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1、数学专业外语第三讲数学专业外语第三讲第1页，本讲稿共21页Key points:useful terms and definitions of Mathematics,equation and geometryDifficult points:Some geometrical terms第2页，本讲稿共21页New Words&Expressions:geometry n.几何学几何学function theory 函数论函数论 trigonometry n.三角学三角学conception n.概念概念algebra n.代数学代数学proposition 命题命题equation 方程方程
2、,等式等式definition 定义定义constant n.常量常量notation 符号符号,记号记号mathematical analysis 数学分析数学分析 logical deduction 逻辑推理逻辑推理,推论推论Roman-number n.罗马数字罗马数字2.1 数学、方程与比例Mathematics,Equation and Ratio第3页，本讲稿共21页New Words&Expressions:higher mathematics 高等数学高等数学differential equation 微分方程微分方程equation of condition 条件等式条件等式
3、 linear equation 线性等式线性等式variable adj.变化的变化的,n.变量变量dimension 大小大小,维数维数,尺寸尺寸formula 公式公式even number 偶数偶数arithmetic 算术算术,算术的算术的numerical 数值的数值的,数字的数字的fraction 分数分数 identity 恒等式恒等式 term 项项,术语术语,命名为命名为algebraic 代数的代数的operation 运算运算第4页，本讲稿共21页Mathematics comes from mans social practice,for example,indust
4、rial and agricultural production,commercial activities,military operations and scientific and technological researches.And in turn,mathematics serves the practice and plays a great role in all fields.No modern scientific and technological branches could be regularly developed without the application
5、 of mathematics.数数学学来来源源于于人人类类的的社社会会实实践践,比比如如工工农农业业生生产产,商商业业活动活动,军事行动和科学技术研究军事行动和科学技术研究.反反过过来来,数数学学服服务务于于实实践践,并并在在各各个个领领域域中中起起着着非非常常重重要要的的作作用用.没没有有应应用用数数学学,任任何何一一个个现现在在的的科科技技的的分支都不能正常发展分支都不能正常发展.1A What is mathematics第5页，本讲稿共21页From the early need of man came the concepts of numbers and forms.Then,g
6、eometry developed out of problems of measuring land,and trigonometry came from problems of surveying.To deal with some more complex practical problems,man established and then solved equation with unknown numbers,thus algebra occurred.很很早早的的时时候候,人人类类的的需需要要产产生生了了数数和和形形式式的的概概念念.接接着着,测测量量土土地地的的需需要要形形成成
7、了了几几何何，出出于于测测量量的的需需要要产产生了三角几何生了三角几何.为为了了处处理理更更复复杂杂的的实实际际问问题题，人人类类建建立立和和解解决决了了带带未未知参数的方程知参数的方程,从而产生了代数学从而产生了代数学.第6页，本讲稿共21页Before 17th century,man confined himself to the elementary mathematics,i.e.,geometry,trigonometry and algebra,in which only the constants are considered.The rapid development of
8、industry in 17th century promoted the progress of economics and technology and required dealing with variable quantities.17世世纪纪前前,人人类类局局限限于于只只考考虑虑常常数数的的初初等等数数学学，即即几几何，三角几何和代数。何，三角几何和代数。17世纪工业的快速发展推动了经济技术的进步世纪工业的快速发展推动了经济技术的进步,从而从而遇到需要处理变量的问题。遇到需要处理变量的问题。第7页，本讲稿共21页The leap from constants to variable
9、 quantities brought about two new branches of mathematics-analytic geometry and calculus,which belong to the higher mathematics.Now there are many branches in higher mathematics,among which are mathematical analysis,higher algebra,differential equations,function theory and so on.从从常常数数带带变变量量的的跳跳跃跃产产
10、生生了了两两个个新新的的数数学学分分支支-解解析析几何和微积分几何和微积分,他们都属于高等数学他们都属于高等数学.现现在在高高等等数数学学里里面面有有很很多多分分支支,其其中中有有数数学学分分析析,高高等等代数代数,微分方程微分方程,函数论等函数论等.第8页，本讲稿共21页Mathematicians study conceptions and propositions.Axioms,postulates,definitions and theorems are all propositions.Notations are a special and powerful tool of math
11、ematics and are used to express conceptions and propositions very often.数数学学家家研研究究的的是是概概念念和和命命题题,公公理理,公公设设,定定义义和和定定理理都是命题都是命题.符符号号是是数数学学中中一一个个特特殊殊而而有有用用的的工工具具,常常用用于于表表达达概概念和命题念和命题.第9页，本讲稿共21页Formulas,figures and charts are full of different symbols.Some of the best known symbols of mathematics are t
12、he Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition,subtraction,multiplication,division and equality.公式公式,图表都是不同的符号图表都是不同的符号.阿阿拉拉伯伯数数字字1,2,3,4,5,6,7,8,9,0和和加加,减减,乘乘,除除,等式都是一些常见的符号等式都是一些常见的符号.第10页，本讲稿共21页The conclusions in mathematics are obtained mainly by logical deductions and computa
13、tion.For a long period of the history of mathematics,the centric place of mathematics methods was occupied by the logical deductions.数学结论主要由逻辑推理和计算得到数学结论主要由逻辑推理和计算得到.在在数数学学发发展展历历史史的的很很长长时时间间内内,逻逻辑辑推推理理一一直直占占据据着着数学方法的中心地位数学方法的中心地位.第11页，本讲稿共21页Now,since electronic computers are developed promptly and
