Originally Pascal’s Triangle was developed by the Chinese of long ago. But then the French mathematician Blaise Pascal was officially the first person to discover the importance of the patterns it had within itself. But how exactly does it work??? In this research paper, I will explain how to make the Pascal’s Triangle and why it is so special. Construction:

Pascal’s Triangle is basically a triangle of numbers. “At the tip of the triangle is the number 1, which makes up row zero. Then the second row has two 1’s by adding the 2 numbers above them to the left and right, 1 and 0 (all numbers outside the triangle are zeros). Now do the same for the second row.” 0+1= 1, 1+2=3, 2+1=3, 1+0=1. Then the results become the third row. 0+1=1, 1+3=4, 3+3=6, 3+1=4, and 1+0=1. Then the pattern continues on infinitely.

￼

There seems to be many patterns in this triangle. For example: The Sums of the Rows.

The Sums of the Rows:
“The sum of the numbers in any row is equal to 2 to the nth power or 2^n when n represents the number of the row.” For example: 20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16

Prime Numbers:
“If the 1st element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1's) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.”

The Hockey Stick:
“If a diagonal of numbers of any length is selected starting at any of the 1's bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself.” Look at the example on the next page if you don’t understand. ￼

Magic 11’s:
“If a row is made into a single number by using each element as a digit of the number (carrying over when an element itself has more than one digit), the...

...Language comparison of FORTRAN and Pascal
By: Peter Ryan
Richard Zultak
Brendt Lozen
University Of Phoenix
Dan Cohen
POS370
Programming Concepts
Sep 12th 2007
Abstract
The following is a research paper regarding two programming languages called FORTRAN and Pascal.
Pascal [mathematician/philosopher Blaise Pascal] was designed primarily as a tool for teaching good programming skills, but - thanks largely to the availability of Borland's inexpensive Pascal compiler for the early IBM PC - it has become popular outside of the classroom. Unlike many languages, Pascal requires a fairly structured approach, which prevents the kinds of indecipherable "spaghetti code" and easily-overlooked mistakes that plague programmers using languages such as Fortran or C. Free and commercial tools are available from various sources for DOS, Windows, Mac, OS/2, AmigaOS, and Unix-like systems. The web site editor BBEdit is written in Pascal.
Fortran ["FORmula TRANslation"] is the oldest language still in general use, dating back to 1957, the year the Space Age began. It excels at the first task computers were called on for: number-crunching. This is the language that literally put a man on the moon, and some of the features it developed in the process of that project (and other less glamorous ones) have yet to be duplicated in other, more "modern" languages.
Table...

...Pascal’s Triangle
The Pascal’s Triangle is a triangular array of the binomial coefficients. The system after French mathematician Blaise Pascal. The set of numbers that form Pascal's triangle were known before Pascal. However, Pascal developed many uses of it and was the first one to organize all the information together in his treatise, Traité du triangle arithmétique (1653). The numbers originally arose from Hindu studies of combinatorics and binomial numbers and the Greeks' study of figurate numbers.
The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an Ancient Indian book on Sanskrit prosody written by Pingala in or before the 2nd century BC.While Pingala's work only survives in fragments, the commentator Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru". It was also realised that the shallow diagonals of the triangle sum to the Fibonacci numbers. In 1068, four columns of the first sixteen rows were given by the mathematician Bhattotpala, who realized the combinatorial significance.
At around the same time, it was discussed in Persia (Iran) by the Persian mathematician, Al-Karaji (953–1029).It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám...

...
Pascal Programming
Arieus Green
Professor Gary Smith
Sam Houston State University
Pascal was designed in 1968, but was no published until the 1970 by the mind of a man named Niklaus Wirth. Niklaus Wirth was born in Winterthur, Switzerland in 1934 were he attended Swiss Federal Institute of Technology Zurich. Where he soon earns his degree in Electronic Engineering by the mid 1960’s. Pascal was named based off the memory of the late Baise Pascal, a famous French Philosopher as well as a major mathematician (Bill Catambay). This particular language was inspired by Algol along with Simula 67. Although pascal resembles Algol, It far surpasses it in run precision and capabilities. Pascal was designed to be a straight forward block Structured programming. Pascal structurally sound functionality provided the way for several new languages we use today for example, Ada, Java, Modula and so many others. Pascal was design to farther educate the development of a systematically discipline construct.
Pascal was initially designed to influence the practice of good or better program design. The language in particular is an imperative and procedural programming language (Bill Catambay). Imperative programming language simply describes the computation of each term as a statement. This important detail makes it easier to produce...

