Pascal's Triangle

History

Blaise Pascal was born at Clermont-Ferrand, in the Auvergne region of France on June 19, 1623. In 1653 he wrote the Treatise on the Arithmetical Triangle which today is known as the Pascal Triangle. Although other mathematicians in Persia and China had independently discovered the triangle in the eleventh century, most of the properties and applications of the triangle were discovered by Pascal. This triangle was among many of Pascal’s contributions to mathematics. He also came up with significant theorems in geometry, discovered the foundations of probability and calculus and also invented the Pascaline-calculator but he is best known for his contributions to the Pascal triangle.

Construction of the Triangle

The easiest way to construct the triangle is to start at row zero and write only the number one. From there in order to obtain the numbers in the following rows, add the number directly above and to the left of the number with the number above and to the right to acquire the new value. If there are no numbers on the left or the right just replace a zero for that missing number and proceed with the addition. Here is an illustration of rows zero to four.

Pascals triangle

Definition of Pascal’s Triangle

Most people are introduced to Pascal’s triangle by means of an arbitrary-seeming set of rules. Begin with a 1 on the top and with 1’s running down the two sides of a triangle. Each additional number lies between 1 two numbers and below them, and its value is the sum of the two numbers above it. The theoretical triangle is infinite and continues downward forever, but only the first 6 lines appear in figure 1. In the figure, each number has arrows pointing to it from the numbers whose sum it is. More rows of Pascal’s triangle are listed on the final page of this article. A different way to describe the triangle is to view the first line is an infinite sequence of zeros except for a single 1. To obtain successive lines,

The theoretical triangle is infinite and continues downward forever, but only the first 6 lines appear in figure 1. In the figure, each number has arrows pointing to it from the numbers whose sum it is. More rows of Pascal’s triangle are listed on the final page of this article. A different way to describe the triangle is to view the first line is an infinite sequence of zeros except for a single 1. To obtain successive lines,

A different way to describe the triangle is to view the first line as an infinite sequence of zeros except for a single 1. To obtain successive lines, add every adjacent pair of numbers and write the sum between and below them. The non-zero part is Pascal’s triangle.

Patterns & Applications

1) Addition of the Rows: One of the interesting properties of the triangle is that
the sum of its rows is equal to 2n

where n corresponds to the number of the row:

  1 = 1 = 20

  1 + 1 = 2 = 21

  1 + 2 + 1 = 4 = 22

  1 + 3 + 3 + 1 = 8 = 23

  1 + 4 + 6 + 4 + 1 = 16 = 24

2) Prime Numbers in the Triangle: Another pattern visible in the triangle deals with prime numbers. If a row starts with a prime number or is a prime numbered row, all the numbers that are in that row (not counting the 1’s) are divisible by that prime. If we look at row 5 (1 5 10 10 51), we can see that 5 and 10 are divisible by 5. However, for a composite numbered row, such as row 8 (1 8 28 56 70 56 28 8 1), 28 and 70 are not divisible by 8.

3) Fibonacci Sequence in the Triangle: By adding the numbers in the diagonals of the Pascal triangle the Fibonacci sequence can be obtained as seen in the figure given below.

Fibonacci series

There are various ways to show the Fibonacci numbers on the Pascal triangle. R. Knott was able to find the Fibonacci appearing as sums of “rows” in the Pascal triangle. He moved all the rows over by one place and here the sums of the columns would represent the Fibonacci numbers.

Some Patterns in Pascal’s Triangle

  • Each number is the sum of the two numbers above it.
  • The outside numbers are all 1.
  • The triangle is symmetric.
  • The first diagonal shows the counting numbers.
  • The sums of the rows give the powers of 2.
  • Each row gives the digits of the powers of 11.
  • Each entry is an appropriate “choose number.”
  • And those are the “binomial coefficients.”
  • The Fibonacci numbers are in there along diagonals.

Here is a 18 lined version of the pascals triangle;

Pascal's Triangle


Practise This Question

Multiplying a given integer by 1 gives