What if Archimedes Had Met Taylor?
What if Archimedes Had Met Taylor?
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如果阿基米德遇见泰勒怎么办？
DOI：
10.1080/0025570x.2008.11953563
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Jason Slowbe
中科院分区：
文献类型：
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作者：
Jason Slowbe
甚至在阿基米德之前，数学家就知道圆的直径与其周长成正比，圆的面积与其半径的平方成正比。然而，是阿基米德首先提供了严格的证明，证明这两个比例常数是相同的；我们现在称它们为π(见[1，p.31])。他还指出，增加内切和外接于圆的正多边形边的数量会产生与圆的周长变得任意接近的周长。阿基米德从内切和外接的正六边形开始，围绕一个单位直径的圆，然后将边数增加一倍，达到12、24、48和96，阿基米德计算了它们的周长，从而产生了一个区间的下端点和上端点，该区间的长度缩小到零，但始终包含已知为π的圆的周长。然而，这种方法不会产生π的单一数值近似值。有什么方法可以提高他的方法生成实际π数字的能力吗？答案是肯定的，泰勒会伸出援手。人们认为阿基米德还考虑了这些内切和外接多边形的面积，以证明从圆的周长和面积得出的常数是相同的[2，第281页]。增加n“填满单位圆”，从而产生一个类似的算法来生成π的小数近似值。我们将从分析基于区域的算法开始。这导致了一种更有效地近似π的S值的方法。类似的基于周长的结果进一步提高了其效率。我们将看到这两种方法之间有几个很好的联系。
Even before Archimedes, mathematicians knew that a circle’s diameter is proportional to its circumference, and that the area of a circle is proportional to the square of its radius. It was Archimedes, however, who first supplied a rigorous proof that these two proportionality constants were the same; we now call them π (see [1, p. 31]). He also showed that increasing the number of sides of regular polygons inscribed and circumscribed about a circle creates perimeters that become arbitrarily close to the circle’s perimeter. Starting with inscribed and circumscribed regular hexagons about a circle of unit diameter, then doubling the number of sides to 12, 24, 48, and 96, Archimedes calculated their perimeters and thus created lower and upper endpoints of an interval that shrinks in length to zero yet always contains the circle’s perimeter known to be π . This method, however, does not produce a single numeric approximation of π . Is there some way to improve his method’s ability to generate actual digits of π? The answer is yes, and Taylor will lend a hand. It is thought that Archimedes also considered areas of these inscribed and circumscribed polygons in his quest to prove that the constants derived from the perimeters and areas of circles were the same [2, p. 281]. Increasing n “fills the unit circle,” thereby producing a similar algorithm for generating decimal approximations of π . We will begin by analyzing the areas-based algorithm. This leads to a more efficient way to approximate π’s value. An analogous perimeters-based result further improves its efficiency. We will see several nice connections between the two methods.