Fibonacci numbers date back to an 800-year-old problem concerning the number of offspring born in a single year to a pair of rabbits. This book offers the solution and explores the occurrence of Fibonacci numbers in number theory, continued fractions, and geometry. A discussion of the "golden section" rectangle, in which the lengths of the sides can be expressed as a ration of two successive Fibonacci numbers, draws upon attempts by ancient and medieval thinkers to base aesthetic and philosophical principles on the beauty of these figures. Recreational readers as well as students and teachers will appreciate this light and entertaining treatment of a classic puzzle.
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Yes, you can access Fibonacci Numbers by Nikolai Nikolaevich Vorob'ev in PDF and/or ePUB format, as well as other popular books in Mathematics & Number Theory. We have over one million books available in our catalogue for you to explore.
where q1, q2, …, qn are whole positive numbers and q0 is a whole non-negative number. Thus in contrast to the numbers q1q2 …, qn, the number q0 can equal zero. We shall keep this somewhat special position of the number q0 in mind, and not mention it specially on each occasion.
The expression (24) is called a continued fraction and the numbers q0, q1, …, qn are called the partial denominators of this fraction.
Sometimes continued fractions are also known as chain fractions. They are of use in a wide assortment of mathematical problems. The reader who wants to study them in greater detail is referred to A.Ya. Khinchin, “Chain Fractions”*.
The process of transformation of a certain number into a continued fraction is called the development of this number into a continued fraction.
Let us see how we can find the partial denominators of such an expansion of the ordinary fraction
.
We consider the Euclidean algorithm, as applied to the numbers a and b.
The first of these equations gives us
But it follows from the second equation of set (25) that
so that
From the third equation of (25) we deduce
and therefore
Continuing this process to the end (induction!) we arrive, as is seen easily, at the equation
By the very sense of the Euclidean algorithm, qn > 1. (If qn were equal to unity then rn−1 would equal rn and rn−2 would have been divisible by rn−1, exactly, i.e. the whole algorithm would have terminated one step earlier.) This means that in place of qn we can consider the expression (qn –1) +
i.e. consider (qn−1) the last but one partial denominator, and 1 the last. Such a convention turns out to be convenient for what follows.
The Euclidean algorithm as applied to a given pair of natural numbers a and b is realized in a completely definite and unique way. The partial denominators of the development of
into a continuous fraction are also defined in a unique way by the system of equations describing this algorithm. Any rational fraction
, th...
Table of contents
Cover
Title Page
Copyright Page
Contents
Foreword
Introduction
I. The Simplest Properties of Fibonacci Numbers
II. Nmber-theoretic Properties of Fibonacci Numbers
