Free Essay

Fibonacci Numbers

In: Other Topics

Submitted By bwood2511
Words 278
Pages 2
Fibonacci Numbers In the 13th century a man named Leonardo of Pisa or Fibonacci founded Fibonacci Numbers. Fibonacci Numbers are “a series of numbers in which each number is the sum of the two preceding numbers” (Burger 57). His book “Liber Abaci” written in 1202 introduced this sequence to Western European mathematics, although they had been described earlier in Indian mathematics. He proved that through spiral counts there is a sequence of numbers with a definite pattern. The simplest series is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…and so on. When looking at this series the pattern proves that adding the previous number to the next will give you the following number in the series. For example, (1+1=2), (2+3=5), and etc. In order to ensure accuracy when using Fibonacci Numbers a formula was created. The formula or rule that follows the Fibonacci sequence is Fn = Fn-1 + Fn-2. By plugging in any numbers in a problem to this equation a student can find the right answer. This gives students the ability to calculate any Fibonacci Number. In modern times society uses these numbers to calculate numerous things. For instance, like the sizes of our arms relative to our torso and even the structure of hurricanes. On another note, Fibonacci Numbers can also be found in patterns in nature. It is truly astonishing to think about how relations in Fibonacci Numbers may possibly be represented in our lives. Works Cited
Burger, Edward B., and Michael P. Starbird. The Heart of Mathematics: An Invitation to Effective Thinking. Emeryville, CA: Key College Pub. in Cooperation with Springer, 2000. Print.

Similar Documents

Free Essay

Technical Process

...Vineet Iyer Process Decryption Fibonacci. 1. The Mathematics Mathematicians, scientists and naturalists have known this ratio for years. It's derived from something known as the Fibonacci sequence, named after its Italian founder, Leonardo Fibonacci (whose birth is assumed to be around 1175 AD and death around 1250 AD). Each term in this sequence is simply the sum of the two preceding terms (1, 1, 2, 3, 5, 8, 13, etc.). But this sequence is not all that important; rather, it is the ratio of the adjacent terms that possess great qualities, roughly 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, PHI and the divine proportion, among others. Almost everything that has dimensional properties adheres to the ratio of 1.618. 2. Prove It! Take honeybees, for example. If you divide the female bees by the male bees in any given hive, you will get 1.618. Sunflowers, which have opposing spirals of seeds, have a 1.618 ratio between the diameters of each rotation. This same ratio can be seen in relationships between different things throughout nature. Still don't believe it? Try measuring from your shoulder to your fingertips, and then divide this number by the length from your elbow to your fingertips. Or try measuring from your head to your feet, and divide that by the length from your belly button to your feet. The golden ratio is seemingly unavoidable. The markets have the very same mathematical base as these......

Words: 702 - Pages: 3

Free Essay

The Da Vinci Dode

...our class, sacred geometry, was the shape of the museum that the dead body was found. The building that the body was found in was the Louvre museum. The Louvre museum is located in Paris, France and was established in 1793. In front of the actual museum there is something that is known as the Louvre pyramid. The Louvre pyramid is a large glass and metal pyramid surrounded by three smaller pyramids. The large pyramid serves as the entrance to the Louvre museum and was opened in 1989. The reason that I can connect this to our course in sacred geometry is the actually structure of the pyramid. There are many small triangles on it, which are also known as the triad. The triad is a three-sided shape and also is known as the first and oldest number. Also this is a pyramid and has many pyramids throughout it. The pyramid is something that we talked about in class to as one of the Zimmer 2 platonic solid. But the pyramid is known as the tetrahedron when we talked about the platonic solids. Another aspect sacred geography that is found in this pyramid is also known as the Sierpinski triangle. The sierpinski triangle is when there is triangle inside of triangles, and then inside those triangles there are more triangles. So this was the first thing that I saw...

Words: 950 - Pages: 4

Free Essay

The Parthenon Peristasis

...| The Parthenon Peristasis | | Ates Gulcugil Abstract Two golden ratio models will be constructed for the peristasis of the Parthenon and their dimensions compared with the actual one. Definitions Φ: The golden ratio, 1.618... Golden Numbers: Integer powers of Φ. Interval: Distance between two, neighboring, parallel line segments. Golden Interval: An interval which is a golden number. Normalizing: Dividing each dimension of a structure by a selected one of its dimensions. Aspect Ratio: Ratio of longer-to-shorter side of a rectangle. The Parthenon Peristasis The dimensions (in feet) of the Parthenon peristasis as measured by Francis Penrose are shown in the following diagram. There are two different kinds of intervals in the peristasis: The corner spaces (total 8), and the intercolumnia (total 38). These are shown below. Substituting the (scaled up by 101.361/101.341) intervals of the left hand side for the unmeasured right hand side, the mean values are figured out as: Mean corner space: 15.448 ft Mean intercolumnium: 14.090 ft The peristasis with the mean values is shown below. Golden Ratio Relations If the Parthenon was designed around the golden ratio there must be a golden ratio relationship between the corner space and the intercolumnium. The following diagram shows the analysis of these two intervals. The intercolumnium, because it is uninterrupted, will be considered as a golden interval, (interval a). When normalized with respect...

