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 ebook takes a singular 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 suite ofmathematical pebbles which are vital in addition to appealing.

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**Example text**

Definition 4 implies that each multiple of a magnitude—double, triple, or any integral multiple—is also a magnitude of the same kind. In other words, if A is a geometric magnitude and n is a natural number, it makes sense to speak of the magnitude nA, the n-fold multiple of A. For example, 2A is the magnitude of a figure consisting of two copies of a figure of magnitude A. 3. The subtraction of the smaller from the larger of two like magnitudes. If the geometric magnitude A is larger than B, then A — B is the magnitude of a geometric object of magnitude A from which an object of magnitude B has been removed.

The bulging of the basilar The Music of the Ratios 45 membrane activates nerves contained in the organ of Corti, sending information to the hearing center of the brain. The foregoing explanation is called the place theory of pitch perception because it explains how different simple tones are associated with different sites on the basilar membrane. Despite some difficulties, the place theory is considered the primary explanation of pitch perception. Although the ear is a wonderfully sophisticated instrument, it shares a certain limitation with all frequency analyzers—a limit of resolution.

Nevertheless, the Pythagorean ratios apply to frequency as well as the length of a string. For example, doubling the length of a string halves the frequency of vibration. However, the sounds heard from a vibrating string are not simple tones, but rather a mixture of simultaneous simple tones. This happens because a string has not just one but a sequence of modes of vibration. Sometimes the higher modes are used for musical effect—for example, when a violinist places a finger lightly on a node of the vibrating part of a string.