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 publication assembles a set ofmathematical pebbles which are vital in addition to attractive.

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**Example text**

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. We can now restate Definition 4 in a form that is known as the Axiom of Archimedes. 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).

This is because the remainders (e. , 65, 13 in the above example) become smaller at each step; a decreasing sequence of natural numbers can contain only finitely many elements. On the other hand, a BAFS of geometric magnitudes can be an infinite process —as we will see in the next section. 6) is called a continued fraction. More specifically, since all the numerators are equal to 1, it is an example of a simple continued fraction. 11 There is no direct statement in Euclid's Elements that the ancient Greeks also used BAFS to defineethe concept of ratio; however, some scholars12 believe that there is ample indirect evidence to support their claim that Greek mathematicians of the fourth century BCE, Theaetetus and others, used BAFS to define ratio.

In the preceding chapter, we have seen the practical, unsophisticated number concepts of the Egyptians and Babylonians (thesis). In this chapter, we will see how the Greeks introduced new concepts (antithesis), and foreshadowed the modern concept of real number (synthesis). The Heresy It is said that the Pythagoreans punished those who divulged their secrets. This may be a calumny promulgated by outsiders suspicious of this secret brotherhood. Truth or legend, it is said that Hippasus of Metapontum (400?