Definition of Fibonacci numbers or the Fibonacci sequence - a sequence pipette of numbers, which ha

Leonardo Fibonacci - one of the greatest mathematicians of the Middle Ages. In one of his works, and "The Book of computing" Fibonacci described Hindu-Arabic number pipette system and the benefits of its use before Roman.
Definition of Fibonacci numbers or the Fibonacci sequence - a sequence pipette of numbers, which has a number of properties. For example, pipette the programming of two adjacent numbers of the sequence yields the value next to them (e.g., 1 +1 = 2 2 +3 = 5 etc.) that confirms the existence of so-called coefficients Fibonacci i.e. constant relations. Fibonacci sequence pipette begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ... Properties of the Fibonacci sequence: 1. The ratio of each number to the next more and more committed to 0.618 by increasing atomic number. The attitude among each of the previous pipette approaches 1.618 (inverse of 0.618). pipette The number 0,618 is a (FI). 2. When you divide each number by following it through one obtained by 0.382, on the contrary - 2.618 respectively. 3. By choosing such relations, we obtain a core set of factors fibonachchievskih ... 4.235, 2.618, 1.618, 0.618, 0.382, 0.236. Contact the Fibonacci sequence and the "golden section" Consistency Fibonachchm pipette asymptotically (ppiblizhayas slower and slower) to earn, every day SOME constant ratio. However, this ratio ippatsionalno, is right-that is, an infinite number of, unpredictable sequence of decimal pipette digit in dpobnoy part. It can not be expressed exactly. If a member of the Fibonacci sequence are separated by at ppedshestvuyuschy to him (eg, 13:8), inferences will be the value that fluctuates around ippatsionalnogo value 1.61803398875 ... and then chepez paz ppevoskhodyaschaya does not achieve it. Ho even zatpativ this eternity, it is impossible to know exactly sotnoshenie to the last decimal digit. Kpatkosti padi, we are quoted him as 1.618. Specific names of this relationship began to give more before pipette Luca Pacioli (spednevekovy mathematician) called it the divine proportion, pipette to fill. MEDIA, it is of Contemporary names such as the golden ratio, golden s.pedney and otnoshenie veptyaschihsya kvadpatov. Keplep called this relationship a "sokpovisch geometpii." In algebpe obscheppinyato its designation gpecheskoy letter phi F = 1.618 Consider the example of the golden section of the segment. Consider a segment with endpoints A and B. Let the point C divides the segment AB so, AC / CB = CB / AB or AB / CB = CB / AC. You can imagine it like this: A ----- C -------- B Golden section - it is proportional to the length of the division into unequal parts, in which the entire segment as applies to most, as she refers to most of the less, or in other words, a smaller segment of attitude toward greater as more of that. Segments of the golden ratio expressed an infinite irrational fraction 0.618 ... if AB is taken as unity, AC = 0,382 .. Kak we already know the number of 0.618 and 0.382 are the coefficients of the Fibonacci sequence. Fibonacci ratio and the golden section in nature and history is important to note that as the Fibonacci sequence would remind his humanity. She was known to the ancient Greeks and Egyptians. Indeed, since in nature, pipette architecture, fine arts, mathematics, physics, astronomy, biology, and many other areas were found patterns described by Fibonacci ratios. It's amazing how much you can constantly calculate DURING using the Fibonacci sequence, and how its members are manifested in a huge number of combinations. However, it is no exaggeration to say that this is not just a numbers game, and the most important mathematical expression of all natural phenomena ever discovered. THE FOLLOWING examples show some interesting applications of this mathematical sequence. 1. Vanity is wound in a spiral. If you expand it, then we get a long, slightly inferior to the length of the snake. Desyatisantimetrovymi small spiral shell has a length of 35 cm form a spiral curled shells attracted the attention of Archimedes. pipette The fact that the ratio of the measurement curls sink is constant and equal 1.618. Archimedes studied spiral shells and derived an equation of the spiral. Cpiral, drawn by this equation is called by his name. The increase its pitch is always evenly. Currently Archimedes spiral is widely used in the art. 2. Plants and animals. Yet Goethe emphasized the tendency of nature to the helicity. Helical and spiral arrangement of leaves pipette on the trees have noticed for a long time. Cpiral seen in the arrangement of sunflower seeds in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians shed light on these amazing natural phenomenon. It turned out that in the arrangement of leaves on a branch of sunflower seeds, pine cones proves the Fibonacci series and, therefore, proves the law of the golden section. The spider weaves a web spiral. Cpiralyu spins