Blaise Pascal “Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.” “We are usually convinced.

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Presentation transcript:

Blaise Pascal “Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.” “We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others. Pensées “(1670)

1 1 1 Pascal's Triangle 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 1 1 Zeroth Row  1st Row  1 2 1 2nd Row  1 3 3 1 3rd Row  1 4 6 4 1 4th Row  1 5 10 10 5 1 1 6 15 20 15 6 1

Binomial Coeff. for a group of 4 1 1 1 Binomial Coeff. for a group of 4 1 2 1 1 3 3 1 1 4 6 4 1 4th  4C0 1 5 10 10 5 1 4C1 4C2 4C3 4C4

Binomial Coeff. for a group of 4 1 1 1 1 2 1 1 3 3 1 These are the coefficients in the expansion of (x+y)4 1 4 6 4 1 4th  1 5 10 10 5 1

Other Interesting Patterns 1 1 1 1 2 1 Counting Numbers 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

Other Interesting Patterns 1 1 1 1 2 1 Triangular Numbers 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

Other Interesting Patterns 1 1 1 sum = 1 Powers of 2 sum = 2 1 2 1 sum = 4 1 3 3 1 sum = 8 1 4 6 4 1 sum = 16 1 5 10 10 5 1 sum = 32 1 6 15 20 15 6 1 sum = 64

Other Interesting Patterns 1 1 1 1 6 15 20 15 6 1 1 5 10 10 5 1 1 4 6 4 1 1 3 3 1 1 2 1 sum = 2 sum = 3 sum = 5 sum = 8 sum = 13 Fibonacci Numbers

Other Interesting Patterns Pascal’s Flowers The gray cell is surrounded by 6 “petals.” If you multiply the yellow petals you get 2100. If you multiply the orange petals, you get 2100.

Other Interesting Patterns 1 1 1  11 0  11 1 1 2 1  11 2 Powers of 11 1 3 3 1  11 3 1 4 6 4 1  11 4 1 5 10 10 5 1 1 6 15 20 15 6 1