Wednesday, March 11, 2009
Wednesday Math, Vol. 63: Patterns in Pascal's Triangle
I talked last week about my first website, Pascal's Triangle From Top To Bottom. There are a lot of patterns in the numbers, which are known as identities. There are literally hundreds, possibly thousands of identities using the numbers, but I want to show two, and give what I hope are simple proofs of both.
The numbers are known as the binomial coefficients. While that sounds intimidating, it just means that if you raise a math term like (x + y) or (2a - 1) or (3mn - k) to the power of 4, for example, the numbers in the fourth row will be contained in the coefficients of the expanded polynomial. Each of the things inside the parentheses is called a binomial because they consist of two terms, either added together or one subtracted from the other. (The subtraction could be thought of as adding a negative to a positive, and that way all of them would be cases of adding.)
In the first five instances, rows 0 through 4, the sum of the numbers is equal to 2 raised to the power of the row number. For example, 1+4+6+4+1 = 16, and 16 = 2*2*2*2. This pattern is not just coincidence, it continues on in all the rows of Pascal's Triangle indefinitely. The reason is fairly simple. If instead of expanding the general binomial (x + y), change that to (1 + 1). All the x and y terms, raised to various powers, become 1 raised to powers, and 1 raised to any power is 1. All we will be left with is the sum of the binomial coefficients in that row, 1+4+6+4+1, and since 1+1 = 2, the left side of the equation is 2 to the fourth power.
Okay, here's another pattern. If the binomial is (x - y), all that changes is the plus and minus signs alternate. In the third row, we would get 1 - 3 + 3 - 1 = 0. In the fourth row, the alternating sum is 1 - 4 + 6 - 4 + 1 = 0. Once we get past row 0, the alternating sum of every row of Pascal's Triangle is zero.
The method of proof is the same as above, but instead of 1+1 = 2, we use 1 - 1 = 0. Zero raised to any power is 0, except 0 to the zeroth power, which mathematicians have decided is undefined, since it could be 0 or it could be 1, depending on what rule you want to invoke.
I realize that this is more math than most people want to follow, but these kinds of proofs are the Holy Grail for mathematicians. To find patterns that are always true and to prove why they always work is what we get paid for, and we get paid in the brain and in the heart as well as in the pocketbook when they work really nicely.