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Blaise Pascal: Numbers
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At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1) contains two 1's, both formed by adding the two numbers above them to the left and the right, in this case 1 and 0 (all numbers outside the Triangle are 0's). Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. In this way, the rows of the triangle go on infinitly. A number in the triangle can ... be found by nCr (n Choose r) where n is the number of the row and r is the element in that row.
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Pascal devised an interesting method for the computation of combinations. He formulated a triangular arrangement of numbers such that each number in the triangle is the sum of the two numbers above it. This triangle came to be known as the Pascals triangle. The numbers, called elements are arranged in rows. Each element of the row has its own place, which is determined by counting from left to right. Thus, 20 appears in the fourth place of the seventh row of the triangle. Pascal found that the element in the (x+1)th place of the (y+1)th row is the same as the number of combinations of y things taken x at a time, denoted by Cyx.
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Pascal's device, capable of adding two decimal numbers, was based on a design described in Greek antiquity by Hero of Alexandria. It employed the principle of a one tooth gear engaging a ten-tooth gear once every time it revolved. Thus, it took ten revolutions of the first gear in order to make next gear rotate once. The train of gears produced mechanically an answer equivalent to that obtained using manual arithmetic.
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Pascal's device could only add and subtract, while multiplication and division operations were implemented by performing a series of additions or subtractions. In fact the Arithmetic Machine could really only add, because subtractions were performed using complement techniques, in which the number to be subtracted is first converted into its complement, which is then added to the first number. Interestingly enough, modern computers employ similar complement techniques.
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Fibonnacci's Sequence can ... be located in Pascal's Triangle. The sum of the numbers in the consecutive rows shown in the diagram are the first numbers of the Fibonnacci Sequence. The Sequence can also be formed in a more direct way, very similar to the method used to form the Triangle, by adding two consecutive numbers in the sequence to produce the next number. The creates the sequence: 1,1,2,3,5,8,13,21,34, 55,89,144,233, etc . . . . The Fibonnacci Sequence can be found in the Golden Rectangle, the lengths of the segments of a pentagram, and in nature, and it decribes a curve which can be found in string instruments, such as the curve of a grand piano. The formula for the nth number in the Fibonnacci Sequence is
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Each number in a Pascal triangle is calculated by adding together the two adjacent numbers in the wider adjacent row. The sum bf the numbers in any row gives the total arrangement of combinations possible within that group. The numbers at the end of each row give the the "odds" of the least likely combinations, with each succeeding pair of triangles giving the chances of combinations which are increasingly likely.
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