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数学专业英语-The Real Number System

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发表于 2004-5-6 09:22:04 | 显示全部楼层 |阅读模式
< ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
< ><B><FONT face="Times New Roman">The Real-Number System<p></p></FONT></B></P>
< ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>The real-number system is collection of mathematical objects, called real number, which acquire mathematical life by virtue fundamental principles, or rules, that we adopt. The situation is somewhat similar to a game, like chess, for example. The chess system, or game, is a collection of objects, called chess pieces, which acquire life by virtue of the rules of the game, that is, the principles that are adopted to define allowable moves for the pieces and the way in which they may interact.</FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>  Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system. However, to establish a common ground of understanding and avoid certain errors that have become very common, we shall explicitly state and illustrate many of these principles.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">  The real-number system includes such numbers as –27,-2,2/3,</FONT>…<FONT face="Times New Roman"> It is worthy of note that positive numbers, 1/2, 1, for examples, are sometimes expressed as +(1/2), +1. The plus sign, “+”, used here does not express the operation of  addition, but is rather part of the symbolism for the numbers themselves. Similarly, the minus sign, “-“, used in expressing such numbers as -(1/2), -1, is part of the symbolism for these numbers.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">  Within the real number system, numbers of various kinds are identified and named. The numbers 1, 2, 3, 4,</FONT>…<FONT face="Times New Roman"> which are used in the counting process, are called natural numbers. The natural numbers, together with–1,-2,-3,-4,</FONT>…<FONT face="Times New Roman">and zero, are called integers. Since 1,2,3,4,</FONT>…<FONT face="Times New Roman">are greater than 0, they are also called positive integers; -1,-2,-3,-4,</FONT>…<FONT face="Times New Roman">are less than 0, and for this reason are called negative integers. A real number is said to be a rational number if it can be expressed as the ratio of two integers, where the denominator is not zero. The integers are included among the rational numbers since any integer can be expressed as the ratio of the integer itself and one. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">  One of the basic properties of the real-number system is that any two real numbers can be compared for size. If a and b are real numbers, we write a&lt;b to signify that a is less than b. Another way of saying the same thing is to write b&gt;a, which is read “b is greater than a “.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">   Geometrically, real numbers are identified with points on a straight line. We choose a straight line, and an initial point f reference called the origin. To the origin we assign the number zero. By marking off the unit of length in both directions from the origin, we assign positive integers to marked-off points in one direction (by convention, to the right of the origin ) and negative integers to marked-off point in the other direction. By following through in terms of the chosen unit of length, a real number is attached to one point on the number line, and each point on the number line has attached to it one number.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">  Geometrically, in terms of our number line, to say that a&lt;b is to say that a is to the left of b; b&gt;a means that b is to the right of a.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><B><FONT face="Times New Roman">Properties of Addition and Multiplication</FONT></B></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">  Addition and multiplication are primary operations on real numbers. Most, if not all, of the basic properties of these operations are familiar to us from experience.</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>(a)</FONT>      <FONT size=3>Closure property of addition and multiplication.</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>Whenever two real numbers are added or multiplied, we obtain a real number as the result. That is, performing the operations of addition and multiplication leaves us within the real-number system.</FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>(b)</FONT>      <FONT size=3>Commutative property of addition and multiplication.</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>The order in which two real numbers are added or multiplied does not affect the result obtained. That is, if a and b are any two real numbers, then we have (i) a+ b=b+ a and (ii) ab = ba. Such a property is called a commutative property. Thus, addition and multiplication of real numbers are commutative operations.</FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>(c)</FONT>      <FONT size=3>Associative property of addition and multiplication.</FONT></FONT></P>
