Pascal’s triangle is a visual model of numbers known as binomial coefficients. Simply put, each number in the interior of the triangle is the sum of the two directly above it. The exterior border is always a series of 1’s. It is a common beginning topic in number theory, and understanding the structure of the triangle facilitates more advanced applications such as the binomial theorem.
It is believed that the first set of this famous triangle was developed in China in the B.C. era. The French mathematician Blasé Pascal introduced it to the Western world in the 1600’s. The single number 1 at the very top of triangle represents what is called the zeroth row. The first row is directly below and contains two 1’s. Moving diagonally downward, the outward border of the triangle is only ever lined with 1’s. This is true regardless of how big or complex the inner numbers become as the triangle widens.
Formulas have been created to locate specific number in Pascal’s triangle with the use of factorials, represented by an exclamation point (!) To apply these, mathematicians assign variables as follows: n = number and r = row. The syntax for this formula reads “number Choose row”, or nCr. The exact formula for nCr reads: n! divided by r! multiplied by the number subtracted from the row enclosed in parentheses. Recall that a factorial following any number represents the number multiplied by every number between itself and 1. This is also known as the systematic calculation of combinations.
Pascal’s triangle is also instrumental in understanding the concept of the sums of rows and of locating specific prime numbers. It is also possible to locate the famous Fibonacci sequence in the triangle by adding together numbers across rows from left to right. It almost seems like magic. Other concepts that mathematicians have derived from triangle include patterns of squares and polygons that occur with astounding regularity, no matter how large or small the numbers can get in either direction.
Numerous graphical designs have also been based on Pascal’s triangle, and many of them are quite elaborate and their concepts are applied to many areas. Some scientists have calculated models of this triangle with the largest numbers exceeding eight or nine digits by themselves. Newer computer technology has been applied to create three-dimensional models of the triangle and has also modeled it into what is known as Pascal’s tetrahedron, a pyramid-like structure that has many different architectural applications.
Understanding the concepts behind the triangle is essential to the study of various mathematical subjects such probability theory and various topics in math analysis. Most of us recall from algebra class an introduction to the expansion of (x+y) ^2 = x^2 +2xy + y^2. The actual mechanics of this expansion come from the patterns found in Pascal’s triangle; these patterns actually predict which coefficients would come next if this expression were to be taken a step further. The result would be 1x^2y^0 +2x^1y^1+1x^0y^2.
When the coefficient n is substituted for the square, the result of the expansion matches up to the exact numbers in a given specific row of the triangle; this row is again designated by the variable r. This principle can be combined with combination calculations in order to determine the location of diagonal rows as well within the triangle, based on at least two given numbers. Creating certain diagonal patterns in Pascal's triangle also introduces the concept of fractals, which are defined as split geometric shapes that can be formed into parts that are simply reductions in size of the whole original shape.
An appealing aspect of this famous triangle is that it can be made as simplified or complex as needed in order to explain any of these above-described mathematical concepts at different educational levels, ranging from primary to college. It can be used to teach things as simple as addition and multiplication or things that complex as cellular structures in advanced science classes and in complex game design theory.