Azeena Hassan
Math 301
Project #1 - Fibonacci Sequence
Fibonacci also known as Leonard of Pisa was born in the early 1770’s AD, and has had a huge impact on today’s math world. He made his mathematical discoveries along the Meditterainean coast by learning from the locals. With inspiration from the Hindi-Arabic numerical system, Fibonnacci created the decimal system that we still use today. One of his most famous of discoveries is known as the Fibonnacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. He discovered this sequence through the analysis of a rabbit population. His ananlysis lead him to realizing that if you add the last two numbers together you get the next one.
The Fibonnacci sequence can be found almost anywhere including in nature. For example shells follow the Fibonacci sequence in how they are formed. Seeds are arranged in flowers to prevent overcrowding, this is easily seen in the sunflower and in pinecones. In the body the DNA strands are 34 by 21 angstroms. A finger is broken up into three different parts. From the tip of the finger to the first joint is two fingernails. The rest of the finger is five fingernail widths. Lastly, the palm is the length of 8 fingernails. In music, Mozart uses Fibonnacci numbers in how he composes his sonatas by how many measures he puts in each section of his music. In architcture, you can see Fibonnacci’s contributions in the Great Giza Pyramids. The Fibonacci sequence is also used in the pascal triangle. The sum of each diagonal row as well as the right sequence is a Fibonacci number.
Fibonnacci first showed his great discoveries in his first published book, Liber Abaci in 1202. It was here that the Fibonacci number’s were first discussed. It was based on ...
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...plifying the results, we obtain: . We then multiply the numerator by the reciprocal of the denominator. Giving us, . After distributing, Binet’s formula is obtained: .
Fibonacci sequence has had a lasting affect, with his exponential growth, explanation of nature, golden growth, and golden ratio. The goden ratio takes the ratio of the two successive numbers in Fibonacci’s series, dividing each by the number before it. If you plot a graph with the results of this, you will see that they seem to be tending to a limit, which we call the golden ratio or can be refered to by the greek letter phi. This number accounts for the spirals in the seedheads and the arrangement of leaves in many plants. So based on Fibonacci’s sequence, we have been able to explain natural phenominas, it lead to the understanding of phi, and it has lead to many other mathematical equations.
The first and primary contribution to solving the DNA structure was the relationship of Crick and Watson. Without their teamwork and determination, another scientist would have discovered the structure before them. One of Crick’s bigger contributions was discovering the gene is self-replicating. After talking with John Griffith, Crick came up with the idea that the gene is self-replicating, meaning the gene has the ability “to be exactly copied when the chromosome number doubles during cell division”(126). With further discussion with Griffith, Francis believed that DNA replication involved specific attractive forces between the flat surfaces of the bases (128). One of Watson’s major contributions was after seeing the B form of DNA by Franklin, Watson knew that the structure of DNA was two-chained and that led to the building of the model of DNA (171). Also through research, Watson became aware that adenine and thymine pair together and are held by two hydrogen bonds that were identical in shape to the guanine and cytosine pair held together by at least two hydrogen bonds (194). This discovery showed that the two chains of DNA are complementary to each other. With these individual contributions coming together, Watson and Crick successfully were able to piece together the structure of DNA.
Yang Hui has been found to be the oldest user of Pascal’s Triangle. But it is Blaise Pascal who around the year 1654 was credited for his extensive work on the many patterns of this triangle. Because of this people began to call it Pascal’s Triangle.
Leonardo was known as Fibonacci around his local community, and as a result the name is “Fibonacci sequence”. Using the Fibonacci betting system is incredibly easy to comprehend and to implement and eventually win. You start off with a bet of one because it is the first number in the Fibonacci sequence. Every time you win a bet, the next bet you play should correspond to the next number in the sequence. If you look according to the sequence you will always know what to bet, although if you lose you must start the sequence over
Blaise Pascal has contributed to the field of mathematics in countless ways imaginable. His focal contribution to mathematics is the Pascal Triangle. Made to show binomial coefficients, it was probably found by mathematicians in Greece and India but they never received the credit. To build the triangle you put a 1 at the top and then continue placing numbers below it in a triangular pattern. Each number is the two numbers above it added together (except for the numbers on the edges which are all ‘1’). There are patterns within the triangle such as odds and evens, horizontal sums, exponents of 11, squares, Fibonacci sequence, and the triangle is symmetrical. The many uses of Pascal’s triangles range from probability (heads and tails), combinations, and there is a formula for working out any missing value in the Pascal Triangle: . It can also be used to find coefficients in binomial expressions (put citation). Another staple of Pascal’s contributions to projective geometry is a proof called Pascal’s theore...
In mathematics, Pascal’s triangle is taught everywhere throughout schools. He also started probability theory that many if not all mathematicians today use. Pascal even changed science by his experiments on atmospheric pressure and later had units of pressure named after him for his study. Pascal also, has a law in physics named after him. His inventions were just as impactful. Pascal created one of the first digital calculators. Pascal also invented the core principles of the roulette machine when study a perpetual motion theory.
