Pascal’s Triangle is a visual represenation a series of binomial expansions. The triangle emerges as a result of the function (x + y) ^n where n is an integer greater than or equal to zero. As n increases, the quantity of terms in the result increases:
1. (x + y)^0 = 1………………………………………………………………………………. one term
2. (x + y)^1 = x + y………………………………………………………………………… two terms
3. (x + y)^2 = x^2 + 2xy + y2……………………………………………………………. .three terms
Additionally, the integers represented on the triangle are found as the coefficients of the expansion. For instance, the third row of Pascal’s Triangle (0, 1, 2) is “1 2 1”, which corresponds with the coefficients of 3 above. The triangle itself is simply a visualization of this pattern.
The discovery of Pascal’s Triangle is widely considered to have taken place centuries prior to Pascal’s lifetime. However, Blaise Pascal was the first to publish the triangle, which he did in his 1654 work, “Traite du Triangle Arithmetique”. (Kazimir, 2014) Originally, interest in the concept was confined to gamblers as it provided a convenient method to calculate probabilities of an event such as a coin flip or game of dice. (Witchita, 2014) Since then, interest has tended toward the methods and applications of the triangle. The methods of use are essentially number theoretical and the applications are wide. Many fields such as algebra, probability, and combinatorics may find use in Pascal’s Triangle, and additional applications include identifying number sequences such as triangular and tetrahedral numbers.
Each application of Pascal’s Triangle can be solved using all methods available. The convenience of the triangle then is not necessarily in any particular method of examination, but in the variety of method...
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...head: head, tail; or tail, head. This yields a 2/4 or 50% chance. Using Pascal’s Triangle for 10 flips, we found the sum of the elements of the 10th row equals 1024. Using combinations, (10 C 5) = 252. Given these results, there are exactly 252 ways to have exactly 5 heads in 10 flips of a coin or (252/1024) = 24.6% chance. Our application of Pascal’s Triangle had given a result, which to some would be a priori counterintuitive.
It is clear to us that Pascal’s Triangle is a very interesting source of useful information. The study of binomial expansions has proven to be a fount of interesting patterns and its allure is the suggestion of more hidden within it. We hope that this cursory examination provides insight for those unfamiliar with Pascal’s Triangle and a renewed interest in those who have experience with it. That is the result which it had on these authors.
On the second day of class, the Professor Judit Kerekes developed a short chart of the Xmania system and briefly explained how students would experience a number problem. Professor Kerekes invented letters to name the quantities such as “A” for one box, “B” for two boxes. “C” is for three boxes, “D” is for four boxes and “E” is for five boxes. This chart confused me because I wasn’t too familiar with this system. One thing that generated a lot of excitement for me was when she used huge foam blocks shaped as dice. A student threw two blocks across the room and identified the symbol “0”, “A”, “B”, “C”, “D”, and “E.” To everyone’s amazement, we had fun practicing the Xmania system and learned as each table took turns trying to work out problems.
Blaise Pascal was born on 19 June 1623 in Clermont Ferrand. He was a French mathematician, physicists, inventor, writer, and Christian philosopher. He was a child prodigy that was educated by his father. After a horrific accident, Pascal’s father was homebound. He and his sister were taken care of by a group called Jansenists and later converted to Jansenism. Later in 1650, the great philosopher decided to abandon his favorite pursuits of study religion. In one of his Pensees he referred to the abandonment as “contemplate the greatness and the misery of man”.
-The wager is part of Apologetic philosophy, meaning that Pascal is defending his position/belief in an argument with the use of information
Blaise Pascal was born on June 19, 1623. Pascal was a mathematician along with a Christian philosopher who wrote the Pensees which included his work called Pascal’s wager. The crucial outline of this wagers was that it cannot be proved or disprove that God does exists. There are four main parts to the wager that include his reasoning to that statement. It has been acknowledged that Pascal makes it clear that he is referring to the Christian God in his wager. This is the Christian God that promises his people will be rewarded with eternal life along with infinite bliss.
Pascal’s Triangle falls into many areas of mathematics, such as number theory, combinatorics and algebra. Throughout this paper, I will mostly be discussing how combinatorics are related to Pascal’s Triangle.
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.
... Pascal was such a brilliant man because he could do both of these. Pascal was one of the only men that wrote about his beliefs in God and was an accredited scientist and mathematician too. He was a true man of the scientific revolution.
Leonardo Fibonacci was one of the great mathematicians of his time. His lifestyle allowed him to travel and study math in various countries, and he ended up combining his cultural knowledge to discover the most effective ways of doing mathematics. He is most famous for his contributions to the European number system and for his sequence of numbers known as the Fibonacci numbers. Starting with 0 and 1 as the first two numbers, each number in the sequence is the sum of the two preceding numbers. He came across these numbers as a solution to a problem that he used as an example in one of his many publications. He was not aware of the importance of his findings at the time. Many uses have been found for these numbers since Leonardo’s death and many mathematicians have used this sequence in their own theories.
Deep within the realm of fractal math lies a fascinating triangle filled with unique properties and intriguing patterns. This is the Sierpinski Triangle, a fractal of triangles with an area of zero and an infinitely long perimeter. There are many ways to create this triangle and many areas of study in which it appears.
Lily’s use of a purple triangle to represent Mrs. Ramsay and James (Woolf 52) could symbolize many different things. Lily says, “It was a question [of] how to connect the mass on the right hand with that on the left hand” (Woolf 53). To connect one with two requires three, the completion of a triangle, the third stroke. Mrs. Ramsay is a representation of that third stroke. She brings people together through her d...
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.
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
One of Polya’s most noted problem solving techniques can be found in “How to Solve it”, 2nd ed., Princeton University Press, 1957.
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