Pascals Triangle Essay, Research Paper
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 way to send no children) K=1 4 ways ( A; B; C; D) K=2 6 ways (like above with Annie, Bob, Carlos, Danny) K=3 4 ways (above with triplets) K=4 1 way (there is only one way to send a group of four) The above numbers (1 4 6 4 1) are the fourth row of numbers in Pascal Triangle (Ladja, 5). “If we extend Pascal’s triangle to infinitely many rows, and reduce the scale of our picture in half each time that we double the number of rows, then the resulting design is called self-similar — that is, our picture can be reproduced by taking an subtriangle and magnifying it,” Granville notes.The pattern becomes more evident if the numbers are put in cells and the cells colored according to whether the number is 1 or 0 (Peterson’s, 5).Similar, though more complicated designs appear if one replaces each number of the triangle with the remainder after dividing that number by 3. So, I get: 1 1 1 1 2 1 1 0 0 1 1 1 0 1 1 1 2 1 1 2 1 1 0 0 2 0 0 1 This time, one would need three different colors to reveal the patterns of triangles embedded in the array. One can also try other prime numbers as the divisor (or modulus), again writing down only the remainders in each position (Freedman, 5). Actually, there’s a simpler way to try this out. With the help of Jonathan Borwein of Simon Fraser University in Burnaby, British Columbia, and his colleagues, Granville has created a “Pascal’s Triangle Interface” on the web. One can specify the number of rows (up to 100), the modulus (from 2 to 16), and the image size to get a colorful rendering of the requested form.It’s a neat way to explore the fractal side of Pascal’s triangle. Here’s one example that I tried out, using 5 as the modulus (Petetson’s, 5).