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Pascal's Triangle

A mathematical triangle made by numbers
by

Nafiul Islam

on 17 May 2010

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Transcript of Pascal's Triangle

Pascal's Triangle By: Nafiul Islam What is It? Pascal’s triangle is basically a triangle that goes on until infinity The triangle is basically an adding triangle starting from 0 and going on and on but the zero is never visible I will now show you how to make the first six rows of Pascal's Triangle. If a pair of numbers cannot be formed the second number will be substituted with a zero. That is why the sides of the triangle are always 1’s since they do not form a pair. History Pascal wasn’t the one to discover this number triangle. This concept was known well before by the Greeks and Hindus but Blaise Pascal was the first one to get deeper into the concept and look at the rules. The earliest sign of Pascal’s triangle was seen in an ancient Indian book in the 10th century. It was also seen in India between the 5th and 2nd century B.C. During the same time it was discovered in Persia by the Persian mathematician called Al-Karaji. In the 13th century a Chinese mathematician named Yang Hui discovered it. To this day, china calls Pascal’s triangle Yang Hui’s triangle and below is the Chinese version of Pascal’s Triangle and the numbers are shown as rods.

Relationships If you color in the odd number in Pascal’s triangle you will get something close to the Sierpinski Trianlge which is a triangle where you put a white triangle in each black one until you get the big triangle.

Here Is the Sierpinski Triangle Patterns All the numbers on the sides are always 1’s. All number beside 1’s are natural numbers if you go in a diagonal pattern. Natural number are basically numbers that go up by 1’s (1,2,3,4,5…) After the natural number( moving inwards) you get triangular numbers which are numbers you can arrange into a triangle if each digit was a dot. Here are some of the Trianglular Numbers More Patterns In Pascal’s triangles there is a relationship between the dots and the lines. As you can see in the picture below in each node (knots), (going upward) shows how many paths there are to the first “1”. For example: From “4” to the first “1” there are 4 pathways. These are all the bands of the Pascal’s triangle. All the diagonal colored bands add up to successful Fibonacci numbers. Fibonacci is when the number before adds up to the next number and so on. Basically the sum of the last two numbers like in the numbers below. Class Activities Fill In this Pascals Triangle Was Pascal The first one to discover this triangle? The Numbers after the Diagonal "Ones" are
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Thank You for Watching
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