<P  align=left><FONT size=3><FONT face="Times New Roman">Parentheses, brackets, and the like, we recall, are used in algebra to group together whatever terms are within them. Thus 2+(3+4) means that 2 is to be added to the sum of 3 and 4 yielding 2+7 =9 whereas (2+3)+4 means the sum of 2 and 3 is to be added to 4 yielding also 9. Similarly, 2</FONT>&#8226;<FONT face="Times New Roman">(3</FONT>&#8226;<FONT face="Times New Roman">4) yields 2</FONT>&#8226;<FONT face="Times New Roman">(12)=24 whereas (2</FONT>&#8226;<FONT face="Times New Roman">3) </FONT>&#8226;<FONT face="Times New Roman">4 yields the same end result by the route 6</FONT>&#8226;<FONT face="Times New Roman">4=24 . That such is the case in general is the content of the associative property of addition and multiplication of real numbers.</FONT></FONT></P>
<P  align=left><FONT face="Times New Roman"><FONT size=3>(d)</FONT>      <FONT size=3>Distributive property of multiplication over addition.</FONT></FONT></P>
<P  align=left><FONT size=3><FONT face="Times New Roman">We know that 2</FONT>&#8226;<FONT face="Times New Roman">(3</FONT>&#8226;<FONT face="Times New Roman">4)=2</FONT>&#8226;<FONT face="Times New Roman">7=14 and that 2</FONT>&#8226;<FONT face="Times New Roman">3+ 2</FONT>&#8226;<FONT face="Times New Roman">4=14 ,thus 2</FONT>&#8226;<FONT face="Times New Roman">(3+4)=2</FONT>&#8226;<FONT face="Times New Roman">3+ 2</FONT>&#8226;<FONT face="Times New Roman">4. That such is the case in general for all real numbers is the content of the distributive property of multiplication over addition, more simply called the distributive property.</FONT></FONT></P>
<P ><B><FONT face="Times New Roman">Substraction and Division</FONT></B></P>
<P  align=left><FONT face="Times New Roman" size=3>The numbers zero and one. The following are the basic properties of the numbers zero and one.</FONT></P>
<P  align=left><FONT face="Times New Roman"><FONT size=3>(a)</FONT>    <FONT size=3>There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a  is any real number.</FONT></FONT></P>
<P  align=left><FONT size=3><FONT face="Times New Roman">There is a unique real number, different from zero, called one and denoted by 1, with the property that a</FONT>&#8226;<FONT face="Times New Roman">1=1</FONT>&#8226;<FONT face="Times New Roman">a=a, where a is any real number.</FONT></FONT></P>
<P  align=left><FONT face="Times New Roman"><FONT size=3>(b)</FONT>    <FONT size=3>If a is any real number, then there is a unique real number x, called the additive inverse of a  , or negative of a, with the property that a+ x = x+ a .If a is any nonzero real number, then there is a unique real number y, called the multiplicative inverse of a, or reciprocal of a, with the property that ay = ya = 1 </FONT></FONT></P>
<P  align=left><FONT face="Times New Roman" size=3>The concept of the negative of a number should not be confused with the concept of a   negative   number; they are not the same. ”Negative of“ means additive inverse of “. On the other hand, a “negative number” is a number that is less than zero.</FONT></P>
<P  align=left><FONT face="Times New Roman" size=3>The multiplicative inverse of a is often represented by the symbol 1/a or a<SUP>-1</SUP>. Note that since the product of any number y and 0 is 0, 0 cannot have a multiplicative inverse. Thus 1/0 does not exist.</FONT></P>
<P  align=left><FONT face="Times New Roman"><FONT size=3>Now substraction is defined in terms of addition in the following way.<p></p></FONT></FONT></P>
<P  align=left><FONT face="Times New Roman" size=3>If a and b are any two real numbers, then the difference a-b is defined by a- b= c where c is such that b+ c=a or c= a+(-b). That is, to substract b from a means to add the negative of b (additive inverse of b) to a.</FONT></P>
<P  align=left><FONT face="Times New Roman" size=3>Division is defined in terms of multiplication in the following way.</FONT></P>
<P  align=left><FONT size=3><FONT face="Times New Roman">If a and b are any real numbers, where b</FONT>≠<FONT face="Times New Roman">0, then a+ b is defined by a +b= a</FONT>&#8226;<FONT face="Times New Roman">(1/b) =a</FONT>&#8226;<FONT face="Times New Roman">b<SUP>-1</SUP>. That is, to divide a by b means to multiply a by the multiplicative inverse ( reciprocal)of b. The quotient a +b is also expressed by the fraction symbol a/b.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
