By Donald C. Benson

ISBN-10: 0195144368

ISBN-13: 9780195144369

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 ebook takes a unique examine the themes of faculty 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 publication assembles a set ofmathematical pebbles which are vital in addition to attractive.

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**Example text**

BCE THE SIXTH CENTURY BCE, PYTHAGORAS and his followers believed that numbers are the language in which the meaning of the universe is written. Numbers are also fundamental in today's world in which digital computers—containing streams of Os and Is—entertain us, perform our accounting needs, and compute the interaction of galaxies. Although the Pythagoreans lacked the digital computer, their belief in the universal importance of number was confirmed by devices familiar to them—musical instruments.

The waveform shown here can be generated by two simple tones played simultaneously. We need to include this sort of waveform in our discussion of consonance and dissonance. The class of periodic waveforms is too confining. We need a wider class of waveforms, the almost periodic waveforms. For our purposes, the class of almost periodic waveforms consists of those that can be approximated to arbitrary precision by sums of sine waves—with possibly incommensurable periods. Unlike a periodic waveform, we cannot construct all of an almost periodic waveform by endlessly repeating part of its graph.

We can now restate Definition 4 in a form that is known as the Axiom of Archimedes. 1 (Axiom of Archimedes). If A is a magnitude of the same kind as B, and A exceeds B (A > B), then a sufficiently large multiple of B exceeds A; that is, there exists a natural number n such that the n-fold multiple of B exceeds A (nB > A). 24 Bridging the Gap The ratio of two magnitudes, A and B, is written A : B. In modern terms, A : B is like the quotient A IB. There are two possible methods for giving meaning to ratios in the ancient context: 1.

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