Words: 1577 - Pages: 7

Free Essay


...the flowers, the number of petals is one of the numbers that occur in the strange sequence 1, 2,3,5,8, 13, 21, 34, 55, 89, 144. They form the beginning of the so-called Fibonacci series, in which each number is the sum of the two that precedes. In order to verify this, we collected 30 different samples of flowers from Somaiya Vidyavihar Campus and counted the number of petals present in each flower. These data were then added to excel from which we observed that 60% of the 30 samples of flowers considered had petal counts in the above Fibonacci sequence. No of flowers | Fibonacci | Non Fibonacci | 30 | 18 | 12 | Proportion | .6 | .4 | Z value | 1.09 | | P value | .1366>.05 | Null acceptance | As we can see the above table, the p value is greater than .05, (level of significance) the alternate hypothesis that the number of petals in flowers follows Fibonacci series is not conclusively proved. However even the proportion of 0.65 would have proved the Fibonacci hypothesis. We feel that sample size of 30 is insufficient. GOLDEN RATIO What is it exactly? Think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. When you have written down about 20 numbers, calculate the ratio of the last to the second from last. The answer should be close to 1.6180339887. What makes the golden ratio special is the number of mathematical properties it possesses. The golden ratio is the only number whose square can...

Words: 593 - Pages: 3

Free Essay

The Golden Ratio

...(Isolate zero by moving one to the other side of the equation) b=-b±b2-4ac2a Where ab2+bb+c=0 b=-1±(1)2-41(-1)2(10 b=-1+52 b=.618034 1+.6180341= 1.618034 a=1.618034. The Golden Ratio has been used throughout history. Discovered in 1200 AD, this equation can be found in art, architecture, nature, geometry, and the human body. “The Golden Ratio, also known as the divine proportion, Golden Mean, or golden section, is a number produced when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon, and dodecahedron. The Golden Ratio also happens to be denoted using the symbol, or sometimes using the symbol ” (Golden Mean/Ratio, 2013, p. 1). “Leonardo Fibonacci discovered a simple mathematical sequence that is the foundation for the mathematical relationship of Phi” (Golden Mean/Ratio, 2013 p. 1). “Starting with zero and one, each new number in the sequence is the sum of the two before it. For example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610. The ratio of each consecutive pair of numbers in the sequence approximates phi (1.618. . .) as 5 divided by 3 is 1.66- and 8 divided...

Words: 1190 - Pages: 5

Free Essay


...Chapter 3 Complete coding questions 14, 15, and 16 from chapter 3. Each question should be commented and created as a self-named function (fourteen, fifteen, and sixteen) in a file called 14. Write a program that finds the average of a series of numbers entered by the user. As in the previous problem, the program will first ask the user how many numbers there are. Note: the average should always be a float, even if the user inputs are all ints. 15. Write a program that approximates the value of π by summing the terms of this series: 4/1 − 4/3 + 4/5 − 4/7 + 4/9 − 4/11 + . . . The program should prompt the user for n, the number of terms to sum, and then output the sum of the first n terms of this series. Have your program subtract the approximation from the value of math.pi to see how accurate it is. 16. A Fibonacci sequence is a sequence of numbers where each successive number is the sum of the previous two. The classic Fibonacci sequence begins: 1, 1, 2, 3, 5, 8, 13,. . .. Write a program that computes the nth Fibonacci number where n is a value input by the user. For example, if n = 6, then the result is 8. Chapter 4 Please complete the coding questions 4, 10, and 11 at the end of the chapter. Each question is implemented as its own function using its numbered name (ex question 4 to be called four()) and all three functions to be included in one file called 4. Write a program that draws a winter scene with a Christmas tree and a......