 楼主| 发表于 2004-5-6 09:22:37 | 显示全部楼层
<DIV class=Section1 style="LAYOUT-GRID:  15.6pt none">< 12pt 0cm 0pt; mso-layout-grid-align: none"><FONT face="Times New Roman"> <p></p></FONT></P>< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><A><B><FONT face="Times New Roman">Vocabulary</FONT></B></A><B><p></p></B></P></DIV><BR auto; mso-break-type: section-break" clear=all><DIV class=Section2 style="LAYOUT-GRID:  15.6pt none">< 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">real number     </FONT>实数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">negative     </FONT>负的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">the real number system  </FONT>实数系</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">rational number   </FONT>有理数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">collection   </FONT>集体<FONT face="Times New Roman">,</FONT>总体</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">ratio  </FONT>比<FONT face="Times New Roman">,</FONT>比率</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">object   </FONT>对象<FONT face="Times New Roman">,</FONT>目的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">denominator  </FONT>分母</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">principle  </FONT>原理<FONT face="Times New Roman">,</FONT>规则</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">numerator   </FONT>分子</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">adopt  </FONT>采用</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">irrational number   </FONT>无理数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">define </FONT>定义<FONT face="Times New Roman">(</FONT>动词<FONT face="Times New Roman">)</FONT></P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">signify  </FONT>表示</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">definition   </FONT>定义<FONT face="Times New Roman">(</FONT>名词<FONT face="Times New Roman">)</FONT></P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">geometrical  </FONT>几何的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">establish  </FONT>建立</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">straight line  </FONT>直线</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">explicit   </FONT>清晰的<FONT face="Times New Roman">,</FONT>明显的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">initial point </FONT>初始点</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">illustrate  </FONT>说明</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">point of reference  </FONT>参考点</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">positive  </FONT>正的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">origin   </FONT>原点<FONT face="Times New Roman"> </FONT></P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">express  </FONT>表达</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">assign   </FONT>指定</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">plus     </FONT>加</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">unit     </FONT>单位</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">sign  </FONT>记号<FONT face="Times New Roman">,</FONT>符号<FONT face="Times New Roman">,</FONT>正负号</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">property   </FONT>性质</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">operation    </FONT>运算<FONT face="Times New Roman">,</FONT>操作</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">closure property </FONT>封闭性质</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">addition   </FONT>加法</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">commutative     </FONT>交换的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">multiplication   </FONT>乘法</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">associative    </FONT>结合的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">substraction    </FONT>减法</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">parentheses   </FONT>圆括号</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">division   </FONT>除法</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">brackets    </FONT>括号</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">sum    </FONT>和<FONT face="Times New Roman">,</FONT>总数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">algebra    </FONT>代数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">procuct  </FONT>乘积</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">yield     </FONT>产生</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">difference </FONT>差<FONT face="Times New Roman">,</FONT>差分</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">term     </FONT>术语<FONT face="Times New