的大小比较法则相同:(1)正数大于0，0大于负数，正数大于一切负数，两个负数比较，绝对值大的反而小;(2)实数和数轴上的点一一对应，在数轴上表示的两个数，右边的数总比左边的数大;(3)两个数比较大小常见的方法有:求差法，求商法，倒数法，估算法，平方法。(4)对于一些带根号的无理数，我们可以通过比较它们的平方或者立方的大小。常用有理数来估计无理数的大致范围，要想正确估算需记熟0,20之间整数的平方和0,10之间整数的立方(考点六、实数的运算(1)在实数范围内，可以进行加、减、乘、除、乘方及开方运算(2)有理数的运算法则和运算律在实数范围内仍然成立(3)实数混合运算的运算顺序与有理数的运算顺序基本相同，先乘方、开方、再乘除，最后算加减。同级运算按从左到右顺序进行，有括号先算括号里。(4)在实数的运算中，当遇到无理数时，并且需要求结果的近似值时，可以按照所要求的精确度用相应的近似有限小数去代替无理数，再进行计算。二、典例剖析，综合拓展知识点1:算术平方根1111121.的算术平方根为()(A)(B),(C)?(D)()169131313169算术平方根的定义:12.的算术平方根可表示为，即=169算术平方根的表示方法:(用含a的式子表示)13.,有算术平方根吗,8的算术平方根是,2吗,169a算术平方根具有性，即?被开方数a0，?本身0，必须同时成立5，115,11、已知mn4的小数部分为，的小数部分为，则m，n,跟踪练习:x，3?式子有意义，x的取值范围x,55,x3,a，b,4,0?已知:y=++3,求xy的值，求a+b的值thedrawings,firsttobecomefamiliarwiththedrawings,mainlyaboutthefollowingaspects:1)pagesofdrawingsanddiagrams,maps;2)acomprehensiveunderstandingofdrawing;3)findingofdesign-drivendimensions(),donotresizeandadjustthesize;4)questionedthedrawings;5)developburiesaconstructionprogrammeandtechnicalclarification;6)clearcornersandspecialtreatment;7)controlledconstructiondrawingstoverifyconstructionschemeanddesign.(2)attheconstructionsitetofindsubjectplantedarea:variousengineering,firstonthescenetofindplantedarea,itmustfirstunderstandthecurtainwallinstallationsection,andsomeprojectsareallcurtainwall,theentireprojectisaregion-wide,onlypartialwalls,thezoneislocal.(3)identifytheaxispositioning:positioningthedrawingsshowninaxiscomparedwiththeactualconstructionsitetofindoutexactlywheretheaxispositioning,axispositioningfunctionare:1)tohelpdeterminetheoriginofcurtainwalls;2)acceptanceoftheinstallationisaccurate;3)adjusterrors,anddeterminetheerrorrange.(4)findanchorpoints:accordingtositetofindtheexactlocationoftheaxis,accordingtothedrawingsprovidedbythecontentdeterminethepositioningpoint;numberofanchorpointsshallnotbelessthantwopoints.Repeatedmeasurementsdeterminethelocationpointsmustensurethatthepositioningiscorrect.(5)takelevel(level):level(minorworksavailableinhorizontalpipe),thetwoanchorpointsdeterminethehorizontalposition.Levelaccordingtouse-3-知识点2:平方根1.49的平方根是，算术平方根是，它的平方根可表示为;2、的平方根是93、快速地表示并求出下列各式的平方根92?1?|,5|?0.81?(,9)16平方根的定义:平方根的表示方法(用含a的式子表示)平方根的性质:a，12a,74、如果一个数的平方根是和，求这个数5.用平方根定义解方程22?16(x+2)=81?4x-225=06、下列说法正确的是(),616A、的平方根是B、表示6的算术平方根的相反数,42,aC、任何数都有平方根D、一定没有平方根知识点3:立方根1.,8的立方根是，表示为立方根的表示方法:(用含a的式子表示)2.说出下列各式表示的意义并求值:333333(,2)8,0.512,729?=?,=?=?()=3x,23.如果有意义，x的取值范围为4.用立方根的定义解方程33?x-27=0?2(x+3)=512拓展提高:3,1.73230,5.477300,0.3,1、已知，，(1);(2);x,54.77x,(3)0.03的平方根约为;(4)若，则33333,1.44230,3.107300,6.6940.3,2、已知，，，求(1);3x,31.07x,(2)3000的立方根约为;(3)，则知识点4:重要公式thedrawings,firsttobecomefamiliarwiththedrawings,mainlyaboutthefollowingaspects:1)pagesofdrawingsanddiagrams,maps;2)acomprehensiveunderstandingofdrawing;3)findingofdesign-drivendimensions(),donotresizeandadjustthesize;4)questionedthedrawings;5)developburiesaconstructionprogrammeandtechnicalclarification;6)clearcornersandspecialtreatment;7)controlledconstructiondrawingstoverifyconstructionschemeanddesign.(2)attheconstructionsitetofindsubjectplantedarea:variousengineering,firstonthescenetofindplantedarea,itmustfirstunderstandthecurtainwallinstallationsection,andsomeprojectsareallcurtainwall,theentireprojectisaregion-wide,onlypartialwalls,thezoneislocal.