Even the smallest tasks can impact the world in a significant way. Math, despite its trivial appearance, is large in grandeur that governs our world from the inside and the outside. The many twists and turns that exist in Mathematics make its versatility unparalleled and continues to awe the many Mathematicians today and the many more to come. The Binomial Theorem is one such phenomenon, which was founded by the combined efforts of Blaise Pascal, Isaac Newton and many others. This theorem is mainly algebraic, which contains binomial functions, arithmetic sequences and sigma notation. I chose the Binomial Theorem because of its complexity, yet simplicity. Its efficiency fascinates me and I would like to share this theorem that can be utilized to solve things in the Mathematical world that seem too daunting to be calculated by normal means.
In 1665, the Binomial Theorem was born by the highly appraised Isaac Newton, who at the time was just a graduate from Cambridge University. He came up with the proof and extensions of the Binomial Theorem, which he included it into what he called “method of fluxions”. However, Newton was not the first one to formulate the expression (a + b)n, in Euclid II, 4, the first traces of the Binomial Theorem is found. “If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle of the segments” (Euclid II, 4), thus in algebraic terms if taken into account that the segments are a and b:
(a + b)2 = a2 + b2 + 2ab
Then in Euclid II, 7, it goes farther to explain that “if a straight line be cut at random, the square on the whole and that on one of the segments both together, are equal to twice the rectangle contained by the whole and said segm...
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... and there are 4 trials (4 customers). Using the Binomial Theorem, I can substitute these numbers into the formula.
(.4+(1-.4))^4=∑_(k=0)^4▒〖(4¦2) 〖.4〗^2 〖(1-.4)〗^(4-2) 〗
Therefore the probability that 3 people will purchase an item is .0576. A business has to compensate these numbers for the amount of products that they will have in stock.
A world without the Binomial Theorem is a world destined for the almost seemingly unlimited amount of numbers and calculations. However, the positive side is that this world does have the Binomial Theorem! Therefore, any long binomial problem is compressed into a simplistic form that any student can exercise and get the right answer-and maybe enjoy even if it’s a little. So are you tired of foiling (a + b)n to those seemingly endless amount of variables and numbers? Never fear for the Binomial Theorem is here.
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b. The amount of molasses and byproduct shipped to seven customers (a majority of which are internal and therefore don't generate profit accounted for in this model).
two lines of different lengths, while the lines are the same size. This illustrates the fact
For centuries, mathematicians tried to contradict Euclid's Postulate V, and determine that there was more than one line parallel to that of another. It was declared impossible until the 19th century when Non-Euclidean Geometry was developed. Non-Euclidean geometry was classified as any geometry that differed from the standards of Euclidean geo...
When most people hear the name Isaac Newton, they think of various laws of physics and the story of the apple falling from the tree; in addition, some may even think of him as the inventor of calculus. However, there was much more to Newton’s life which was in part molded by the happenings around the world. The seventeenth century was a time of great upheaval and change around the world. The tumultuousness of this era was due mostly to political and religious unrest which in effect had a great impact on the mathematics and science discoveries from the time Newton was born in 1646 until the early 1700’s.
As I was looking for a theorem to prove for my Mathematics SL internal assessment, I couldn’t help but read about Fermat’s Little Theorem, a theorem I never heard of before. Looking into the theorem and reading about it made me develop an interest and genuine curiosity for this theorem. It was set forth in the 16th century by a French lawyer and amateur mathematician named Pierre de Fermat who is given credit for early developments that led to infinitesimal calculus. He made significant contributions to analytic geometry, probability, and optics. Fermat is best known for Fermat’s last theorem. Nevertheless, for the purpose of this investigation I will study his little theorem one of the beautiful proofs in Mathematics.
The quartic equation is used by geometry teachers around the world and in computer graphics. This formula originated in Italy in the 1500’s. It was rare for someone to find a solution and achieve fame in doing so. The chances of that happening were slim to none due to the lack of education during this period. A mathematician named Lodovico Ferrari beat those odds and created a formula that still has applications today.
The cafeteria serves about one hundred and fifty residents of Cambridge Hall and approximately one hundred residents from Nottingham Hall. The cafeteria serves hot foods, salads, snacks, sandwiches, and beverages. The data has given me information on the percentage of customers that preferred a hot meal (interactive and precooked) to snacks, the ratio of customers that prefer precooked hot meals over an interactive hot meal, line formation, service times at the different stations, arrival times and the location of the different stations. I also learned that the peak hours of operations are from 5:00 p.m. to 6:30 p.m. and that the cafeteria has two cash registers available but only one is being utilized during the peak hours. If customers decide on a hot meal there is a 2 to 1 ratio that customers will purchase a precooked meal over an interactive meal. Through an informal customer survey, reasonable waiting times were established for the precooked line (5 minutes), the interactive line (10 minutes), and the cashier payment line (1minute).
I decided to do my math exploration and internal assessment on the different patterns in the Pascal triangle. My aim is to discover and elaborate on the many different patterns exhibited in the Pascal’s triangle. One of the main reasons this choice of topic spoke to me is because it relates to a lot of things we do in math class, such as Pascal’s triangle, probability, sequences and series, binomial theory, and negative coefficients. Another reason I choose this topic is because I am very interested in patterns and I find myself intrigued by patterns and puzzles. It will be interesting to see what I discover.