Roman">,</FONT>项</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">quotient  </FONT>商</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">distributive  </FONT>分配的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">symbolism  </FONT>符号系统</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">unique   </FONT>唯一的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">minus  </FONT>减</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">additive inverse  </FONT>加法逆运算</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">identify  </FONT>使同一</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">multiplicative inverse  </FONT>乘法逆运算</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">count   </FONT>计数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">reciprocal  </FONT>倒数<FONT face="Times New Roman">,</FONT>互逆</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">natural number  </FONT>自然数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">concept  </FONT>概念</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">zero  </FONT>零</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">fraction   </FONT>分数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">integer  </FONT>整数</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">arithmetic   </FONT>算术的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">greater than   </FONT>大于</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">solution    </FONT>解<FONT face="Times New Roman">,</FONT>解法</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">less than    </FONT>小于</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">even   </FONT>偶的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">be equal to    </FONT>等于</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">odd  </FONT>奇的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">arbitrary   </FONT>任意的</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">square  </FONT>平方</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">absolute value   </FONT>绝对值</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">square root  </FONT>平方根</P></DIV><BR always; mso-break-type: section-break" clear=all><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; tab-stops: 100.5pt; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">cube   </FONT>立方<FONT face="Times New Roman"> </FONT></P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">induction   </FONT>归纳法</P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman"> <p></p></FONT></P>
 楼主| 发表于 2004-5-6 09:23:00 | 显示全部楼层
< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><B><FONT face="Times New Roman">Note</FONT></B></P>< 0cm 0cm 0pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">1. Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system.</FONT></P>< 0cm 0cm 0pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left>意思是<FONT face="Times New Roman">:</FONT>我们对数的实际工作经验使我们大家对支配着实数系的各原则早已有了某些熟悉<FONT face="Times New Roman">,</FONT>这里<FONT face="Times New Roman">working</FONT>作<FONT face="Times New Roman">”</FONT>实际工作的<FONT face="Times New Roman">”</FONT>解<FONT face="Times New Roman">,govern</FONT>作<FONT face="Times New Roman">”</FONT>支配<FONT face="Times New Roman">”</FONT>解<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">2.The plus sign,”+”, used here not express the operation of addition, but is rather part of the symbolism for the numbers themselves. </FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 21pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left>意思是<FONT face="Times New Roman">:</FONT>这里的正符号<FONT face="Times New Roman">”+”</FONT>不是表示加法运算<FONT face="Times New Roman">,</FONT>而是数本身的符号系统的一部分<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">3. A real number<U> is said to</U> be a rational number <U>if</U> it can be expressed as the ratio of two integers, <U>where</U> the denominator is not zero.</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 21pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left>这是定义数学术语的一种形式<FONT face="Times New Roman">.</FONT>下面是另一种定义数学术语的形式<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">   A matrix <U>is called</U> a square matrix <U>if</U> the number of its rows equals the number of its columns.                 </FONT></P><P 0cm 0cm 0pt 15.9pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left>这里<FONT face="Times New Roman">is called</FONT>与<FONT face="Times New Roman">is said to be </FONT>可以互用<FONT face="Times New Roman">,</FONT>注意<FONT face="Times New Roman">is called</FONT>后面一般不加<FONT face="Times New Roman">to be</FONT>而<FONT face="Times New Roman">is said</FONT>后面一般要加<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt; TEXT-ALIGN: left; mso-layout-grid-align: none" align=left><FONT face="Times New Roman">4. A real number <U>that cannot be expressed as the ratio of two integers</U> is said to be an irrational number.