By Donald C. Benson

Книга A Smoother Pebble: Mathematical Explorations A Smoother Pebble: Mathematical Explorations Книги Математика Автор: Donald C. Benson Год издания: 2003 Формат: pdf Издат.:Oxford college Press Страниц: 280 Размер: 11,1 ISBN: 0195144368 Язык: Английский0 (голосов: zero) Оценка:This booklet takes a unique examine the themes of college mathematics--arithmetic, geometry, algebra, and calculus. during this walk at the mathematical beach we are hoping to discover, quoting Newton, "...a smoother pebble or a prettier shell than ordinary..." This ebook assembles a set ofmathematical pebbles which are vital in addition to attractive.

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**Example text**

The sine curve is a graph of the time dependence of the vertical height of a point fixed to the rotating circle — a point that is initially at PQ. 2, respectively. The dashed lines show how these points on the circle determine points on the sine curve. The amplitude is equal to the maximum height of the sine curve —the radius of the circle. The period is the distance between successive peaks of the sine curve —the time for one revolution of the circle. The frequency (in Hz) is the angular velocity of the moving point measured in revolutions per second.

2) is divisible by 4. Since C is even and there is no common factor between C and A, A must be odd. The square of an odd number is odd. 2) is divisible by 2 but not by 4. We are finished. 2) is divisible by 4 and the left side is not. 1) holds must be false. 1 is equivalent to the assertion that \/2 is an irrational number. In geometry, two line segments J and J are called commensurable—that is, they have a "common measure"—if there exists a unit segment U. such that both X and J can be covered by an integral number of nonoverlapping copies of U.

We can see this in the history of the concept of number. In the preceding chapter, we have seen the practical, unsophisticated number concepts of the Egyptians and Babylonians (thesis). In this chapter, we will see how the Greeks introduced new concepts (antithesis), and foreshadowed the modern concept of real number (synthesis). The Heresy It is said that the Pythagoreans punished those who divulged their secrets. This may be a calumny promulgated by outsiders suspicious of this secret brotherhood.