By Andreev P. D.

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In the second half of the 20th century multiple interesting mathematical discoveries in the area of golden mean applications in computer science and mathematics had been made [86 119]. In 1956, the young American mathe matician George Bergman made an important mathematical discovery in the field of number systems [86]. We are talking about the number system with irrational base (the golden mean) described in [86]. Modern mathematicians had been so anxious of overcoming the crisis in the basis of mathematics that they simply had not noticed Bergman’s discovery, which is, without doubt, one of the greatest mathematical discoveries in the field of number systems after the discovery by Babylonians of the positional principle of number repre Alexey Stakhov MATHEMATICS OF HARMONY xxxii sentation.

I would like to express to him great gratitude for scientific col laboration. “ Lastly, this book would never have been written without self denying sup port of my wife Antonina, who always creates perfect conditions for my sci entific work in any countries I have been; and who has been sailing with me for more than 47 years on my “Golden” journey. In addition, I would like to express my special appreciation to my daughter Anna Sluchenkova for her critical remarks, and her invaluable help in the English translation and edit ing of the book, and, especially, for her work in preparing illustrations, and coordination and final preparation of camera ready manuscript.

The Algebraic Equation (in the standard form) is a statement written with algebraic designations that some polynomial function is equal to zero for some values of a variable x. For example, ( ) Alexey Stakhov THE MATHEMATICS OF HARMONY 12 x 2 − 5 x + 6 = 0 is an algebraic equation. The values of the variable, for which the polynomial becomes equal to 0, are named roots of this polynomial. For example, the polynomial x 2 − 5 x + 6 has two roots, 2 and 3, because 22 − 5 × 2 + 6 = 0 and 32 − 5 × 3 + 6 = 0.