</FONT></P><P 0cm 0cm 0pt 15.75pt; tab-stops: 284.85pt">与注<FONT face="Times New Roman">3</FONT>比较<FONT face="Times New Roman">,</FONT>这里用定语从句界定术语<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt; tab-stops: 284.85pt"><FONT face="Times New Roman">5. There is a unique real number,<U> called</U> zero and <U>denoted</U> by 0, with the property that a+0=0+a, <U>where</U> a  is any real number.</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 21.75pt; tab-stops: 284.85pt">意思是<FONT face="Times New Roman">:</FONT>存在唯一的一个实数<FONT face="Times New Roman">,</FONT>叫做零并记为<FONT face="Times New Roman">0,</FONT>具有性质<FONT face="Times New Roman">a+0=0+a,</FONT>这里<FONT face="Times New Roman">(</FONT>其中<FONT face="Times New Roman">)a</FONT>是任一实数<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt 42pt; TEXT-INDENT: -20.25pt; tab-stops: list 42.0pt left 284.85pt; mso-list: l5 level1 lfo4"><FONT face="Times New Roman">1)        </FONT>这里<FONT face="Times New Roman">called</FONT>和<FONT face="Times New Roman">denoted</FONT>都是过去分词<FONT face="Times New Roman">,</FONT>与后面的词组成分词短语<FONT face="Times New Roman">,</FONT>修饰<FONT face="Times New Roman">number.</FONT></P><P 0cm 0cm 0pt 42pt; TEXT-INDENT: -20.25pt; tab-stops: list 42.0pt left 284.85pt; mso-list: l5 level1 lfo4"><FONT face="Times New Roman">2)        with the property</FONT>是前置短语<FONT face="Times New Roman">,</FONT>修饰<FONT face="Times New Roman">number.</FONT><v:shapetype><FONT face="Times New Roman"> <v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></FONT></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape></P><P 0cm 0cm 0pt 42pt; TEXT-INDENT: -20.25pt; tab-stops: list 42.0pt left 284.85pt; mso-list: l5 level1 lfo4"><FONT face="Times New Roman">3)        </FONT>注意本句和注<FONT face="Times New Roman">3.</FONT>中<FONT face="Times New Roman">where</FONT>的用法<FONT face="Times New Roman">,</FONT>一般当需要附加说明句子中某一对象时可用此结构<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt 21.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman"> <p></p></FONT></P>
 楼主| 发表于 2004-5-6 09:23:21 | 显示全部楼层
< 0cm 0cm 0pt 21.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman"> <p></p></FONT></P>< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><B><FONT face="Times New Roman">Exercise</FONT></B></P>< 0cm 0cm 0pt 78.75pt; TEXT-INDENT: -36pt; tab-stops: list 78.75pt left 284.85pt; mso-list: l5 level2 lfo4"><FONT face="Times New Roman">I.                    Turn the following arithmetic expressions into English:</FONT></P><P 0cm 0cm 0pt 78.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman">i) 3+(-2)=1           ii) 2+3(-4)=-10</FONT></P><P 0cm 0cm 0pt 78.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman">iii) </FONT><v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">= -5</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman">     iv) </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">=3</FONT></P><P 0cm 0cm 0pt 78.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman">v)2/5-1/6=7/30</FONT></P><P 0cm 0cm 0pt 78.75pt; TEXT-INDENT: -36pt; tab-stops: list 78.75pt left 284.85pt; mso-list: l5 level2 lfo4"><FONT face="Times New Roman">II.                 Fill in each blank the missing mathematical term to mark the following sentences complete.</FONT></P><P 0cm 0cm 0pt 78.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman">i) The<U>             </U> of two real numbers of unlike signs is negative.</FONT></P><P 0cm 0cm 0pt 78.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman">ii) An integer n is called<U>             </U>if n=2m for some integer m.</FONT></P><P 0cm 0cm 0pt 78.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman">iii) An solution to the equation x<SUP>n</SUP>=c is called the n is <U>    </U>of  c.</FONT></P><P 0cm 0cm 0pt 78.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman">iv) If x is a real number, then the          of x is a nonnegative real number denoted by |x| and defined as follows</FONT></P><P 0cm 0cm 0pt 78.75pt; tab-stops: 284.85pt"><FONT face="Times New Roman">                                x,  if x</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman">0</FONT></P><P 6pt auto 0pt 78.75pt"><FONT face="Times New Roman">                                          </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">|x|= </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">-x, if x &lt;0</FONT></P><P 0cm 0cm 0pt 78.75pt; TEXT-INDENT: -36pt; tab-stops: list 78.75pt left 284.85pt; mso-list: l5 level2 lfo4"><FONT face="Times New Roman">III.               