Words: 1038 - Pages: 5

Free Essay


...Name: Nilofer Neubert Student ID: E1271319 NRIC NO: S8790069Z Course Title: ELT373: Film and Gender Assignment Number: Tutor- Marked Assignment 01 The ten-minute film sequence that this essay will be focusing on is between 23:00 to 28:55 of Double Indemnity. The mise-en-scene of this film sequence communicates many key masculine desires of Neff such as having strong will to remain in strict of control of situations that lead him to achieve his desires for money as well as to get Phyllis Dietrichson. A close look at the mise-en-scene of his sequence also reveals his deep and intense feelings of love towards Phyllis Dietrichson. This intense love for her makes him want to control her. Therefore, this essay will be focusing on how the setting, the editing processes used in the post-production stage of the film sequence, the costumes, lighting, and staging of the selected sequence communicates the above mentioned masculine desires of Neff, the protagonist and anti-hero of Double Indemnity. When looking at the setting and the editing processes used in the post-production of this film sequence, it is important to take note of the location of this film sequence, the close up shots that take place in this film sequence and “the order in which scenes are arranged as a chain of events occurring within a given duration” (German SU1-3 & 4). “One of the interpretative strategies that we can employ when analysing Double Indemnity is......

Words: 1300 - Pages: 6

Premium Essay

Fibonacci's Sequence

...The Fibonacci Sequence is one of the most famous, if not the most famous numerical sequence in history. "In the year 1202, mathematician Leonardo of Pisa (also known as Fibonacci) posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on? The total number of pairs, month by month, forms the sequence 1,1,2,3,5,8,13,21,34,55,89, and so on. Each new term is the sum of the previous two terms." (1) As mentioned previously, "the solution of this problem leads to a sequence of numbers known as the Fibonacci sequence. Here are the first fifteen terms of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 (equivalent to 1.618). After the first two terms, in the sequence, each term is obtained by adding the two previous terms. For example, the third term is obtained by adding 1 + 1 to get 2, the fourth term is obtained by adding 1 + 2 to get 3, and so on." (2) Leonardo of Pisa, born in 1170, grew up and traveled throughout Africa and the Mediterranean during much of his childhood and early adult life, learning different mathematical styles and formulas, which were of great interest to him. During these travels, Fibonacci (nicknamed many years after his death; 'Bonacci' means "son of good fortune". While living, Fibonacci went by Leonardo of Pisa, his hometown in Italy, or Leonardo Pisano) Fibonacci......

Words: 972 - Pages: 4

Free Essay

Qso510 Module Five Paper

...Mathematical symbols are all around us yet most people to not seem to realize this usage of what many have dreaded as they made their way through various levels of the education system. Some uses of symbols are right in plain sight every time we drive our vehicles to various locations. For example, think about what greets you at your steering wheel as you enter the highway and decide to engage your cruise control. There right in front of your face is the “+” and “-“ symbols that we all equate with increase or decrease. Two of the most basic of mathematical symbols from algebra and they stare at us virtually every day. Another area that struck me as I was recently searching for a chemical for my work was how often mathematical symbols make their way into the logos for companies. I first noticed it while looking for some chemicals and came across Sigma-Aldrich which uses the uppercase Greek letter “” prominently as part of their logo. Of course as anyone with even a minor exposure to calculus and statistics knows this symbol is used extensively to indicate a summation of values. Another work related logo that just about poked me in the eye is a SPC software package I use on a regular basis from Infinity QS. This company uses the familiar symbol “∞” as part of its’ logo on Proficient QS software. I decided to do some further research into companies that incorporate mathematical symbols in their corporate logos. What I quickly found was the spectrum of......

Words: 755 - Pages: 4

Premium Essay

Exp 105 Personal Strategy Card

...EXP 105: Week 4 Personal Strategy Card Name: A. LCI Scores Sequence Precision Technical Reasoning Confluence Record your LCI scores in the boxes provided. 29 33 27 24 B. Carefully describe the degree to which you use each of your Learning Patterns. (Refer to the Personal Learning Profile you developed for your Week Two assignment and any feedback provided by your instructor to determine if you need to refine your responses as you complete this section.) Sequence: My whole life has been built on structure and going step by step. My whole family being Army veterans as enabled me to have a structure about things. Precision: Again, the Army has helped me with precision as well. Gaining the biggest understanding with every decision I make so I can complete a task correctly. Technical Reasoning: I use this as needed. I sit and analyze every situation and their possible outcomes. Confluence: I like to figure things out mostly by brainstorming. Thinking outside the box and different ways that I may not have used first before. C. Identify all verbs and specific terms from the assignment instructions and describe how each Learning Pattern will be used to effectively complete the Week 5 assignment. (Critically review the Final Reflection assignment in Week Five and decode it.) Sequence: Thinking of what needs to be completed. Brainstorming and recording all info that is thought of. Organize them from start to finish. Read over it and make......

