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By Steven G. Krantz

An Episodic historical past of Mathematics offers a sequence of snapshots of the background of arithmetic from precedent days to the 20th century. The reason isn't to be an encyclopedic background of arithmetic, yet to provide the reader a feeling of mathematical tradition and historical past. The booklet abounds with tales, and personalities play a powerful function. The e-book will introduce readers to a few of the genesis of mathematical rules. Mathematical heritage is fascinating and worthwhile, and is an important slice of the highbrow pie. an exceptional schooling involves studying varied tools of discourse, and positively arithmetic is among the so much well-developed and demanding modes of discourse that we have got. the focal point 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 pupil with many avenues for exploration and plenty of new entrees into the topic.

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The next regular polygon in our study has 192 sides. It breaks up naturally into 192 isosceles triangles, each of which has area 2− 2+ A(T ) = 2+ √ 2+3 . 4 Thus the area of the regular 192-gon is 2− A(P ) = 192 · 2+ 2+ 4 √ 2+3 √ = 48 · 2− 2+ 2+ 2+3. 14103 . 3 Archimedes 37 This new approximation of π is accurate to nearly three decimal places. Archimedes himself considered regular polygons with nearly 500 sides. His method did not yield an approximation as accurate as ours. But, historically, it was one of the first estimations of the size of π.

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. 20, is √ √ 1 1 3 3 1 = . A(T ) = · (base) · (height) = · · 2 2 2 2 8 Therefore the area√of the full equilateral triangle, with all sides equal to 1, is twice this or 3/4. 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 = .

Thus the area is A(T ) = = 1 · (base) · (height) 2 1 · 2 2− 2+ 2+ √ 3· √ 2− √ 2+ 3 . 4 The polygon comprises 48 such triangles, so the total area of the polygon is √ 2− 2+ 3 √ A(P ) = 48 · = 12 2 − 2 + 3 . 1326 . This is obviously a better approximation to π than our last three attempts. It is accurate to one decimal place, and the second decimal place is close to being right. And now it is clear what the pattern is. The next step is to examine a regular polygon with 96 sides. The usual calculations will show that this polygon breaks up naturally into 96 isosceles triangles, and each of these triangles has area 2− 2+ A(T ) = 2+ 4 √ 3 .

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