Download PDF by Steven G. Krantz: An Episodic History of Mathematics: Mathematical Culture

By Steven G. Krantz

An Episodic heritage of Mathematics offers a sequence of snapshots of the background of arithmetic from precedent days to the 20th century. The rationale isn't really to be an encyclopedic background of arithmetic, yet to provide the reader a feeling of mathematical tradition and heritage. The publication abounds with tales, and personalities play a powerful position. The booklet will introduce readers to a few of the genesis of mathematical rules. Mathematical heritage is intriguing and lucrative, and is an important slice of the highbrow pie. a great schooling includes studying assorted tools of discourse, and positively arithmetic is among the so much well-developed and demanding modes of discourse that we have got. the point of interest during this textual content is on getting concerned with arithmetic and fixing difficulties. each bankruptcy ends with an in depth challenge set that might give you the scholar with many avenues for exploration and lots of new entrees into the topic.

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Since the sum of the angles in a triangle is 180◦ , and since each of these triangles certainly has two equal sides and hence two equal angles, we may now conclude that all the angles in each triangle have measure 60◦ . 19. But now we may use the Pythagorean theorem to analyze one of the triangles. 20. Thus the triangle is the union of two right triangles. 3 Archimedes 27 hexagon—is 1 and the base is 1/2. Thus the Pythagorean theorem tells √ us that the height of the right triangle is 12 − (1/2)2 = 3/2.

To do this one must reach the 1/4 point, to do this one must reach the 1/8 point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum 1 1 1 + + + ··· = 1 2 4 8 On the one hand Zeno can argue that the sum 1/2 + 1/4 + 1/8 + . . never actually reaches 1, but more perplexing to the human mind is the attempts to sum 1/2 + 1/4 + 1/8 + . . backwards. Before traversing a unit distance we must get to the middle, but before getting to the middle we must get 1/4 of the way, but before we get 1/4 of the way we must reach 1/8 of the way etc.

Now of course the full regular hexagon is made up of six of these equilateral triangles, so the area inside the hexagon is √ √ 3 3 3 = . 21. Thus the area inside the circle is very roughly the area inside the hexagon. Of course we know from other considerations that the area inside this circle is π · r2 = π · 12 = π. 598 . . 14159265 . .. So our approximation is quite crude. The way to improve the approximation is to increase the number of sides in the approximating polygon. In fact what we shall do is double the number of sides to 12.

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An Episodic History of Mathematics: Mathematical Culture through Problem Solving by Steven G. Krantz

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