Pascal`s Triangle Essays - Blaise Pascal, Combinatorics,

Pascal`s Triangle Blas? 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. 3 * 2 * 1 * 3 * 2 * 1 The triangular numbers and the Fibonacci numbers can be found in Pascal's triangle. The triangular numbers are easier to find: starting with the third one on the left side go down to your right and you get 1, 3, 6, 10, etc (Swarthmore, 5) 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 The Fibonacci numbers are harder to locate. To find them you need to go up at an angle: you're looking for 1, 1, 1+1, 1+2, 1+3+1, 1+4+3, 1+5+6+1 (Dr. Math, 4). Another thing I found out is that if you multiply 11 x 11 you will get 121 which is the 2nd line in Pascal's Triangle. If you multiply 121 x 11 you get 1331 which is the 3rd line in the triangle (Dr. Math, 4). If you then multiply 1331 x 11 you get 14641 which is the 4th line in Pascal's Triangle, but if you then multiply 14641 x 11 you do not get the 5th line numbers. You get 161051. But after the 5th line it doesn't work anymore (Dr. Math, 4). Another example of probability: Say there are four children Annie, Bob, Carlos, and Danny (A, B, C, D). The teacher wants to choose two of them to hand out books; in how many ways can she choose a pair (ladja, 4)? 1.A & B 2.A & C 3.A & D 4.B & C 5.B & D 6.C & D There are six ways to make a choice of a pair. If the teacher wants to send three students: 1.A, B, C 2.A, B, D 3.A, C, D 4.B, C, D If the teacher wants to send a group of "K" children where "K" may range from 0-4; in how many ways will she choose the children K=0 1 way (There is only one

Ludvig Von Mises essays

Ludvig Von Mises essays Ludwig von Mises: Defender of the Free Market Ludwig von Misis thoughts on human behavior, socialism, and money and credit have had a major impact on economic thought. He championed true free markets and is seen as a defender of liberty. Former President of the United States Ronald Reagan said Ludwig von Mises was one of the greatest economic thinkers in the history of Western Civilization. Through his seminal works, he rekindled the flames of liberty. As a wise and kindly mentor, he encourages all who sought to understand the meaning of freedom. We owe him an incalculable debt(Mises Institute). The remainder of this paper will outline the life of Ludwig von Mises. This will be accomplished by describing the social, political, technical, and economic environment that influenced his ideas. A description of his major ideas in economic thought will be presented. Next, the people and ideas that influenced his approach to economics will be addressed. Finally, the paper will conclude with an assessment of Ludwig von Mises co ntributions to economic thought. Overview of the Life of Ludwig von Mises Ludwig von Misis was born on September 29, 1881 in Lemberg, Austria. He attended a private elementary school, the public Akademishe Gymnasium in Vienna from1892 to 1900. In 1900 Mises entered the University of Vienna. On February 20,1906 he received a Dr. Jur degree, a Doctor of both Canon and Roman Laws, from the University of Vienna. When Mises attended the University, it had no separate economics department; the only way to study economics was through law (Mises Institute). From 1907 to 1914 Mises was employed as an advisor to the Austrian Chamber of Commerce. His first major thesis, the Theory of Money and Credit was published in 1912. In 1913 Mises was awarded the position of Privatdozent (unsalaried lecturer) at the University of Vienna (Mises Institute). Mises academic pursuits were interrupted from ...