Fibonacci

Фибоначи<bg>

The Fibonacci numbers form a series in mathematics, which is defined recursively as follows:

F(0) = 0 F(1) = 1

F(n) = F(n-1) + F(n-2)It begins with 0 and 1, and each member of the

sequence is obtained as the sum of the previous two. The first few Fibonacci numbers are

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Here are some of the basic properties of Fibonacci numbers:

* (F (n), F (m)) = 1

* (F (n), F (m)) = F ((m, n)) ie DEPB numbers F (n) and F (m) e Fibonacci number of index DEPB

(m, n)

* F (n + k) = F (k-1) * F (n) + F (k) * F (n +1)

* F (k) / F (kn) for arbitrary n

Relations associates are fractions of golden mean φ and in particular in Bulgaria- knowledge that are used in different constructions of such churches, bridges, castles.

Italian mathematician Leonardo Fibonacci in 1202 published a series of numbers, each of which is

obtained as the sum of the previous two, the first two numbers are 1 and 2: 1, 2, 3, 5, 8, 13, 21, It is ...

learned about this series of numbers while traveling in the countries of East and then the line was named

after him because it was popularized.

It turns out that the greater are the numbers of the sequence of Fibonacci, the attitude of the last two numbers are close to the 'golden mean' and border transition (with an infinite number of numbers in a

row) becomes equivalent to the 'golden mean'.

Often the sequence of Fibonacci is also associated with the following task: A pair of rabbits (males and females) can produce per unit time (eg one month) a new pair of rabbits, which continue to proliferate (in the classical problem of Fibonacci pair of newborn

rabbits it needs two months to give its first generation, and then continue to grow each month). How many pairs of rabbits live after a certain time if no one destroys the rabbit? The answer is given by the last number in the sequence of Fibonacci. Of

course, this task is purely illustrative.

However, it appears that too many regularities observed in nature and human behavior can be

described, albeit with a lesser or greater error with numbers in the sequence of Fibonacci, although in some cases, this explanation may seem intentional.

In fact, the algorithm to form the next number of the sequence of Fibonacci is the fact that the

investigation (the last day of the order) depends on the background (causes) in this way for this series,

namely: the latter figure is the sum of the two previous numbers.

So this algorithm is included in the category of so-called recurrent formulas. How a key of 'golden

mean' can not explain natural and human phenomena that is depends on how far these phenomena are

governed by a simple top, while conforming well to the 'common sense' recurrent dependence on the

investigation of the causes that produce it. Fibonacci algorithms to describe the main players of formulas

were geometric and arithmetic progression

Creasing Leonardo was born in 1170 in Pisa, Italy. His most notable contribution to our lives is the top

row of your keyboard. Although not as famous as his other fellow Italians or ninja turtles Leonardovtsi has much to thank him. While traveling through northern

Africa, Leo discovers that the local numbering system 0-9 is far better and superior imposed before

the millennium Roman XV, designed to confuse primary school pupils in later generations. Leonardo

brings this new system in Europe and eventually manage to invent it, Sudoku.

Orthogonal (Orthogons) design system for centuries has enabled artists and craftsmen in Bulgaria to establish harmonious work without complex calculations.

10th century birth of Christ, Bulgarians have an understanding of arithmetic, such as features and used the abacus. That After that described by Fibonacci.

Until the early 20th century in Bulgaria abacus was used to calculate

shestzvuchie ratios majeure cause fluctuations in nice snails of the inner ear -

a body which is hardly accidental form of logarithmic spiral

Fibonacci numbers are undoubtedly part of the natural harmony that is pleasant to feel, nice looks and even sounds nice.

The music is based on 8-speed octave as 1, 3, and 5th notes create the basis of all chords. Euphonious, harmonious chords are not random. The most important-sounding harmonic intervals can be obtained using the relationship of the numbers 1, 2, 3, 4. If the length of the string or the length of the flute is to halve, then their tone will be increased by one octave. If we reduce the ratio to 3:2 or 4:3, then this will correspond to musical intervals quint or quanta. When strings are three harmonic chord is obtained when the ratio of the lengths of these strings are close to 3:4:6.

On piano octave is represented by 8 white and 5 black keys - all 13. It is no accident that the musical harmony, which seems, brings the greatest pleasure is shestzvuchie majeure. Note is (I *) ratio sounds like 0625 to the note (to *)

Veronika Tosheva 11b Plamen Bekyarov 11astudents 10aSOU”Zheleznik” Stara ZagoraBulgaria2010