数学与应用数学英文文献及翻译

.数学与应用数学英文文献及翻译-勾股定理（外文翻译从原文第一段开始翻译，翻译了约2000字） 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 one 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 mathematics 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 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 root 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 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 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 demonstrated 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 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 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 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 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 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 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 this 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 Pythagorean 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 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, 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 the 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 square 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 using 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. 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 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 area 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 seen 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. This 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 Pythagorean 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 comparing the similarities of triangles 2 and 3 we get: From Figure 4, AB = CD. By substitution,Cross-multiplication gives:Finally, by adding equations 1