Translate the following exercises into Chinese:</FONT></P><P 0cm 0cm 0pt 99.75pt; TEXT-INDENT: -36pt; tab-stops: list 99.75pt left 284.85pt; mso-list: l5 level3 lfo4"><FONT face="Times New Roman">i)                    If x is an arbitrary real number, prove that there is exactly one integer n such that x&lt;n&lt;x+1.</FONT></P><P 0cm 0cm 0pt 99.75pt; TEXT-INDENT: -36pt; tab-stops: list 99.75pt left 284.85pt; mso-list: l5 level3 lfo4"><FONT face="Times New Roman">ii)                   Prove that there is no rational number whose square in 2.</FONT></P><P 0cm 0cm 0pt 99.75pt; TEXT-INDENT: -36pt; tab-stops: list 99.75pt left 284.85pt; mso-list: l5 level3 lfo4"><FONT face="Times New Roman">iii)                 Given positive real numbers a1,a2,a3,</FONT>…<FONT face="Times New Roman">such that an&lt;ca<SUB>n-1</SUB> for all n&gt;2, where c is a fixed positive number, use induction to prove that an&lt;c<SUP>n-1</SUP>a<SUB>1</SUB>, for all n&gt;1.</FONT></P><P 0cm 0cm 0pt 99.75pt; TEXT-INDENT: -36pt; tab-stops: list 99.75pt left 284.85pt; mso-list: l5 level3 lfo4"><FONT face="Times New Roman">iv)                 Determine all positive integers n for which 2<SUP>n</SUP>&lt;n!</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt; tab-stops: 284.85pt">Ⅳ<FONT face="Times New Roman">  Translate the following passage into Chinese:</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt; tab-stops: 284.85pt"><FONT face="Times New Roman">     There are many ways to introduce the real number system. One popular method is to begin with the positive integers 1,2,3,</FONT>…<FONT face="Times New Roman">and use them as building blocks to construct a more comprehensive system having the properties desired. Briefly, the idea of this method is to take the positive integers as undefined concepts, state some axioms concerning them, and them use the positive integers to build a larger system consisting of the positive rational numbers. The positive irrational numbers, in turn, may then be used as basis for constructing the positive irrational numbers. The final step is the introduction of the negative numbers and zero. The most difficult part of the whole process is the transition from the rational numbers to the irrational num</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt; tab-stops: 284.85pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt; tab-stops: 284.85pt">Ⅴ<FONT face="Times New Roman">. Translate the following theorems into English:</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt; tab-stops: 284.85pt"><FONT face="Times New Roman">   1. </FONT>定理<FONT face="Times New Roman">A: </FONT>每一非负数有唯一一个非负平方根<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt; tab-stops: 284.85pt"><FONT face="Times New Roman">   2. </FONT>定理<FONT face="Times New Roman">B: </FONT>若<FONT face="Times New Roman">x&gt;0, y</FONT>是任意一实数<FONT face="Times New Roman">,</FONT>则存在一正整数<FONT face="Times New Roman">n</FONT>使得<FONT face="Times New Roman">nx &gt; y.</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt; tab-stops: 284.85pt">Ⅵ<FONT face="Times New Roman">. 1. Try to show the structure of the set of real numbers graphically.</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt; tab-stops: 284.85pt"><FONT face="Times New Roman">   2. List and state the laws that operations of addition and multiplication of real numbers obey.</FONT></P><P 0cm 0cm 0pt; tab-stops: 284.85pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt; tab-stops: 284.85pt"><FONT face="Times New Roman"> <p></p></FONT></P>
发表于 2004-11-27 20:23:46 | 显示全部楼层
< align=center center? TEXT-ALIGN: 0pt; 0cm><B><FONT face="Times New Roman">Exercise</FONT></B></P>< 0cm TEXT-INDENT: 0pt list tab-stops: mso-list: lfo4? l5 284.85pt; left level2 78.75pt -36pt; 78.75pt;><FONT face="Times New Roman">I.                    Turn the following arithmetic expressions into English:</FONT></P>< 0cm 0pt tab-stops: 284.85pt? 78.75pt;><FONT face="Times New Roman">i) 3+(-2)=1           ii) 2+3(-4)=-10</FONT></P><P 0cm 0pt tab-stops: 284.85pt? 78.75pt;><FONT face="Times New Roman">iii) </FONT><v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path gradientshapeok="t" connecttype="rect" extrusionok="f"></v:path><LOCK v:ext="edit" aspectratio="t"></LOCK></v:shapetype><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">= -5</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman">     iv) </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">=3</FONT></P><P 0cm 0pt tab-stops: 284.85pt? 78.75pt;><FONT face="Times New Roman">v)2/5-1/6=7/30</FONT></P><P 0cm TEXT-INDENT: 0pt list tab-stops: mso-list: lfo4? l5 284.85pt; left level2 78.75pt -36pt; 78.75pt;><FONT face="Times New Roman">II.                 Fill in each blank the missing mathematical term to mark the following sentences complete.