The sunflower is a beautiful flower that grows wild in the most parts of the United States and many other countries throughout the world. However, when we look at the flower normally we don’t see anything other than a flower. To the mathematician however, there is more, much more. Inside the sunflower flows a sequence. This sequence is known as the Fibonacci sequence. Here we will discuss the Fibonacci sequence going back to the origins, its uses, and where we can find it in the everyday world.
...ocity. On the other hand, Leibniz had taken a geometrical approach, basing his discoveries on the work of previous thinkers like Fermat and Pascal. Though Newton had been the first to derive calculus as a mathematical approach, Leibniz was the first one to widely disseminate the concept throughout Europe. This was perhaps the most conclusive evidence that Newton and Leibniz were both independent developers of calculus. Newton’s timeline displays more evidence of inventing calculus because of his refusal to use theories or concepts to prove his answers, while Leibniz furthered other mathematician’s ideas to collaborate and bring together theorems for the application of calculus. The history of calculus developed as a result of sequential events, including many inventions and innovations, which led to forward thinking in the development of the mathematical system.
The prominence of numeracy is extremely evident in daily life and as teachers it is important to provide quality assistance to students with regards to the development of a child's numeracy skills. High-level numeracy ability does not exclusively signify an extensive view of complex mathematics, its meaning refers to using constructive mathematical ideas to “...make sense of the world.” (NSW Government, 2011). A high-level of numeracy is evident in our abilities to effectively draw upon mathematical ideas and critically evaluate it's use in real-life situations, such as finances, time management, building construction and food preparation, just to name a few (NSW Government, 2011). Effective teachings of numeracy in the 21st century has become a major topic of debate in recent years. The debate usually streams from parents desires for their child to succeed in school and not fall behind. Regardless of socio-economic background, parents want success for their children to prepare them for life in society and work (Groundwater-Smith, 2009). A student who only presents an extremely basic understanding of numeracy, such as small number counting and limited spatial and time awareness, is at risk of falling behind in the increasingly competitive and technologically focused job market of the 21st Century (Huetinck & Munshin, 2008). In the last decade, the Australian curriculum has witness an influx of new digital tools to assist mathematical teaching and learning. The common calculator, which is becoming increasing cheap and readily available, and its usage within the primary school curriculum is often put at the forefront of this debate (Groves, 1994). The argument against the usage of the calculator suggests that it makes students lazy ...
Pascal's Triangle Blasé Pacal was born in France in 1623. He was a child prodigy and was fascinated by mathematics. When Pascal was 19 he invented the first calculating machine that actually worked. Many other people had tried to do the same but did not succeed. One of the topics that deeply interested him was the likelihood of an event happening (probability). This interest came to Pascal from a gambler who asked him to help him make a better guess so he could make an educated guess. In the coarse of his investigations he produced a triangular pattern that is named after him. The pattern was known at least three hundred years before Pascal had discover it. The Chinese were the first to discover it but it was fully developed by Pascal (Ladja , 2). Pascal's triangle is a triangluar arrangement of rows. Each row except the first row begins and ends with the number 1 written diagonally. The first row only has one number which is 1. Beginning with the second row, each number is the sum of the number written just above it to the right and the left. The numbers are placed midway between the numbers of the row directly above it. If you flip 1 coin the possibilities are 1 heads (H) or 1 tails (T). This combination of 1 and 1 is the firs row of Pascal's Triangle. If you flip the coin twice you will get a few different results as I will show below (Ladja, 3): Let's say you have the polynomial x+1, and you want to raise it to some powers, like 1,2,3,4,5,.... If you make a chart of what you get when you do these power-raisins, you'll get something like this (Dr. Math, 3): (x+1)^0 = 1 (x+1)^1 = 1 + x (x+1)^2 = 1 + 2x + x^2 (x+1)^3 = 1 + 3x + 3x^2 + x^3 (x+1)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 (x+1)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 ..... If you just look at the coefficients of the polynomials that you get, you'll see Pascal's Triangle! Because of this connection, the entries in Pascal's Triangle are called the binomial coefficients.There's a pretty simple formula for figuring out the binomial coefficients (Dr. Math, 4): n! [n:k] = -------- k! (n-k)! 6 * 5 * 4 * 3 * 2 * 1 For example, [6:3] = ------------------------ = 20.
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...
This evaluation has not only allowed me explore calculus more in depth, but also physics, and the way the world works. This has personally allowed me to explore the connections between math and real-world situations, which is hard to find in textbooks.
Strangely, the Fibonacci numbers appear in nature too. One familiar way in which the Fibonacci numbers appear in nature is the rabbit family line (and bee family line as well). Another strange way in which the Fibonacci numbers relate to nature is the plant kingdom. Because of these strange relationships, I ask the question: How and why do the Fibonacci numbers appear in nature? In this paper, I will attempt to answer this question. Pascal?s Triangle - Golden Rectangle