Words: 593 - Pages: 3

Premium Essay

Fibonacci Sequence- Introduction

...FIBONACCI SEQUENCE Shafira Chairunnisa | 11 Blue INTRODUCTION Before we get into the details of Fibonacci sequence, let us go back to the basic definition of a sequence in Mathematics. According to The Free Dictionary by Farlex, a sequence is an ordered set of mathematical quantities called terms. There are two types of sequence: arithmetic and geometric sequence. An arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms. 1, 3, 5, 7, 9,…. The numbers in the sequence are called terms. Thus, 1 is the first term, 3 is the second term, 5 is the third term, and so forth. The symbol Un denotes the first term of a sequence. Since the first term is 1, we can express it as Un= 1. Each consecutive terms has a difference (denoted as d). Meanwhile, a geometric sequence is a sequence in which the same number is multiplied or divided by each term to get the next term in the sequence. 3, 9, 27, 81, 243,…. This is an example of geometric sequence. The quotient of a term with its previous term is called ratio. In geometric sequence, the ratio between each successive terms is constant. What is a Fibonacci sequence? Fibonacci sequence is a special sequence. It is founded by an Italian mathematician named Leonardo Pisano Bigollo, known commonly as Fibonacci. According to, it is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number......

Words: 1504 - Pages: 7

Premium Essay

Fibonacci Sequence

...How many pairs will there be in one year? The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers: the number of existing pairs is the sum of the two previous numbers of pairs in the sequence. The Fibonacci numbers or, sequences as you may call it was named after an Italian mathematician Fibonacci and was first introduced to Western European society in his 1202 book Liber Abaci. In mathematics, the Fibonacci sequence are the pattern of sequences (as you can see in these two sequences) that always start with numbers 1 and 1, or 0 and 1. 1,1,2,3,5,8,13,21,34,55,89,144 or 0,1,1,2,3,5,8,13,21,34,55,89,144 Although many people are not aware of it, Fibonacci numbers appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. And here are some of the famous examples. 1. Flower petal The number of petals in a flower consistently follows the Fibonacci sequence. The lily, which has three petals, buttercups, which have five, the chicory's 21, the daisy's 34, and so on. 2. Pinecones Pinecones are another great example. The seed pods on a pinecone are arranged in a spiral pattern and each cone consists of a pair of spirals, each one spiraling upwards in opposing directions. 3. Tree Branches The Fibonacci sequence can also be seen in the way tree branches form or split. A main trunk of a tree grows until it......

Words: 367 - Pages: 2

Free Essay

Maths in Nature

...Maths in nature "The laws of nature are but the mathematical thoughts of God" - Euclid Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted  in nature. SYMMETRY Symmetry is everywhere you look in nature .  Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry.  There are two kinds of symmetry.  One is bilateral symmetry in which an object has two sides that are mirror images of each other.  The human body would be an excellent example of a living being that has bilateral symmetry. The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn.  The most obvious geometric example would be a circle. Shapes Sphere: A sphere  is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.  The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator. The wee electron has gotten its most thorough physical examination yet, and scientists report that it is almost, almost a......

Words: 921 - Pages: 4

Free Essay

Life, Death, and the Critical Transistion

...Life, Death, and the Critical Transition: Finding Liveness Bugs in Systems Code Charles Killian, James W. Anderson, Ranjit Jhala, and Amin Vahdat University of California San Diego {ckillian,jwanderson,jhala,vahdat} Abstract finding bugs with model checking currently requires the programmer to have intimate knowledge of the low-level Modern software model checkers find safety violations: actions or conditions that could result in system failure. breaches where the system has entered some bad state. For We contend that for complex systems the desirable bemany environments however, particularly complex con- haviors of the system may be specified more easily than current and distributed systems, we argue that liveness identifying everything that could go wrong. Of course, properties are both more natural to specify and more im- specifying both desirable conditions and safety assertions portant to check. Liveness conditions specify desirable is valuable; however, current model checkers do not have system conditions in the limit, with the expectation that any mechanism for verifying whether desirable system they will be temporarily violated, perhaps as a result of properties can be achieved. Examples of such properties failure or during system initialization. include: i) a reliable transport eventually delivers all mesExisting software model checkers cannot verify live- sages even in the face of network losses and delays, ii) all ness because doing......

Words: 19579 - Pages: 79

Free Essay

Mth 110

...middle ages, mathematicians were coming out and at one point competing without knowing in discovering new techniques. Archimedes had one of the greatest impacts on its work in mathematics but he was known later in the 16th century when Federico Commandino in 1558 translation into Latin most of his printing texts and spread it out with other mathematicians and physics of the time, that includes Johannes Kepler and Galileo Galilei (Toomer, 2014). Many other mathematicians were able to join the works of Archimedes but for other mathematicians, Arabic and Renaissance the methods were unknown until the 19th century. Mathematics has played a huge role in the technological and social history of humankind (National Academy of Sciences, 1968). The number of mathematicians has grown rapidly and so the quality of research, it has become the central part on most...

Words: 1412 - Pages: 6