毕业论文外文翻译-勾股定理.doc

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1、数学与应用数学英文文献及翻译-勾股定理 勾股定理是已知最早的古代文明定理之一。这个著名的定理被命名为希腊的数学家和哲学家毕达哥拉斯。毕达哥拉斯在意大利南部的科托纳创立了毕达哥拉斯学派。他在数学上有许多贡献，虽然其中一些可能实际上一直是他学生的工作。毕达哥拉斯定理是毕达哥拉斯最著名的数学贡献。据传说，毕达哥拉斯在得出此定理很高兴，曾宰杀了牛来祭神，以酬谢神灵的启示。后来又发现2的平方根是不合理的，因为它不能表示为两个整数比，极大地困扰毕达哥拉斯和他的追随者。他们在自己的认知中，二是一些单位长度整数倍的长度。因此2的平方根被认为是不合理的，他们就尝试了知识压制。它甚至说，谁泄露了这个秘密在海上被淹

2、死。毕达哥拉斯定理是关于包含一个直角三角形的发言。毕达哥拉斯定理指出，对一个直角三角形斜边为边长的正方形面积，等于剩余两直角为边长正方形面积的总和 图1根据勾股定理，在两个红色正方形的面积之和A和B，等于蓝色的正方形面积，正方形三区 因此，毕达哥拉斯定理指出的代数式是： 对于一个直角三角形的边长a，b和c，其中c是斜边长度。虽然记入史册的是著名的毕达哥拉斯定理，但是巴比伦人知道某些特定三角形的结果比毕达哥拉斯早一千年。现在还不知道希腊人最初如何体现了勾股定理的证明。如果用欧几里德的算法使用，很可能这是一个证明解剖类型类似于以下内容：六维-论文.网“一个大广场边a+ b是分成两个较小的正方形的边

3、a和b分别与两个矩形A和B，这两个矩形各可分为两个相等的直角三角形，有相同的矩形对角线c。四个三角形可安排在另一侧广场a+b中的数字显示。 在广场的地方就可以表现在两个不同的方式：1。由于两个长方形和正方形面积的总和： 2。作为一个正方形的面积之和四个三角形： 现在，建立上面2个方程，求解得因此，对c的平方等于a和b的平方和（伯顿1991）有许多的勾股定理其他证明方法。一位来自当代中国人在中国现存最古老的含正式数学理论能找到对Gnoman和天坛圆路径算法的经典文本。这勾股定理证明是一个鼓舞人心的数字证明，被列入书Vijaganita，（根计算），由印度数学家卜哈斯卡瑞。卜哈斯卡瑞的唯一解释是他

4、的证明，简单地说，“看”。这些发现证明和周围的几何定理的毕达哥拉斯是导致在作为Pythgorean数论问题的最早的问题之一。 毕达哥拉斯问题：找到所有的边的长度为直角三角形边长的组成，从而找到在毕达哥拉斯方程的正整数所有的解决方案：有三个整数（x，y，z）满足这个方程，则称为勾股数。部分勾股数：x y z3 4 55 12 137 24 259 40 4111 60 61该公式将产生所有勾股数最早出现在书欧几里德的元素x：1794数学与应用数学英文文献及翻译-勾股定理其中n和m是.正整数，且不同为奇数或偶数在他的书中算术，丢番图证实，他能利用这个公式直角三角形，虽然他给了一个不同的论证。勾股定

5、理可在初中向学生介绍。在高中这个定理变得越来越重要。仅仅这样还不够，为勾股定理代数公式，学生需要看到的几何连接以及在教学和学习中的勾股定理，可丰富和通过使用增强点纸，geoboards，折纸，和计算机技术，以及许多其他的教学材料。通过对教具和其他教育资源的使用，毕达哥拉斯定理可能意味着更多的学生不仅仅是插上数字的公式。 以下是对勾股定理的证明包括欧几里德一个品种。这些证明，随着教具和技术提高，可以大大提高学生对勾股定理的理解。下面是一个由欧几里德其中最有名的数学家之一证明的总结。这个证明可以在书欧几里德的元素中找到。命题：直角三角形上斜边的平方等于在直角边的平方和。 图2欧几里德开始在上面图2