Pierre de Fermat Pierre de Fermat was born in the year 1601 in Beaumont-de-Lomages, France. Mr. Fermat's education began in 1631. He was home schooled. Mr. Fermat was a single man through his life. Pierre de Fermat, like many mathematicians of the early 17th century, found solutions to the four major problems that created a form of math called calculus. Before Sir Isaac Newton was even born, Fermat found a method for finding the tangent to a curve. He tried different ways in math to improve the system. This was his occupation. Mr. Fermat was a good scholar, and amused himself by restoring the work of Apollonius on plane loci. Mr. Fermat published only a few papers in his lifetime and gave no systematic exposition of his methods. He had a habit of scribbling notes in the margins of books or in letters rather than publishing them. He was modest because he thought if he published his theorems the people would not believe them. He did not seem to have the intention to publish his papers. It is probable that he revised his notes as the occasion required. His published works represent the final form of his research, and therefore cannot be dated earlier than 1660. Mr. Pierre de Fermat discovered many things in his lifetime. Some things that he did include: -If p is a prime and a is a prime to p then ap-1-1 is divisible by p, that is, ap-1-1=0 (mod p). The proof of this, first given by Euler, was known quite well. A more general theorem is that a0-(n)-1=0 (mod n), where a is prime...
Carl Friedrich Gauss was born April 30, 1777 in Brunswick, Germany to a stern father and a loving mother. At a young age, his mother sensed how intelligent her son was and insisted on sending him to school to develop even though his dad displayed much resistance to the idea. The first test of Gauss’ brilliance was at age ten in his arithmetic class when the teacher asked the students to find the sum of all whole numbers 1 to 100. In his mind, Gauss was able to connect that 1+100=101, 2+99=101, and so on, deducing that all 50 pairs of numbers would equal 101. By this logic all Gauss had to do was multiply 50 by 101 and get his answer of 5,050. Gauss was bound to the mathematics field when at the age of 14, Gauss met the Duke of Brunswick. The duke was so astounded by Gauss’ photographic memory that he financially supported him through his studies at Caroline College and other universities afterwards. A major feat that Gauss had while he was enrolled college helped him decide that he wanted to focus on studying mathematics as opposed to languages. Besides his life of math, Gauss also had six children, three with Johanna Osthoff and three with his first deceased wife’s best fri...
Fibonacci Numbers originated from India hundreds of years ago. Though Fibonacci Numbers came from India, Leonardo of Pisa, better known as Fibonacci, made it known to the world. Leonardo came from a wealthy Italian family and traveled to North America to join his father. He was educated by the Moors and sent on business trips. “After returning to Pisa around 1200, Leonardo wrote his most famous literature, Liber Abaci” (Pearson). Leonardo featured a rabbit question in the book. The question was asked in a mathematical competition, he appeared in when he was young. Leonardo Fibonacci used the Fibonacci Numbers to solve it. Fibonacci Numbers is now used throughout our society.
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the numbers, which is entirely his own independent discovery.
Named after the Polish mathematician, Waclaw Sierpinski, the Sierpinski Triangle has been the topic of much study since Sierpinski first discovered it in the early twentieth century. Although it appears simple, the Sierpinski Triangle is actually a complex and intriguing fractal. Fractals have been studied since 1905, when the Mandelbrot Set was discovered, and since then have been used in many ways. One important aspect of fractals is their self-similarity, the idea that if you zoom in on any patch of the fractal, you will see an image that is similar to the original. Because of this, fractals are infinitely detailed and have many interesting properties. Fractals also have a practical use: they can be used to measure the length of coastlines. Because fractals are broken into infinitely small, similar pieces, they prove useful when measuring the length of irregularly shaped objects. Fractals also make beautiful art.
There are many people that contributed to the discovery of irrational numbers. Some of these people include Hippasus of Metapontum, Leonard Euler, Archimedes, and Phidias. Hippasus found the √2. Leonard Euler found the number e. Archimedes found Π. Phidias found the golden ratio. Hippasus found the first irrational number of √2. In the 5th century, he was trying to find the length of the sides of a pentagon. He successfully found the irrational number when he found the hypotenuse of an isosceles right triangle. He is thought to have found this magnificent finding at sea. However, his work is often discounted or not recognized because he was supposedly thrown overboard by fellow shipmates. His work contradicted the Pythagorean mathematics that was already in place. The fundamentals of the Pythagorean mathematics was that number and geometry were not able to be separated (Irrational Number, 2014).
The history of the computer dates back all the way to the prehistoric times. The first step towards the development of the computer, the abacus, was developed in Babylonia in 500 B.C. and functioned as a simple counting tool. It was not until thousands of years later that the first calculator was produced. In 1623, the first mechanical calculator was invented by Wilhelm Schikard, the “Calculating Clock,” as it was often referred to as, “performed it’s operations by wheels, which worked similar to a car’s odometer” (Evolution, 1). Still, there had not yet been anything invented that could even be characterized as a computer. Finally, in 1625 the slide rule was created becoming “the first analog computer of the modern ages” (Evolution, 1). One of the biggest breakthroughs came from by Blaise Pascal in 1642, who invented a mechanical calculator whose main function was adding and subtracting numbers. Years later, Gottfried Leibnez improved Pascal’s model by allowing it to also perform such operations as multiplying, dividing, taking the square root.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...
The Fibonacci Series was discovered around 1200 A.D. Leonardo Fibonacci discovered the unusual properties of the numeric series, that’s how it was named. It is not proven that Fibonacci even noticed the connection between the Golden Ratio meaning and Phi.
The man behind the Fibonacci numbers, Leonardo Fibonacci, was born in Pisa in 1175 A.D. During his life, he was a customs officer in Africa and businessman who traveled to various places. During these trips he gained knowledge and skills which enabled him to be recognized by Emperor Fredrick II. Fredrick II noticed Fibonacci and ordered him to take part in a mathematical tournament. This place would eventuall...