Throughout this semester I began to research Pascal's triangle. At first after leaving office hours, I felt that doing 16 hours of research of Pascal's triangle would be nearly impossible. I quickly realized that I had been very, very wrong about the lack of information on the triangle. I already knew that Pascal's triangle contained a great deal of math patterns and sequences, and that it could be used for a few different mathematical purposes. I had no idea to what extent how awesome this triangle actually was. I began by googling "Pascal's triangle patterns" and thought that maybe a few things would pop up. The third site seemed the most interesting, so I chose that. I had expected a few patterns, most of which I had already known, to be displayed before me. I scrolled through the first three and thought "yap! I was right." Then a scrolled through the fourth and looked over it for about ten minutes trying to understand the pattern that was trying to be conveyed. Finally, I got it. I checked the pattern for a few numbers on the top row of the triangle and then moved my way down around the 10th row. The pattern still worked. Now I was pretty certain that the pattern was true, because I would be utterly shocked if I just happened to stumble upon and disprove a well known pattern in Pascal's triangle, but I still had to continue to look. I found Pascal's triangle filled in to the 27th row and checked for some of the lowest numbers. Check! It still works. I kept scrolling and found that there were more patterns than I could even wrap my head around.
What is Pascal's triangle, you might be wondering. As you have probably already gathered, it is a triangle in which numbers are placed in each row. The point of the triangle contains the number one, and is row n = 0. The next row contains two numbers that are staggered under the first, the third row three numbers staggered under the row above, and so on. This triangle can go on forever. To construct the triangle, each number is the sum of the two numbers directly above it. Below I will show the beginning of the construction of the triangle:
1 2 1
1 3 3 1
1 4 6 4 1
You can see that this pattern will continue for as high as one can count. Although this triangle is named after mathematician Blaise Pascal, the set of numbers were known about before Pascal was even around. The triangle was named after Pascal because he was the first to organize the numbers in this way and he also discovered many different ways that the information could be used. The actual idea of these numbers is thought to have first been known through studies of binomial expansion and combinatorics by Indian mathematicians as well as the Greeks. A number of different mathematicians used these numbers before Pascal. But in 1665, Pascal published Traité du triangle arithmétique, which was his version of the triangle. He used this to solve many probability theory problems. The Name Pascal's Triangle was given Pierre Raymond de Montmort, with the name Table de M. Pascal pour les combinaisons (Table of Mr. Pascal for combinations) and by Abraham de Moivre with the name Triangulum Arithmeticum PASCALIANUM (Pascal's Arithmetic Triangle). As we know, the latter is most commonly used now in Western culture.
While I can't even begin to describe all the different ways that the triangle can be used, I will touch on a few that I find particularly interesting and important.
The first use that I find fascinating is binomial expansions. Each row of the triangle contains the coefficients that are found in binomial expansions. For example, when we expand (x + y)^2 we get the following
Notice that the coefficients are 1,2,1 which is row n = 2 on the triangle above. Essentially, this is the binomial theorem in a nutshell.
The second use for Pascal's Triangle that I find interesting is the calculation of combinations. For two numbers n and k, you can find n choose k by looking at the kth element in the nth row. To put in simpler terms, if you have n objects and you are taking k objects at one time, the number of different combinations that are possible is n choose k.
The last feature that I am going to touch on is the many different patterns that are found I the triangle. Each of these can be checked and tested in your free time. (I've spent hours doing so already! :) )
- The sum of every row is twice that of he preceding row.
- The sum of the square of the elements in the nth row is the middle element of the row 2n.
- The diagonals second from the outside contain the natural numbers in order.
- The diagonals next to the natural numbers contain the triangular numbers.
- Next to the triangular numbers are the tetrahedral numbers, and then the pentatope numbers.
- The triangle is symmetrical.
These are just a few of many different patterns that can be found in Pascal's triangle. Further investigating will surely amaze you at all the this set of numbers contains. It sure amazed me.