(3)identifytheaxispositioning:positioningthedrawingsshowninaxiscomparedwiththeactualconstructionsitetofindoutexactlywheretheaxispositioning,axispositioningfunctionare:1)tohelpdeterminetheoriginofcurtainwalls;2)acceptanceoftheinstallationisaccurate;3)adjusterrors,anddeterminetheerrorrange.(4)findanchorpoints:accordingtositetofindtheexactlocationoftheaxis,accordingtothedrawingsprovidedbythecontentdeterminethepositioningpoint;numberofanchorpointsshallnotbelessthantwopoints.Repeatedmeasurementsdeterminethelocationpointsmustensurethatthepositioningiscorrect.(5)takelevel(level):level(minorworksavailableinhorizontalpipe),thetwoanchorpointsdeterminethehorizontalposition.Levelaccordingtouse222===公式一:?324222(,2)(,3)(,4)===2?=a122(,)有关练习:1.==199972(a,3)2.如果=a-3，则a的取值范围是;2(a,3)如果=3-a,则a的取值范围是2(a,b)3.数a,b在数轴上的位置如图:化简:+|c+a|ab0C2229254公式二:?()=()=()=2?=(a?0)(a)22a综合公式一和二，可知，当满足a条件时，=(a)3333333公式三:?===24333333(,2)(,3)(,4)===33a?=;233(1,a)(a,3)随堂练习:化简:当1,a,3时，+333333827125公式四:?()=()=()=33?=(a)3333a综合公式三和四，可知，当满足a条件时，=(a)3,a公式五:=thedrawings,firsttobecomefamiliarwiththedrawings,mainlyaboutthefollowingaspects:1)pagesofdrawingsanddiagrams,maps;2)acomprehensiveunderstandingofdrawing;3)findingofdesign-drivendimensions(),donotresizeandadjustthesize;4)questionedthedrawings;5)developburiesaconstructionprogrammeandtechnicalclarification;6)clearcornersandspecialtreatment;7)controlledconstructiondrawingstoverifyconstructionschemeanddesign.(2)attheconstructionsitetofindsubjectplantedarea:variousengineering,firstonthescenetofindplantedarea,itmustfirstunderstandthecurtainwallinstallationsection,andsomeprojectsareallcurtainwall,theentireprojectisaregion-wide,onlypartialwalls,thezoneislocal.(3)identifytheaxispositioning:positioningthedrawingsshowninaxiscomparedwiththeactualconstructionsitetofindoutexactlywheretheaxispositioning,axispositioningfunctionare:1)tohelpdeterminetheoriginofcurtainwalls;2)acceptanceoftheinstallationisaccurate;3)adjusterrors,anddeterminetheerrorrange.(4)findanchorpoints:accordingtositetofindtheexactlocationoftheaxis,accordingtothedrawingsprovidedbythecontentdeterminethepositioningpoint;numberofanchorpointsshallnotbelessthantwopoints.Repeatedmeasurementsdeterminethelocationpointsmustensurethatthepositioningiscorrect.(5)takelevel(level):level(minorworksavailableinhorizontalpipe),thetwoanchorpointsdeterminethehorizontalposition.Levelaccordingtouse-5-知识点五:实数定义及分类无理数的定义:实数的定义:实数与上的点是一一对应的1、判断下列说法是否正确:(1)实数不是有理数就是无理数。()(2)无限小数都是无理数。()(3)无理数都是无限小数。()(4)根号的数都是无理数。()一、选择题(每题3分,共36分)1.下列等式正确的是()2(,3)(A),,3(B),?12(C),8,,2(D),25,,51442.算术平方根等于3的是()39(A)(B)3(C)9(D)23.下列说法:(1)任何数都有算术平方根;(2)一个数的算术平方根一定是正数;(3)a的算术平方根是a;2(4)(,4)的算术平方根是,4;(5)算术平方根不可能是负数。其中不正确的有(),,(A)5个(B)4个(C)3个(D)2个4.若一个数的平方根与它的立方根完全相同，则这个数是()(A)0(B)1(C),1(D)?1,0x5.若=x，则实数x是()(A)负实数(B)所有正实数(C)0或1(D)不存在26.若a=-a，则实数a在数轴上的对应点，一定在()(A)原点左侧(B)原点右侧(C)原点或原点左侧(D)原点或原点右侧,76.在实数0(3，0，，，0(123456„中，无理数的个数是()2(A)2(B)3(C)4(D)57.下列各式中，无意义的是()2,3323,310(A).(B).(C).(D).(,3)(,3)226148.4、、15三个数的大小关系是()2262261414(A).4<15<(B)<15<42262261414(C)4<<15(D)<4<1549.