14、used widely,the role of computation becomes more and more important.In our times,computation is not only used to deal with a lot of information and data,but also to carry out some work that merely could be done earlier by logical deductions,for example,the proof of most of geometrical theorems.现在现在,
15、由于电子计算机的迅速发展和广泛使用由于电子计算机的迅速发展和广泛使用,计算的计算的地位越来越重要地位越来越重要.现在计算机不仅用于处理大量的信息和数据现在计算机不仅用于处理大量的信息和数据,还可以还可以完成一些之前只能由逻辑推理来做的工作完成一些之前只能由逻辑推理来做的工作,例如大多例如大多数几何定理的证明数几何定理的证明.第12页，本讲稿共21页回顾：回顾：1如如果果没没有有运运用用数数学学，任任何何一一个个科科学学技技术术分分支支都都不不可能正常的发展可能正常的发展。No modern scientific and technological branches could be regul
16、arly developed without the application of mathematics.2符符号号是是数数学学中中一一个个特特殊殊而而有有用用的的工工具具,常常用用于于表表达达概念和命题概念和命题.Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.第13页，本讲稿共21页An equation is a statement of the equality between two equ
17、al numbers or number symbols.Equation are of two kinds-identities and equations of condition.An arithmetic or an algebraic identity is an equation.等式是关于两个数或者数的符号相等的一种描述。等式是关于两个数或者数的符号相等的一种描述。等式有两种恒等式和条件等式。等式有两种恒等式和条件等式。算术或者代数恒等式是等式。算术或者代数恒等式是等式。1B Equation第14页，本讲稿共21页In such an equation either the t
18、wo members are alike,or become alike on the performance of the indicated operation.An identity involving letters is true for any set of numerical values of the letters in it.这这种种等等式式的的两两端端要要么么一一样样，要要么么经经过过执执行行指指定定的的运运算算后变成一样。后变成一样。含有字母的恒等式对其中字母的任一组数值都成立。含有字母的恒等式对其中字母的任一组数值都成立。第15页，本讲稿共21页An equation
19、 which is true only for certain values of a letter in it,or for certain sets of related values of two or more of its letters,is an equation of condition,or simply an equation.Thus 3x-5=7 is true for x=4 only;and 2x-y=0 is true for x=6 and y=2 and for many other pairs of values for x and y.一一个个等等式式若若
20、仅仅仅仅对对其其中中一一个个字字母母的的某某些些值值成成立立，或或对对其其中中两两个个或或着着多多个个字字母母的的若若干干组组相相关关的的值值成成立立，则则它它是一个条件等式，简称方程。是一个条件等式，简称方程。因因此此3x-5=7仅仅当当x=4 时时成成立立，而而2x-y=0，当当x=6,y=2时时成立，且对成立，且对x,y的其他许多对值也成立。的其他许多对值也成立。第16页，本讲稿共21页A root of an equation is any number or number symbol which satisfies the equation.There are various ki
21、nds of equation.They are linear equation,quadratic equation,etc.To solve an equation means to find the value of the unknown term.方程的根是满足方程的任意数或者数的符号。方程的根是满足方程的任意数或者数的符号。方程有很多种，例如：方程有很多种，例如：线性方程，二次方程等。线性方程，二次方程等。解方程意味着求未知项的值解方程意味着求未知项的值.第17页，本讲稿共21页To do this,we must,of course,change the terms about
22、until the unknown term stands alone on one side of the equation,thus making it equal to something on the other side.We then obtain the value of the unknown and the answer to the question.为为了了求求未未知知项项的的值值，当当然然必必须须移移项项，直直到到未未知知项项单单独独在方程的一边，令其等于方程的另一边。在方程的一边，令其等于方程的另一边。从而求得未知项的值，解决了问题。从而求得未知项的值，解决了问题。第
23、18页，本讲稿共21页To solve the equation,therefore,means to move and change the terms about without making the equation untrue,until only the unknown quantity is left on one side,no matter which side.Equation are of very great use.We can use equation in many mathematical problems.因因此此解解方方程程意意味味着着进进行行一一系系列列的
24、的移移项项和和同同解解变变形形，直直到未知量被单独留在方程的一边，无论那一边。到未知量被单独留在方程的一边，无论那一边。方程作用很大。方程作用很大。可以用方程解决很多数学问题。可以用方程解决很多数学问题。第19页，本讲稿共21页We may notice that almost every problem gives us one or more statements that something is equal to something,this gives us equations,with which we may work if we need it.注注意意到到几几乎乎每每一一个个
25、问问题题都都给给出出一一个个或或多多个个关关于于一一个个事事情情与与另另一一个个事事情情相相等等的的陈陈述述，这这就就给给出出了了方方程程，利利用用该方程，如果我们需要的话，可以解方程。该方程，如果我们需要的话，可以解方程。第20页，本讲稿共21页回顾：回顾：1一一个个等等式式若若仅仅仅仅对对其其中中一一个个字字母母的的某某些些值值成成立立，或或对对其其中中两两个个或或着着多多个个字字母母的的若若干干组组相相关关的的值值成成立，则它是一个条件等式，简称方程。立，则它是一个条件等式，简称方程。An equation which is true only for certain values of
26、 a letter in it,or for certain sets of related values of two or more of its letters,is an equation of condition,or simply an equation.2解解方方程程时时要要进进行行一一系系列列移移项项和和同同解解变变形形，最最后后求求出它的根，即未知量的值。出它的根，即未知量的值。To solve the equation,therefore,means to move and change the terms about without making the equation untrue第21页，本讲稿共21页