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# Fibonacci Numbers

## on May 30, 2013

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## Fibonacci NumbersPresentation Transcript

• Fibonacci NumbersA presentation by Era Kraja
• Who was Fibonacci?Also referred to as Leonard of Pisa, Fibonacci wasan Itallian number theorist. It is believed thatLeonardo Pisano Fibonacci was born in the 13thcentury, in 1170 (approximately) and that he died in1250. Fibonacci was born in Italy but obtained hiseducation in North Africa. Fibonacci is consideredto be one of the most talented mathematicians forthe Middle Ages. Few people realize that it wasFibonacci that gave us our decimal number system(Hindu-Arabic numbering system) which replacedthe Roman Numeral system. When he was studyingmathematics, he used the Hindu-Arabic (0-9)symbols instead of Roman symbols which didnthave 0s and lacked place value. In fact, when usingthe Roman Numeral system, an abacus was usuallyrequired. There is no doubt that Fibonacci saw thesuperiority of using Hindu-Arabic system over theRoman Numerals. He shows how to use our currentnumbering system in his book Liber abaci.
• IntroductionIn mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integersequence:By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of theprevious two.In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relationwith seed values0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...Fn= Fn-1 + Fn-2F0=0, F1=1
• The first 21 Fibonacci numbersFo F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F200 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 15972584 4181 6765
• The problemA certain man put a pair of rabbits in a place surrounded onall sides by a wall. How many pairs of rabbits can beproduced from that pair in a year if it is supposed thatevery month each pair begets a new pair, which from thesecond month on becomes productive?
• How he solved it:Fibonaccis experiment shows that over a period of time, a pair of rabbits willreproduce at a rate expressed in his identified sequence of numbers. One pairof rabbits will create a pair of offspring. When those rabbits mature, they willcreate another pair of offspring, and during that time, the older, original pair ofrabbits will have created an additional pair of offspring, etc.
• Uses of Fibonacci numbersPoetryPlantsAnimalsComputer ScienceFinance
• Fibonacci and PoetryIn English, we tend to think of poetry as lines of text that rhyme, that is, lines that end with similar sounds as in this childrens song:Twinkle twinkle little starHow I wonder what you are.Also we have the rhythm of the separate sounds (called syllables). Words like twinkle have two syllables: twin- and -kle whereas words such as star havejust one. Some syllables are stressed more than others so that they sound louder (such as TWIN- in twinkle), whereas others are unstressed and quieter(such as -kle in twinkle).If we let S stand for a stressed syllable and s an unstressed one, then the stress-pattern of each line of the song or poem has the rhythm SsSsSsS.In Sanskrit poetry syllables are are either long or short. All the syllables in the song above take about the same length of time to say whether they arestressed or not, so all the lines take the same amount of time to say. However cloudy sky has two words and three syllables CLOW-dee SKY, but the firstand third syllables are stressed and take a longer to say then the other syllable.Lets assume that long syllables take just twice as long to say as short ones.So we can ask the question: In Sanskrit poetry, if all lines take the same amount of time to say, what combinations of short (S) and long (L) syllables canwe have?For one time unit, we have only one short syllable to say: S = 1 wayFor two time units, we can have two short or one long syllable: SS and L = 2 waysFor three units, we can have: SSS, SL or LS = 3 waysAny guesses for lines of 4 time units? Four would seem reasonable - but wrong! Its five! (SSSS, SSL, SLS, LSS and LL)The general answer is that lines that take n time units to say can be formed in Fib(n) ways. This was noticed by Acarya Hemacandra about 1150 AD or70 years before Fibonacci published his first edition of Liber Abaci in 1202.Prof George Eckel Duckworths book“Structural patterns and proportions in Virgils Aeneid: a study in mathematical composition”argues that Virgilconsciously used Fibonacci numbers to structure his poetry and so did other Roman poets of the time.
• Fibonacci and PlantsThe leaves on this plant are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we will see howit is that this plant exhibits Fibonacci qualities.  In the case of tapered pinecones or pineapples, we see a double set of spirals – one going in a clockwise direction and one in the opposite direction.When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers.  Similarly, sunflowers have a Golden Spiral seed arrangement. This provides a biological advantage because it maximizes the number of seeds that canbe packed into a seed head.  As well, many flowers have a Fibonacci number of petals. Some, like this rose, also have Fibonacci, or Golden Spiral, petal arrangements.
• Fibonacci and AnimalsThe shell of the chambered Nautilus has Golden proportions. It is a logarithmic spiral.The eyes, fins and tail of the dolphin fall at Golden Sections along the body.Humans exhibit Fibonacci characteristics, too. The Golden Ratio is seen in the proportions in thesections of a finger. It is also worthwhile to mention that we have 8 fingers in total, 5 digits on eachhand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand.The cochlea of the inner ear forms a Golden Spiral.
• Fibonacci and Computer ScienceComputer Scientists like the Fibonacci sequence because it is a good example of somethingthat can be programmed easily using what is known as recursion. Recursion just means youdefine something using a simpler version of itself: If we write the 5th Fibonacci number (whichis 8) as fib(5), the 4th (which is 5) as fib(4) and so on then we can calculate it as:That tells a computer to calculate fib(5) by calculating fib(3) and fib(4) first, both simplerFibonacci calculations, and then add them together. fib(4) and fib(3) are worked out in thesame way using simpler calculations again. We can write this to work for any number (letscall it n) as:That just says that for any number n that is bigger than 1, work out the nth Fibonacci numberby first working out the previous two, fib(n-2) and fib(n-1), and adding them. We then justhave to say how to do the simple cases you eventually end up at, when n is either 1 or 0:Deﬁne ﬁb(n) = ﬁb(n-2) + ﬁb(n-1) if n > 1Deﬁne ﬁb(5) = ﬁb(3) + ﬁb(4)Deﬁne ﬁb(n) = ﬁb(n-2) + ﬁb(n-1) if n > 1| ﬁb(1) = 1| ﬁb(0) = 1
• Fibonacci and FinanceIn finance, Fibonacci retracements is a method of technicalanalysis for determining support and resistance levels. They arenamed after their use of the Fibonacci sequence. Fibonacciretracement is based on the idea that markets will retrace apredictable portion of a move, after which they will continue tomove in the original direction.The appearance of retracement can be ascribed to ordinary pricevolatility as described by Burton Malkiel, a Princeton economist inhis book A Random Walk Down Wall Street, who found no reliablepredictions in technical analysis methods taken as a whole. Malkielargues that asset prices typically exhibit signs of random walk andthat one cannot consistently outperform market averages.Fibonacci retracement is created by taking two extreme points ona chart and dividing the vertical distance by the key Fibonacciratios. 0.0% is considered to be the start of the retracement, while100.0% is a complete reversal to the original part of the move.Once these levels are identified, horizontal lines are drawn andused to identify possible support and resistance levels.
• Thank you forwatching!