Pascal's Triangle
Blaise Pascal
Blaise Pascal was born in France in 1623. He was a child prodigy, who was fascinated by mathematics. He was also a physicist and philosopher. When Pascal was nineteen, he invented the first calculating machine which actually worked. He became known for his experiments with fluids in physics and for his work on probability theory in mathematics. In a work called Provincial Letters, Pascal helped create a modern French prose style.
Pascal Discovers his Triangle
One of the topics which interested Pascal was the likelihood of an event occurring. His interest was triggered by a gambler. The gambler asked Pascal to help him make better guesses about which scores would be most likely when two dice were thrown. In the course of his investigations, Pascal produced the triangular pattern of numbers which now bears his name. The pattern was known to the Chinese three hundred years before, but it was Pascal who developed it fully.
Pascal's Triangle
Pascal's Triangle, shown on the left, is created like this:
At the top we place the number 1 all by itself. On the second line, we place two new 1's on either side of the first 1. On succeeding lines, each number that appears is calculated by adding together the two numbers that appear on either side of it on the line above. The two ends of each line of numbers are always 1's. The third line is therefore 1, 2 (1 +1), 1 and the fourth is 1, 3 (1 + 2), 3 (2 + 1), 1. Continuing in this way, we find that the sixth line is 1, 5, 10, 10, 5, 1.
There are many things that make Pascal's Triangle remarkable.
One of these things is that the numbers on the nth line provide the coefficients in the formulas for expanding
(a +b)n-1:(a + b)2 = 1a2 + 2ab + 1b2
(a + b)3 = 1a3 + 3a2b +3ab2 +1b3
(a + b)4 = 1a4 + 4a3b + 6a2b2 +4ab3 + 1b4
(a+b)5=1a5+5a4b+10a3b2+10a2b3+ 5ab4+1b5