What is Pascal’s Triangle?
Pascal’s Triangle was named after Blaise Pascal. Pascal’s triangle starts with the number 1 and goes down the scale. When you start with one, add more numbers in a triangular shape, like a pyramid of some sort. All the numbers on the surrounding right and left sides of the triangle are one. The insides of the triangle are then filled out by finding the sum of the two numbers above it to its left and right (Hosch, 2009, Pierce, 2014). The formula for Pascal’s Triangle is usually written in a form “n choose k” which looks like this: (Pierce, 2014). Pascal’s Triangle is also a never ending triangle of equilaterals (Coolman, 2015). The triangle is symmetric to the other side, with means if you divide the triangle in half, the numbers on the left are the exact same numbers on the right (Pierce, 2014). To find the numbers inside of Pascal’s Triangle, you can use the following formula: nCr = n-1Cr-1 + n-1Cr. Another formula that can be used for Pascal’s Triangle is the binomial formula.
What is the Binomial Theorem?
The binomial theorem is used to find coefficients of each row by using the formula (a+b)n. Binomial means adding two together. According to Rod Pierce, binomial theorem is “what happens when you multiply a binomial by itself… many times.” (2014.) Another way of finding a solution is using binomial distribution, which is like playing a game of heads and tails. The formula for binomial distribution is: .
The binomial formula is (a+b)n. The more complex version would be:
As you can see, the binomial formula equals the “n chooses k” formula (Pierce, 2014). Binomial Distribution has to do with Pascal’s Triangle in the sense that when the nth row (from (a + b)n) is divided by 2n, that nth row becomes the binomial distribution.
Coin Tosses in Relation to Binomial Theorem
When tossing a coin, there are two possible results, head or tails. There is a ½ chance of getting heads and a ½ chance of getting tails. In the event that we flip two coins, there are four (three) conceivable results. We may get two heads, or two tails, or one head and one tail (x2). The possibility of getting two heads is one out of four, or ¼. The shot of getting two tails is ¼. The shot of getting one head and one tail is two out of four, or ½ (Spencer, 1989). As shown in the table below, the toss would represent the row in Pascal’s Triangle.
The heads and tails method for row one is like flipping two coins and getting two results. The first row is organized, 1, for getting a tails, another 1 for getting heads, and 2 for the number of coins, as explained earlier gets the order of the first row, 1 2 1.
~ Heads/Tails Chart/Diagram
Other Patterns in Pascal’s Triangle
The coin toss might be one pattern, but there are others. Some others are the “horizontal sums” (Pierce, 2014). The horizontal sums pattern is adding up the numbers in each row and getting their sums. If you keep doing this, you see the pattern where the sum doubles at each row (Pierce, 2014).
Another pattern is the “exponents of 11” pattern (Pierce, 2014). In this pattern, first, you raise 11 to 0 (110), then you raise it to the numbers after 0 (for example, 110, 111, 112, 113…). The way this relates to Pascal’s Triangle is that 110 = 1, and the number in the first row in Pascal’s Triangle is 1. 111 = 11, and the numbers in the second row are 1, 1. 112 = 121, and the numbers in the third row are 1, 2, 1. This goes on so on and so forth (Pierce, 2014).
You might be looking at the picture above and wondering, “What do I do when it comes to the power of 5?” What you do it you would need to overlap and add the numbers into the box before the number. The picture to the far left shows visuals.
Pascal’s Triangle: Origins and Evolution
Some might think that Pascal’s triangle was discovered by Blaise Pascal, but they would be incorrect. Pascal’s triangle first appeared, in print, on the title page for the Arithmetic of Petrus Apianus in 1527 which was before Pascal was born. Pascal’s triangle properties were first composed by Chinese mathematician, Jia Xian, in the 11th century. His triangle was further studied and developed, making more widely known, by another Chinese Mathematician, Yang Hui, in the 13th century. Because the triangle was popularized by Yang Hui, it is usually titled Yanghui triangle in China. (See attached page.) Yang Hui made his own version of Pascal’s Triangle by using rod numerals. Rod numerals were a number system used by the ancient Chinese. They called it “筹” (pinyin: chóu). (They were small bars about 3-14 cm long) (Hosch, 2009).
~Yang Hui Triangle
Another way that Pascal’s Triangle had evolved was from 20th century Polish mathematician, Wacław Sierpiński. Sierpiński further developed the thoughts of all the mathematicians that let to a fractal known as the Sierpinski gadget (photo at the far left) (Hosch, 2009).
Conclusion
There are many things to know about Pascal’s Triangle. (This one idea branches off into many more topics not mentioned in this paper.) Pascal’s Triangle was not just the work of one man, but the work of many men on the same idea. This shows that, in a sense, teamwork makes the dream work. In this case, the team was the mathematicians using each others slightly more developed thesis, and the dream was finally coming to the conclusion of what is now known as Pascal’s Triangle
Bibliography
Pierce, R. (2014, November 1). Pascal’s Triangle. Math Is Fun. Retrieved November 27, 2015, from http://www.mathsisfun.com/pascals-triangle.html
Pierce, R. (2014, January 25). Binomial Theorem. Math Is Fun. Retrieved November 26, 2015, from http://www.mathsisfun.com/algebra/binomial-theorem.html
Spencer, D. (1989). Potpourri. Invitation to Number Theory with Pascal (pp. 216-219). Ormond Beach, Florida: Camelot Publishing Co.
Payne, T., Ph.D. (n.d.). Binomial Distribution: Definition, Formula & Examples. Retrieved November 27, 2015, from http://study.com/academy/lesson/binomial-distribution-definition-formula-examples.html (Binomial Theorm Picture)
Hosch, W. (2009, August 11). Pascal’s Triangle. Retrieved November 29, 2015, from http://www.britannica.com/topic/Pascals-triangle
Coolman, R. (2015, June 17). Properties of Pascal’s Triangle. Retrieved November 28, 2015, from http://m.livescience.com/51238-properties-of-pascals-triangle.html
Discovering Patterns. (n.d.). Retrieved January 4, 2016, from http://mathforum.org/workshops/usi/pascal/mo.pascal.html (Cover Page Picture)
McCallum, A. (2011, February 26). Chocolate Pretzel Counting Rods. Retrieved November 29, 2015, from http://annmccallumbooks.com/chocolate-pretzel-counting-rods/ (Rods Photo)
Boost C Libraries. (n.d.). Retrieved January 4, 2016, from http://www.boost.org/doc/libs/1_38_0/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html (GRAPH PICTURE)