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 singular examine the subjects 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 e-book assembles a set ofmathematical pebbles which are vital in addition to attractive.
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Extra resources for A Smoother Pebble: Mathematical Explorations
The method of Eudoxus. 4 As we will see in the next section, this can be done even if we have not defined what a ratio A : B actually is. Ratios can be left undefined just as lines and points in geometry are undefined. 2. Anthyphairesis. Define a ratio A : B in terms of the natural numbers even when A and B are not numerical magnitudes. Further below, we will see how this can be done. There is no inconsistency between method 1 and method 2. Book V of Euclid's Elements develops method 1. The role of method 2 can be ascertained only in part by a literal reading of Euclid.
The Pythagoreans did not have the technology to discover that the pitch of a sound corresponds to the frequency of a vibration. 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.
Beats. 2 sec. ) When the simple tones are sounded simultaneously, the pressures are alternately in and out of phase, producing beats (c), which pulsate at a rate equal the frequency difference of the two tones, in this case 20 Hz. 48 Bridging the Gap out-of-tune versions of these intervals. It is surprising that changing the test tones from simple to complex should make such a radical difference. This striking change occurs because a single tone with many partials is equivalent to an entire chorus of simple tones.
A Smoother Pebble: Mathematical Explorations by Donald C. Benson