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

the effect of europeans on ame essays

the effect of europeans on ame essays The Europeans that settled in America changed the lives of the Indians, slowly robbing them of their culture. The future of the Indians was changed drastically as their children, their income off the land and their spiritual bonds with the land were interfered by the Europeans. The Europeans, in trying to create a white society out of the Indians, stole from them what made them unique as a civilization. The Indians children were affected more than any other generation by the coming of the Europeans. Their whole family structure changed in the short period while the Europeans took over, and gradually the children lost all they ever knew. Indian children were brought up learning the skills they would need as adults. A great importance was placed in training boys to become warriors and teaching them skills such as running, swimming, jumping, building stamina and strength and archery. Girls would stay with their mothers until they were eight and then live with their grandmother, who was considered their most dignified protector. She would take over and teach the girl skills such as weaving, molding clay and domestic skills Indian children were as free as the animals that roamed the forest around them. Young boys were encouraged by elders to engage in sports... Girls busied themselves imitating their mothers. Life of a Shawnee, W.C.Mundell With the arrival of the Europeans, Indian children lost their upbringing, their culture and gradually, their self-esteem. The government attempted to civilise the Indian children and many were sent or forcibly taken to boarding schools, often not situated on the reservations. The children were separated from their families, their mentors. Their clothing was missionary style dresses for the girls, knickers and trousers for the boys. The children werent allowed to speak their native tongue. The family structure collapsed as the fathers died and many women were w...

Dorothy Orem's Nursing Theory Research Paper Example | Topics and Well Written Essays - 1250 words

Dorothy Orem's Nursing Theory - Research Paper Example The backbone of the theory is the concept that from time to time people are affected by limitations that prevent them from meeting their self-care needs. These limitations can be caused by injury or accident, or by external or internal situations such as disease or the natural progression of aging (Hartweg, 1991). Orem describes the nature of the relationships involved with nursing, between the nurse and the patient, and between the nurse and others (such as family members and physicians) that may be involved. She compares this relationship to a friendship, with the nurse being more objective, able-bodied, selfless, and skilled (Orem, 2003). Orem’s theory is set forth in a way that makes it easy to both understand and to implement. As writers Kathleen Sitzman and Lisa Eichelberger (2011) has stated, â€Å"The simplicity of wording, coupled with an uncanny resonance with everyday nursing activities, has ensured its broad popularity and use in many areas of nursing† (p. 94). Orem believed that wholeness is part of what makes a person human. Health helps the person be fully who he or she is, and to operate along with physiological and psychophysiological mechanisms. Good health enables people to interact with others and to have meaningful relationships with those around them (Current Nursing). Nursing is required for patients who need â€Å"direct continuing assistance in self-care† (Orem, 1993, p. 258) caused by health problems. These are needs that all people have, regardless of their health needs, but nurses are required when patients are unable to meet them. Patients tend to become healthier and to recover more quickly from disease, illness, and injury when they are able to participate in and accomplish their own self-care. It is the nurse’s role and responsibility to provide patients and their families