Helpful Tools for Calculus, Chapter 2: Helpful Tools for the Mathematics of Calculus
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Pascal’s Triangle
A common type of problem in mathematics is that of expanding a binomial (a+b)n. Beginning algebra students are shown how to distribute, FOIL and expand polynomials through multiplication, but as we get further into mathematics some of the problems get more challenging.
One helpful trick is to use Pascals Triangle to expand a binomial. Let us first construct Pascals Triangle using the following technique. Begin with a 1 in the center, this is row 0. The next row put two 1’s diagonally away from the first 1 in a triangular shape, this is row 1. For the next row and all of the following rows, begin with a 1 diagonally below and to the left of the previous row. Each additional term is defined by adding the two terms above until you get to the end of the row which ends with 1. Every row should add one new term and always begin and end with a 1.
Binomial Expansion
But, how does this help with algebra? The answer is that these numbers represent the coefficients when expanding a binomial. For (a+b)n we take n as the row number. Then each number in that row represents to coefficient for a different term. To construct the terms begin by taking an b0 then for each term decrease the exponent of a by 1 and increase the exponent of b by 1. By the last term we should have a0 bn. Recall that anything, except 0, raised to the 0th power is equal 1, so you can simplify.
Example 1: Expand the following binomial:
( x+y )5
Solution:
The exponent is 5 so we should refer to the row whose second number is 5: 1 5 10 10 5 1.
These form the coefficients of the expansion:
1x5 y0+5x4 y1+10x3 y2+10x2 y3+5x1 y4+1x0 y5
We often omit the variables whose exponent is 0:
x5+5x4 y+10x3 y2+10x2 y3+5xy4+y5
Example 2: Expand the following binomial:
( 2a-b )4
Solution:
The exponent is 4 so we should refer to the row whose second number is 5: 1 4 6 4 1.
These form the coefficients of the expansion:
Then we substitute the values of x and y:
Simplify:
Other Uses of Pascals Triangle and Fun Facts
The reason that this triangle is so famous is because of the variety of applications, patterns, and curiosities that arise from it. Here we will name a few, though most of this section won’t have much use in this text.
The sum of each row is representable by 2^n where n is the number of the row, beginning with 0. This means that each row’s sum is twice that of the previous row.
The ratio of the products of each row is related to Euler’s number, e!
The diagonals represent the number of points required to make a triangular shape of that many dimensions. For example, 1 3 6 10 15… represents the number of points to make a two-dimensional triangle. These are called triangular numbers:
1 4 10 20… represents the number of points to make a three-dimensional triangle or pyramid, called a tetrahedron. So, these are called tetrahedral numbers:
This pattern continues!
Additionally, if we made a grid with a square for each number in the triangle, then shaded only the odd numbers we would create the Sierpinski triangle, which is a fractal:
Pascal’s triangle can be used to construct the Fibonacci sequence, π and other terms too!
A common use in combinatorics and probability is that each term in the triangle represents a combination, described as n choose k. These are written as
and help to describe the number of ways in which a situation could happen, given some cases. This value can be solved from the triangle by going to the kth spot in the nth row. For example,
and says there are ten ways to choose three objects from a list of five.
Historical Information
Pascal’s Triangle, like many named things, does not actually come from the mathematician Blaise Pascal. In 1654, Pascal was working on his textbook called Conics when a friend of his, named Chevalier de Méré who was a French nobleman and gambler, asked him about variations in rolling a die. Pascal, working with Pierre de Fermat, sort of created the study of probability in attempting to answer these questions and others. However, much of this actually belonged to others long before Pascal even existed.
The arithmetic triangle, now called Pascal’s Triangle, is often called Tartaglia’s Triangle in Italy, named after the mathematician Tartaglia from the 1500’s. Even before him, the triangle was constructed and used by Chinese mathematicians, like Jia Xian and Yang Hui, as early as the year 1010. Prior to these, the Persian mathematician Al-Karaji wrote a book some 50 years earlier. As a result of this, and its repetition by another Persian mathematician, it is sometimes called Khayyam’s triangle in Iran.
The main reason that Pascal’s triangle is referred to by that name so often is because Blaise Pascal discovered and published many new things about it, though he did not create it.