6、所示的毕达哥拉斯配置。然后，他建造了一个垂直线，从C做DJ就关于斜边垂线。这点H和G是本与斜边上的正方形的边垂足。它位于的三角形ABC的高。见图3。下一步，欧几里德表明六维-论文.网，矩形HBDG面积等于BC上正方形的和与矩形的HAJG正方形的面积关系。他证明了这些等式利用相似的概念，三角形ABC，AHC和CHB相似 ，HAJG面积=(HA)(AG),AJ=AB, HAJG面积=(HA)(AB), 三角形ABC与三角形AHC相似，即：。因此，以同样的方式，三角形ABC的和CHG是相似的。所以即由于这两个矩形的面积之和，是对斜边正方形的面积，这样就完成了证明。欧几里德急于把这个结果在他的工作尽快

7、得出结果。然而，由于他的工作与相似联系不大，直至图书第五和第六，他必须与另一种方式来证明了勾股定理。因此，他采用平行四边形的结果是相同的基础上翻一番，并在同一平行线之间的三角形。连接CJ和BE。矩形的AHGJ面积是三角形JAC面积的两倍，以及ACLE面积是三角形BAE面积的两倍。这两个三角形全等采用SAS。在同样的结果如下，为其他类似的方式长方形和正方形。（卡茨，1993年）点击这里，普惠制动画来说明这方面的证据。接下来的三个证据更容易看到了毕达哥拉斯定理证明，将高中数学学生的理想选择。其实，这些都是可以证明，学生可以自己在某个时候兴建he Pythagorean Theorem was on

8、e of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathemati

9、cs although some of them may have actually been the work of his students. The Pythagorean Theorem is Pythagoras most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square

10、root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square r

11、oot of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:The area of the square bu六维-论文.网ilt upon the hypotenuse of a right triangle is equal to

12、 the sum of the areas of the squares upon the remaining sides.Figure 1According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and B, is equal to the area of the blue square, square C.Thus, the Pythagorean Theorem stated algebraically is:for a right triangle with

13、sides of lengths a, b, and c, where c is the length of the hypotenuse.Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras. It is not known how the Greeks originally dem

14、onstrated the proof of the Pythagorean Theorem. If the methods of Book II of Euclids Elements were used, it is likely that it was a dissection type of proof similar to the following:A large square of side a+b is divided into two smaller squares of sides a and b respectively, and two equal rectangles

15、 with sides a and b; each of these two rectangles can be split into two equal right triangles by drawing the diagonal c. The four triangles can be arranged within another square of side a+b as shown in the figures. he area of the square can be shown in two different ways:1. As the sum of the area of

16、 the two rectangles and the squares:2. As the sum of the areas of a square and the four triangles:Now, setting the two right hand side expressions in these equations equal, givesTherefore, the square on c is equal to the sum of the squares on a and b. (Burton 1991)There are many other proofs of the

17、Pythagorean Theorem. One came from the contemporary Chinese civilization found in the oldest extant Chinese text containing formal mathematical theories, the Arithmetic Classic of the Gnoman and the Circular Paths of Heaven. The proof of the Pythagorean Theorem that was inspired by a figure in this

18、book was included in the book Vijaganita, (Root Calculations), by the Hindu mathematician Bhaskara. Bhaskaras only explanation of his proof was, simply, Behold.These proofs and the geometrical discovery surrounding the Pythagorean Theorem led to one of the earliest problems in the theory of numbers

19、known as the Pythgorean problem.The Pythagorean Problem:Find all right triangles whose sides are of integral length, thus finding all solutions in the positive integers of the Pythagorean equation:The three integers (x, y, z) that satisfy this equation is called a Pythagorean triple.Some Pythagorean

20、 Triples:x y z3 4 55 12 137 24 259 40 41 11 60 61The formula that will generate all Pythagorean triples first appeared in Book X of Euclids Elements:where n and m are positive integers of opposite parity and mn.In his book Arithmetica, Diophantus confirmed that he could get right triangles using thi

21、s formula although he arrived at it under a different line of reasoning.The Pythagorean Theorem can be introduced to students during the middle school years. This theorem becomes increasingly important during the high school years. It is not enough to merely state the algebraic formula for the Pytha

22、gorean Theorem. Studen六维-论文.网ts need to see the geometric connections as well. The teaching and learning of the Pythagorean Theorem can be enriched and enhanced through the use of dot paper, geoboards, paper folding, and computer technology, as well as many other instructional materials. Through the

23、 use of manipulatives and other educational resources, the Pythagorean Theorem can mean much more to students than justand plugging numbers into the formula.The following is a variety of proofs of the Pythagorean Theorem including one by Euclid. These proofs, along with manipulatives and technology,

24、 can greatly improve students understanding of the Pythagorean Theorem.The following is a summation of the proof by Euclid, one of the most famous mathematicians. This proof can be found in Book I of Euclids Elements. Proposition: In right-angled triangles the square on the hypotenuse is equal to th

