A Smoother Pebble: Mathematical Explorations by Donald C. Benson

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 e-book 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 booklet assembles a suite ofmathematical pebbles which are very important in addition to attractive.

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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. 2, the intervals T and J are covered by, respectively, four and five nonoverlapping copies of the unit interval U.

7, where the ratio of the periods is 1 : n. This figure does not exhibit any obvious periodicity, but, of course, this does not prove that the waveform is not periodic. 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.

Eudoxus's theory of proportion deals with both numerical and geometric magnitudes. A proportion is a relation of equality between two ratios. The traditional notation for a proportion between two ratios is A : B :: 71: S, but this formula has the same meaning as A : B = 11: S. Magnitudes A and S are called the extremes, and B and 7£ are called the means of the proportion A : B :: 'R,: S. Eudoxus's theory deals with magnitudes in general, but to make the following discussion less abstract, magnitudes are interpreted as line segments.

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