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正方形和其它数学问题的关系,历来都是数学爱好者感兴趣的问题。著名的“几何三大难题”,其中一个就是求作一正方形,使它的面积等于一己知圆的面积.这个化圆为方的古典难题,经过二千多年来很多学者的研究、争论,于1982年,林德曼证明π是超越数后,才肯定尺规作图化圆为方是不行的。关于正方形与中学数学中某些问题的关系,是非常有趣的问题。本文就正方形与无理数2~(1/2)、数列、勾股定理,黄金分割,三等分角线、极值等有关的几个例题作一些介绍。上述的几个方面与正方形有些联系是不足为奇的。正方形的代数表达式是a~2,因此,许多涉及到平方数的问题可以联系正方形。例如1+3+5+7+…+(2n-1)=1/2n〔1+2n一1〕=n~2.故可用正方形表其结果(如图2)。正方形又是计量单位,不但a~2能与之联系,就是代数式ab亦能与正方形联系。例如商高定理的古老证法之一就是如此。如图1。正方形还是矩
The relationship between squares and other mathematical problems has historically been of interest to mathematicians. One of the famous “three major geometric problems” is to create a square so that its area is equal to the area of a known circle. This round-squared classical puzzle has been studied and controversial by many scholars for over 2,000 years. In 1982, Lindeman proved that π is a transcendental number, and it is only certain that the ruler drawing circle is square. The relationship between squares and some problems in middle school mathematics is a very interesting question. This article gives some introductions to some examples related to squares and irrational numbers 2~(1/2), series, Pythagorean theorem, golden section, equidistant angles, extremes, etc. It is not surprising that there are some connections between the above aspects and the square. The algebraic expression of squares is a~2, so many questions involving squares can be linked to squares. For example, 1+3+5+7+...+(2n−1)=1/2n[1+2n−1]=n~2. Therefore, it is possible to use a square table as its result (see FIG. 2 ). The square is a unit of measurement, not only a~2 can be associated with it, it is the algebraic ab can also be associated with the square. This is one of the ancient proofs of the Shang Gao Theorem, for example. Figure 1. Square or moment