25、e sum of the squares on the legs.Figure 2Euclid began with the Pythagorean configuration shown above in Figure 2. Then, he constructed a perpendicular line from C to the segment DJ on the square on the hypotenuse. The points H and G are the intersections of this perpendicular with the sides of the s

26、quare on the hypotenuse. It lies along the altitude to the right triangle ABC. See Figure 3.Figure 3Next, Euclid showed that the area of rectangle HBDG is equal to the area of square on BC and that the are of the rectangle HAJG is equal to the area of the square on AC. He proved these equalities usi

27、ng the concept of similarity. Triangles ABC, AHC, and CHB are similar. The area of rectangle HAJG is (HA)(AJ) and since AJ = AB, the area is also (HA)(AB). The similarity of triangles ABC and AHC meansor, as to be proved, the area of the rectangle HAJG is the same as the areaof the square on side AC

28、. In the same way, triangles ABC and CHG are similar. SoSince the sum of the areas of the two rectangles is the area of the square on the hypotenuse, this completes the proof.Euclid was anxious to place this result in his work as soon as possible. However, since his work on similarity was not to be

29、until Books V and VI六维-论文.网, it was necessary for him to come up with another way to prove the Pythagorean Theorem. Thus, he used the result that parallelograms are double the triangles with the same base and between the same parallels. Draw CJ and BE. The area of the rectangle AHGJ is double the ar

30、ea of triangle JAC, and the area of square ACLE is double triangle BAE. The two triangles are congruent by SAS. The same result follows in a similar manner for the other rectangle and square. (Katz, 1993)Click here for a GSP animation to illustrate this proof.The next three proofs are more easily se

31、en proofs of the Pythagorean Theorem and would be ideal for high school mathematics students. In fact, these are proofs that students could be able to construct themselves at some point. The first proof begins with a rectangle divided up into three triangles, each of which contains a right angle. Th

32、is proof can be seen through the use of computer technology, or with something as simple as a 3x5 index card cut up into right triangles.Figure 4 Figure 5It can be seen that triangles 2 (in green) and 1 (in red), will completely overlap triangle 3 (in blue). Now, we can give a proof of the Pythagore

33、an Theorem using these same triangles.Proof:I. Compare triangles 1 and 3.Figure 6Angles E and D, respectively, are the right angles in these triangles. By comparing their similarities, we have and from Figure 6, BC = AD. So, By cross-multiplication, we get :II. Compare triangles 2 and 3:Figure 7By c

34、omparing the similarities of triangles 2 and 3 we get: From Figure 4, AB = CD. By substitution,Cross-multiplication gives:Finally, by adding equations 1 and 2, we get:From triangle 3, AC = AE + ECsoFigure 8We have proved the Pythagorean Theorem.The next proof is another proof of the Pythagorean Theo

35、rem that begins with a rectangle. It begins by constructing rectangle CADE with B六维-论文.网A = DA. Next, we construct the angle bisector of BAD and let it intersect ED at point F. Thus, BAF is congruent to DAF, AF = AF, and BA = DA. So, by SAS, triangle BAF = triangle DAF. Since ADF is a right angle, A

36、BF is also a right angle.Figure 9Next, since mEBF + mABC + mABF = 180 degrees and mABF = 90 degrees, EBF and ABC are complementary. Thus, mEBF + mABC = 90 degrees. We also know that mBAC + mABC + mACB = 180 degrees. Since mACB = 90 degrees, mBAC + mABC = 90 degrees. Therefore, mEBF + mABC = mBAC + m

37、ABC and mBAC = mEBF.By the AA similarity theorem, triangle EBF is similar to triangle CAB.Now, let k be the similarity ratio between triangles EBF and CAB. Figure 10Thus, triangle EBF has sides with lengths ka, kb, and kc. Since FB = FD, FD = kc. Also, since the opposite sides of a rectangle are con

38、gruent, b = ka + kc and c = a + kb. By solving for k, we have五分钟搞定5000字毕业论文外文翻译，你想要的工具都在这里!在科研过程中阅读翻译外文文献是一个非常重要的环节，许多领域高水平的文献都是外文文献，借鉴一些外文文献翻译的经验是非常必要的。由于特殊原因我翻译外文文献的机会比较多，慢慢地就发现了外文文献翻译过程中的三大利器：Google“翻译”频道、金山词霸（完整版本）和CNKI“翻译助手。具体操作过程如下： 1.先打开金山词霸自动取词功能，然后阅读文献； 2.遇到无法理解的长句时，可以交给Google处理，处理后的结果猛一看，不