</FONT></P><P 0cm 0pt tab-stops: 284.85pt? 78.75pt;><FONT face="Times New Roman">i) The<U>             </U>of two real numbers of unlike signs is negative.</FONT></P><P 0cm 0pt tab-stops: 284.85pt? 78.75pt;><FONT face="Times New Roman">ii) An integer n is called<U>             </U>if n=2m for some integer m.</FONT></P><P 0cm 0pt tab-stops: 284.85pt? 78.75pt;><FONT face="Times New Roman">iii) An solution to the equation x<SUP>n</SUP>=c is called the n is <U>    </U>of  c.</FONT></P><P 0cm 0pt tab-stops: 284.85pt? 78.75pt;><FONT face="Times New Roman">iv) If x is a real number, then the          of x is a nonnegative real number denoted by |x| and defined as follows</FONT></P><P 0cm 0pt tab-stops: 284.85pt? 78.75pt;><FONT face="Times New Roman">                                x,  if x</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman">0</FONT></P><P 0pt 78.75pt? auto 6pt><FONT face="Times New Roman">                                          </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">|x|= </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">-x, if x &lt;0</FONT></P><P 0cm TEXT-INDENT: 0pt list tab-stops: mso-list: lfo4? l5 284.85pt; left level2 78.75pt -36pt; 78.75pt;><FONT face="Times New Roman">III.               Translate the following exercises into Chinese:</FONT></P><P 0cm TEXT-INDENT: 0pt list tab-stops: mso-list: lfo4? l5 284.85pt; left -36pt; level3 99.75pt 99.75pt;><FONT face="Times New Roman">i)                    If x is an arbitrary real number, prove that there is exactly one integer n such that x&lt;n&lt;x+1.</FONT></P><P 0cm TEXT-INDENT: 0pt list tab-stops: mso-list: lfo4? l5 284.85pt; left -36pt; level3 99.75pt 99.75pt;><FONT face="Times New Roman">ii)                   Prove that there is no rational number whose square in 2.</FONT></P><P 0cm TEXT-INDENT: 0pt list tab-stops: mso-list: lfo4? l5 284.85pt; left -36pt; level3 99.75pt 99.75pt;><FONT face="Times New Roman">iii)                 Given positive real numbers a1,a2,a3,</FONT>…<FONT face="Times New Roman">such that an&lt;ca<SUB>n-1</SUB> for all n&gt;2, where c is a fixed positive number, use induction to prove that an&lt;c<SUP>n-1</SUP>a<SUB>1</SUB>, for all n&gt;1.</FONT></P><P 0cm TEXT-INDENT: 0pt list tab-stops: mso-list: lfo4? l5 284.85pt; left -36pt; level3 99.75pt 99.75pt;><FONT face="Times New Roman">iv)                 Determine all positive integers n for which 2<SUP>n</SUP>&lt;n!</FONT></P><P 0pt; 0cm TEXT-INDENT: tab-stops: 284.85pt? 37.5pt;>Ⅳ<FONT face="Times New Roman">  Translate the following passage into Chinese:</FONT></P><P 0pt; 0cm TEXT-INDENT: tab-stops: 284.85pt? 37.5pt;><FONT face="Times New Roman">     There are many ways to introduce the real number system. One popular method is to begin with the positive integers 1,2,3,</FONT>…<FONT face="Times New Roman">and use them as building blocks to construct a more comprehensive system having the properties desired. Briefly, the idea of this method is to take the positive integers as undefined concepts, state some axioms concerning them, and them use the positive integers to build a larger system consisting of the positive rational numbers. The positive irrational numbers, in turn, may then be used as basis for constructing the positive irrational numbers. The final step is the introduction of the negative numbers and zero. The most difficult part of the whole process is the transition from the rational numbers to the irrational num</FONT></P><P 0pt; 0cm TEXT-INDENT: tab-stops: 284.85pt? 37.5pt;><FONT face="Times New Roman">
</FONT><p><P 0pt; 0cm TEXT-INDENT: tab-stops: 284.85pt? 37.5pt;>Ⅴ<FONT face="Times New Roman">. Translate the following theorems into English:</FONT></P><P 0pt; 0cm TEXT-INDENT: tab-stops: 284.85pt? 37.5pt;><FONT face="Times New Roman">   1. </FONT>定理<FONT face="Times New Roman">A: </FONT>每一非负数有唯一一个非负平方根<FONT face="Times New Roman">.</FONT></P><P 0pt; 0cm TEXT-INDENT: tab-stops: 284.85pt? 37.5pt;><FONT face="Times New Roman">   2. </FONT>定理<FONT face="Times New Roman">B: </FONT>若<FONT face="Times New Roman">x&gt;0, y</FONT>是任意一实数<FONT face="Times New Roman">,</FONT>则存在一正整数<FONT face="Times New Roman">n</FONT>使得<FONT face="Times New Roman">nx &gt; y.</FONT></P><P 0pt; 0cm TEXT-INDENT: tab-stops: 284.85pt? 37.5pt;>Ⅵ<FONT face="Times New Roman">. 1. Try to show the structure of the set of real numbers graphically.</FONT></P><P 0pt; 0cm TEXT-INDENT: tab-stops: 284.85pt? 37.5pt;><FONT face="Times New Roman">   2. List and state the laws that operations of addition and multiplication of real numbers obey.</FONT></P><P 0pt; 0cm tab-stops: 284.85pt?><FONT face="Times New Roman"><p></FONT><p><P 0pt; 0cm tab-stops: 284.85pt?><FONT face="Times New Roman"><p></FONT><p>
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