Running head: PASCAL TRIANGLE
The topic selected for this project is “Pascal’s Triangle”. Pascal's triangle is a
mathematical and triangular array of coefficients derived binomially. The triangle is arranged
with the 0th row (n=0) at the top. The entries of the triangle are made from the left-hand side. In
the first row (row 0), the entries are nonzero, which is entry 1 (Lee et al., 2016). The rest of the
entries are made by adding the number above to the left and the right as well and the blank
entries as zero. Pascal's triangle has been used widely and intensively across the world as it's
used spans a wide array.
Pascal's triangle has been used since early days in mathematical contexts especially in
combinatorics and binomial numbers. The triangle has eased the work of mathematicians
especially in the expansion of functions. Pascal's triangle wide usage over years and its
significance has led to setting up of Pascal's rule (Majumdar, 2017). Pascal's triangle with higher
dimensional generalizations has led to the development of Pascal's tetrahedron, while the general
format which is simple is known as Pascal's simplices (Lee et al., 2016). Iranians and Chinese
have had their variants of the same with most of their works being undocumented or put on
Years after it first appeared in Persia and China, the triangle came to be known as
Pascal’s Triangle with Blaise Pascal’s completion of Traité du triangle arithmétique in 1654.
Making use of the already known array of binomial coefficients, French mathematician Pascal
developed many of the triangle’s properties and applications within these writings. Although
Pascal is best known for his work with the arithmetic triangle, he made many other contributions
to mathematics during his lifetime. Throughout his thirty-nine years, Pascal also discovered an
important theorem in geometry, worked with cycloids, invented a calculating machine, laid the
foundations of probability, and planted the seeds of calculus (Eves 242-6). Pascal’s
contributions to mathematics, especially of ‘his’ triangle, were unquestionably brought forth
from the mind of a highly intelligent man.
In mathematical contexts, Pascal's triangle is used in binomial expansions. By using the
triangle, it becomes easy to expand functions up to nth terms. Expanding polynomials is
simplified as it is easy to find the coefficients (Majumdar, 2017). The second mathematical usage
of Pascal's triangle is in binomial probability distributions, which is important to model the
number of successes in a wide sample size that is drawn with instances of replacement.
Mathematics, especially those on polynomials and probability has been improved with the input
of Pascal (Lee et al., 2016). Leibniz's rule of differentiation has also been built and borrowed so
much from the works of Pascal.
The most widely used example of Pascal's triangle in the real-world application is in
gambling and insurance companies. In these contexts, it is used to find out the different
combinations an object may have by using the nCr formulae (Lee et al., 2016). Pascal's triangle
has played a significant role in present-day mathematics with most mathematicians basing and
advancing their work from what Pascal did. In the real world, most mathematical figures have
been derived from the product of the triangle.
Pascal's triangle defined as the set of numbers which precisely arranged in a triangle
containing a certain amount of patterns...