A Theorem on the Golden Section and Fibonacci Numbers

Rolando Zucchini, NEMS

• Publisher: NEMS - Nuova Editrice Medica e Scientifica
• Year: 2021
• eBook ISBN: 9788869492983
• Topic: Mathematics
• Subtopic: Mathematics Essays
• Index: Mathematics

The golden section of a segment is the part of the segment mean proportional between the whole segment and the remaining part.Fibonacci's sequenceAlmost all scholars say that Fibonacci has invented his very famous sequence by observing the reproduction of rabbits or other phenomena occurring in nature.In this text, Rolando Zucchini affirm instead that he discovered it by studying the golden section (golden section), and in particular, as shown, by the theorem that generates it.FibonacciLeonardo Pisano, known as Fibonacci (Pisa, b. 1170-1240 (?)), introduced in Europe the zero and the Hindu-Arabic numeral system and so he started the development of arithmetic as we know it today, when, in 1202, he published his most famous book Liber Abaci. In the incipit of this book he writes: "The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0, that the Arabs call Zefiro, any number may be written, as shown below"Italian mathematicianRolando Zucchini taught mathematics in high schools with innovative teaching methods, linking it to its history and philosophy.

Appendix 1

Zero, as a symbol needed for writing numbers, dates back to the ancient Babylonians. Then it was abandoned, and only in 660 (?) it was recovered by the Indians to make a numbering system with nine digits and the sign of zephyr. This numbering is attributed to the great Indian mathematician Brahmagupta (598-668 (?)), author of the book Brahmasphuta Siddhànta, which offers the first example of systematic arithmetic including negative numbers and zero. In this text, however, the zero is still considered as a symbol by which you can easily write any digit. Only later did the Indian mathematics equated zero to the nine other digits and considered it a real number. The many threads about this conceptual difference should not be considered as purely academic. Suffice it to say that the ten (number ten) lost its uniqueness of being a number composed of two digits, 1 and 0, and, by the finger-counting, starting from zero, you get up to nine. It was not easy to introduce into the common mindset these substantial changes, because, as noted, the decimal numbering system was the favorite because in one to one correspondence with the ten fingers.

 

Appendix 2

Musa al-Khwarizmi (Carasmia or Baghdad; 780-850 (?)) was a Persian mathematician, astronomer and geographer. Head of the famous library Bayt al-Hikma (House of Wisdom) in Baghdad, under his direction works on mathematics...

Table of contents

• A theorem on the Golden Section and Fibonacci Numbers
• Appendix 1
• Appendix 2
• Appendix 3
• Appendix 4
• Bibliography