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 collage Press Страниц: 280 Размер: 11,1 ISBN: 0195144368 Язык: Английский0 (голосов: zero) Оценка:This publication 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 booklet assembles a set ofmathematical pebbles which are vital in addition to attractive.
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Additional info for A Smoother Pebble: Mathematical Explorations
Webb showed that there exists a pair of differently shaped drums that generate the same sequence of partials. In addition to the drum, most percussion instruments produce nonharmonic partials: the xylophone, the bells, and the cymbal. Of course, nonharmonic does not mean dissonant or unpleasant. 3) is an example of a periodic waveform. It is called periodic because it is repetitive over time. It is sufficient to depict the graph over a time interval equal to one period. Shifting the initial period horizontally, we can obtain as much of the curve as needed.
In particular, halving the string length raises the pitch by exactly one octave, and if the ratio of the string lengths is 2 : 3, then the corresponding pitches form an interval called a Pythagorean or perfect fifth — so named because the interval do-sol between the first and fifth notes of the diatonic2 scale is an instance of this interval. Other important Pythagorean intervals correspond to the ratios 3 : 4 (the fourth), 8 : 9 (the major second}, and 9:16 (the minor seventh). The role of numerical ratios in music might seem mildly interesting to us, but to the Pythagoreans it was astonishing—even staggering—and emboldened them to speculate that the same numerical ratios controlled the motion of the heavenly bodies.
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. 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.
A Smoother Pebble: Mathematical Explorations by Donald C. Benson