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 subjects 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 ebook assembles a set ofmathematical pebbles which are very important in addition to attractive.

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**Extra resources for A Smoother Pebble: Mathematical Explorations**

**Example text**

The period is the horizontal distance (time) between two adjacent peaks (or troughs). The reciprocal of the period is the frequency—the number of peaks (or troughs) in unit time. Hence, the frequency of a sine curve contains the same information as the period. Frequency is measured in cycles per second—also called hertz —after Heinrich Rudolf Hertz (1857-94), the German physicist who was the first to transmit and receive radio waves. The higher the frequency, the higher the pitch. There is an analogy between mean pressure (loudness) and frequency.

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. In this case, the ratio of J to J is 4 : 5. We can interpret this ratio as the fraction 4/5. 1 asserts that the side and diagonal of a square are incommensurable. 2. The intervals X and J are commensurable with respect to the unit interval U. In fact, four nonoverlapping copies of U cover J, and five of them cover J, that is, J= 4U and J= 5U.

The Greeks did not have a concept of zero, negative, or infinite magnitudes. Book V of Euclid's Elements,5 attributed to the Greek mathematician Eudoxus (4007-347? BCE), contains the following passage concerning magnitude and ratio. Greek Gifts 23 Definition 3. A ratio is a sort of relation in respect of size between two magnitudes of the same kind. Definition 4. Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. The above definitions set the stage, but they do not enable us to understand completely what is meant by ratio—beyond the idea that a ratio is something that depends on two magnitudes.