ࡱ> MOLc G!jbjbSS !F11?]bbbb8$Lb&(NNNNNNN, TNNNNN"NN&"""NlNN^^N"p" Mbbh Pascals Triangle By: Kelli Nipper What is Pascals Triangle? Pascals Triangle, which was actually being used more than five centuries before Blaise Pascal began investigating the properties of it, is an arithmetic array of numbers that has many patterns and can be used in several branches of mathematics. Among these are: Algebra, Probability/Combinatorics, Number Theory and Geometry. The applications of the patterns found in this triangle of numbers seems to be unlimited. Blaise Pascal describes some of the properties of the arithmetic array of whole numbers in A Treatise on the Arithmetic Triangle, which was written in 1653. This triangle provides solutions to many problems and contains many interesting relationships. The purpose of this study is to introduce the triangle and a few of its applications as well as to show the implications of teaching it in the classroom. How do you construct it? Any number can be found by adding the two numbers diagonally above it. For example: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Adding together the 1 and the 3 attains the 4. You can think of the ones as continuing by adding an understood 0 wherever there is no number. Algebraic: More than five centuries before Pascal investigated the properties of the triangle, this triangle was used as a short cut for tabulating the binomial coefficients. Omar Khayyam used it around 1100, and Chu Shih-Chieh used it in a manuscript in the year 1303. (x+y)0=1 (x+y)1=1x + 1y (x+y)2=1x2 + 2xy + 1y2 (x+y)3=1x3 + 3x2y + 3xy2 + 1y3 (x+y)4=1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4 As you can see, the coefficients correspond to the entries in the triangle. Probability/Combinatorics: Chevalier de Mere posed the following problem: In eight throws of a die, a player is to attempt to throw a one, but after three unsuccessful trials, the game is interrupted. How should he be indemnified? Pascal noticed that the probability of choosing n objects from a group of m indistinguishable objects could be found using the triangle. This was the starting point for the modern theory of probability. Many other patterns were eventually found in this triangle. Number Theory: The numerals formed in each row of Pascals Triangle are actually powers of eleven, while adding the digits of each entry in a row yield powers of 2. For example: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 110=1 20=1 = 1 111=11 21=2 = 1+1 112=121 22=4 = 1+2+1 113=1,331 23=8 = 1+3+3+1 114=14,641 24=16 = 1+4+6+4+1 Geometry: Pythagorean Triplets345512137242594041 There are many applications of geometry in Pascals Triangle including the Pythagorean Triplets. 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In this model, the endpoint the chord CD' can be moved along the circumference of the circle from the     
fixed point C.  Since any angle inscribed in a semicircle is a right     
angle, we have that ACD is a right triangle.  Thus, if  the angle ACD is    
[[pi]]/6, then the angle ADC is 
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