化简的结果是()(,2)(A).,4(B).4(C)?4(D)无意义thedrawings,firsttobecomefamiliarwiththedrawings,mainlyaboutthefollowingaspects:1)pagesofdrawingsanddiagrams,maps;2)acomprehensiveunderstandingofdrawing;3)findingofdesign-drivendimensions(),donotresizeandadjustthesize;4)questionedthedrawings;5)developburiesaconstructionprogrammeandtechnicalclarification;6)clearcornersandspecialtreatment;7)controlledconstructiondrawingstoverifyconstructionschemeanddesign.(2)attheconstructionsitetofindsubjectplantedarea:variousengineering,firstonthescenetofindplantedarea,itmustfirstunderstandthecurtainwallinstallationsection,andsomeprojectsareallcurtainwall,theentireprojectisaregion-wide,onlypartialwalls,thezoneislocal.(3)identifytheaxispositioning:positioningthedrawingsshowninaxiscomparedwiththeactualconstructionsitetofindoutexactlywheretheaxispositioning,axispositioningfunctionare:1)tohelpdeterminetheoriginofcurtainwalls;2)acceptanceoftheinstallationisaccurate;3)adjusterrors,anddeterminetheerrorrange.(4)findanchorpoints:accordingtositetofindtheexactlocationoftheaxis,accordingtothedrawingsprovidedbythecontentdeterminethepositioningpoint;numberofanchorpointsshallnotbelessthantwopoints.Repeatedmeasurementsdeterminethelocationpointsmustensurethatthepositioningiscorrect.(5)takelevel(level):level(minorworksavailableinhorizontalpipe),thetwoanchorpointsdeterminethehorizontalposition.Levelaccordingtouse二、填空题(每题3分,共24分)210.如果=2,那么(x+3)=______x，3211.若与|b+2|是互为相反数，则(a,b)=______.2a,213.一个正方形的面积扩大为原来的100倍，则其边长扩大为原来的倍。16.点A在数轴上和原点相距5个单位，点B在数轴上和原点相距3个单位，且点B在点A左边，则AB之间的距离为________22()______;()abcabc，,,,,14.一个三角形的三边分别是a，b，c，则=________(15.a是一个两位数的十位数字，b是它的个位数字，则这个数可表示为.216.的算术平方根是______，25的平方根是______，—8的立方根是_______(，，,4317.已知?a+2?+4=________(bab,,，100,则三、解答题(共60分)18.计算:(共18分)1666,2，2,1,3,6(1)(2)(,)623113，，2y,3,64,08，0,，,(3)(4)483364x,125,0，，x,1,64(5)(6)x19.(8分)?已知，求的平方根.yy,x,2，2,x，325b5?已知是的整数部分,是的小数部分,求的值.aab(5),22a，8，b,3,0x，，20.(8分)已知a、b满足，解关于的方程。a，2x，b,a,1thedrawings,firsttobecomefamiliarwiththedrawings,mainlyaboutthefollowingaspects:1)pagesofdrawingsanddiagrams,maps;2)acomprehensiveunderstandingofdrawing;3)findingofdesign-drivendimensions(),donotresizeandadjustthesize;4)questionedthedrawings;5)developburiesaconstructionprogrammeandtechnicalclarification;6)clearcornersandspecialtreatment;7)controlledconstructiondrawingstoverifyconstructionschemeanddesign.(2)attheconstructionsitetofindsubjectplantedarea:variousengineering,firstonthescenetofindplantedarea,itmustfirstunderstandthecurtainwallinstallationsection,andsomeprojectsareallcurtainwall,theentireprojectisaregion-wide,onlypartialwalls,thezoneislocal.(3)identifytheaxispositioning:positioningthedrawingsshowninaxiscomparedwiththeactualconstructionsitetofindoutexactlywheretheaxispositioning,axispositioningfunctionare:1)tohelpdeterminetheoriginofcurtainwalls;2)acceptanceoftheinstallationisaccurate;3)adjusterrors,anddeterminetheerrorrange.(4)findanchorpoints:accordingtositetofindtheexactlocationoftheaxis,accordingtothedrawingsprovidedbythecontentdeterminethepositioningpoint;numberofanchorpointsshallnotbelessthantwopoints.Repeatedmeasurementsdeterminethelocationpointsmustensurethatthepositioningiscorrect.(5)takelevel(level