39、堪入目，可是经过大脑的再处理后句子的意思基本就明了了； 3.如果通过Google仍然无法理解，感觉就是不同，那肯定是对其中某个“常用单词”理解有误，因为某些单词看似很简单，但是在文献中有特殊的意思，这时就可以通过CNKI的“翻译助手”来查询相关单词的意思，由于CNKI的单词意思都是来源与大量的文献，所以它的吻合率很高。 另外，在翻译过程中最好以“段落”或者“长句”作为翻译的基本单位，这样才不会造成“只见树木，不见森林”的误导。四大工具： 1、Google翻译： google，众所周知，谷歌里面的英文文献和资料还算是比较详实的。我利用它是这样的。一方面可以用它查询英文论文，当然这方面的帖子很多，

40、大家可以搜索，在此不赘述。回到我自己说的翻译上来。下面给大家举个例子来说明如何用吧比如说“电磁感应透明效应”这个词汇你不知道他怎么翻译，首先你可以在CNKI里查中文的，根据它们的关键词中英文对照来做，一般比较准确。 在此主要是说在google里怎么知道这个翻译意思。大家应该都有词典吧，按中国人的办法，把一个一个词分着查出来，敲到google里，你的这种翻译一般不太准，当然你需要验证是否准确了，这下看着吧，把你的那支离破碎的翻译在google里搜索，你能看到许多相关的文献或资料，大家都不是笨蛋，看看，也就能找到最精确的翻译了，纯西式的！我就是这么用的。 2、CNKI翻译： CNKI翻译助手，这个

41、网站不需要介绍太多，可能有些人也知道的。主要说说它的有点，你进去看看就能发现：搜索的肯定是专业词汇，而且它翻译结果下面有文章与之对应（因为它是CNKI检索提供的，它的翻译是从文献里抽出来的），很实用的一个网站。估计别的写文章的人不是傻子吧，它们的东西我们可以直接拿来用，当然省事了。网址告诉大家，有兴趣的进去看看，你们就会发现其乐无穷！还是很值得用的。 3、网路版金山词霸（不到1M）： 4、有道在线翻译：翻译时的速度：这里我谈的是电子版和打印版的翻译速度，按个人翻译速度看，打印版的快些，因为看电子版本一是费眼睛，二是如果我们用电脑，可能还经常时不时玩点游戏，或者整点别的，导致最终SPPEED变慢

42、，再之电脑上一些词典（金山词霸等）在专业翻译方面也不是特别好，所以翻译效果不佳。在此本人建议大家购买清华大学编写的好像是国防工业出版社的那本英汉科学技术词典，基本上挺好用。再加上网站如：google CNKI翻译助手，这样我们的翻译速度会提高不少。具体翻译时的一些技巧（主要是写论文和看论文方面） 大家大概都应预先清楚明白自己专业方向的国内牛人，在这里我强烈建议大家仔细看完这些头上长角的人物的中英文文章，这对你在专业方向的英文和中文互译水平提高有很大帮助。 我们大家最蹩脚的实质上是写英文论文，而非看英文论文，但话说回来我们最终提高还是要从下大工夫看英文论文开始。提到会看，我想它是有窍门的，个人总

43、结如下： 1、把不同方面的论文分夹存放，在看论文时，对论文必须做到看完后完全明白（你重视的论文）；懂得其某部分讲了什么（你需要参考的部分论文），在看明白这些论文的情况下，我们大家还得紧接着做的工作就是把论文中你觉得非常巧妙的表达写下来，或者是你论文或许能用到的表达摘记成本。这个本将是你以后的财富。你写论文时再也不会为了一些表达不符合西方表达模式而烦恼。你的论文也降低了被SCI或大牛刊物退稿的几率。不信，你可以试一试 2、把摘记的内容自己编写成检索，这个过程是我们对文章再回顾，而且是对你摘抄的经典妙笔进行梳理的重要阶段。你有了这个过程。写英文论文时，将会有一种信手拈来的感觉。许多文笔我们不需要自己再翻译了。当然前提是你梳理的非常细，而且中英文对照写的比较详细。 3、最后一点就是我们往大成修炼的阶段了，万事不是说成的，它是做出来的。写英文论文也就像我们小学时开始学写作文一样，你不练笔是肯定写不出好作品来的。所以在此我鼓励大家有时尝试着把自己的论文强迫自己写成英文的，一遍不行，可以再修改。最起码到最后你会很满意。呵呵，我想我是这么觉得的。