...irrational
B.
negative and rational
C.
positive and irrational
D.
positive and rational
2]
The value of the polynomial x2 – x – 1 at x = -1 is
[Marks:1]
A.
Zero
B.
-1
C.
-3
D.
1
3]
The remainder when x2 + 2x + 1 is divided by (x + 1) is
[Marks:1]
A.
1
B.
4
C.
-1
D.
0
4]
In fig., AOB is a straight line, the value of x is:
[Marks:1]
A.
60°
B.
20°
C.
40°
D.
30°
5]
The number of line segment determined by three given non - collinear points is:
[Marks:1]
A.
Two
B.
infinitely many
C.
Four
D.
Three
6]
The area of a right triangle with base 5 m and altitude 12 m is
[Marks:1]
A.
50 m2
B.
15 m2
C.
9 m2
D.
30 m2
7]
Evaluate: 53 - 23 - 33
[Marks:1]
A.
80
B.
60
C.
120
D.
90
8]
The area of an equilateral triangle of side 14 cm is
[Marks:1]
A.
B.
C.
D.
9]
Simplify:
[Marks:2]
10]
Check whether (x + 1) is a factor of x3 + x + x2 + 1.
[Marks:2]
11]
In fig., OQ bisects AOB. OP is a ray opposite to ray OQ. Prove that POA=POB.
OR
In fig., AOC and BOC form a linear pair. If a - b = 80°, find the values of a and b.
[Marks:2]
12]
[Marks:2]
13]
The perpendicular distance of a point from the x - axis is 2 units and the perpendicular distance from the y - axis is 5 units. Write the coordinates of such a...

...
5C Problems involving triangles
cQ1. The diagram shows a sector AOB of a circle of radius 15 cm
and centre O.
The angle at the centre of the circle is 115.
Calculate (a) the area of the sector AOB.
(b) the area of the shaded region. (226 , 124
nQ2. Consider a triangle and two arcs of circles.
The triangle ABC is a right-angled isosceles
triangle, with AB = AC = 2.
The point P is the midpoint of [BC].
The arc BDC is part of a circle with centre A.
The arc BEC is part of a circle with centre P.
(a) Calculate the area of the segment BDCP.
(b) Calculate the area of the shaded region BECD.
cQ3. In the following diagram, O is the centre of the circle
and (AT) is the tangent to the circle at T.
If OA = 12 cm, and the circle has a radius of 6 cm,
ﬁnd the area of the shaded region.
cQ4. The diagram shows a circle, centre O, with a radius 12 cm.
The chord AB subtends at an angle of 75° at the centre.
The tangents to the circle at A and B meet at P.
(a) Using the cosine rule, show that the length of AB
is
(b) Find the length of BP.
(c) Hence find
(i) the area of triangle OBP;
(ii) the area of triangle ABP.
(d) Find the area of sector OAB.
(e) Find the area of the shaded region.
Miscellaneous Problems
Q5. The diagram below...

...Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle.It is also known as the figurate triangle, the combinatorial triangle, and the binomial triangle. Even though Pascal’s Triangle is named after seventeenth century mathematician, Blaise Pascal, several other mathematicians knew about and applied their knowledge of the triangle hundreds of years before the birth of Pascal in 1623. The Arabs, Persians and the Chinese discovered Pascal’s triangle in earlier centuries.
The earliest depictions of a triangle of binomial coefficients occur in the tenth century in commentaries on the Chandas Shastra. The Chandas Shastra was an ancient Indian book on Sanskrit prosody written by Pingala between the fifth and second centuries BC. Though Pingala's work was found in fragments, Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru". It was also realised that the shallow diagonals of the triangle were sums to the Fibonacci numbers. The Indian mathematician Bhattotpala later gave rows 0-16 of the triangle.
Around the same time, the traingle was discussed by mathematicians in Persia, Al-Karaji,and mathematician Omar Khayyám, reffered to the traingle as the "Khayyam triangle" in Iran. Many...

...Research Paper on Pascal’s Law
Blaise Pascal’s findings and contributions to the behavior of fluid in an enclosed space have been an invaluable and important concept in fluid mechanics and its applications especially in the automotive industry, mechanical engineering, and hydraulics.
Pascal's law or the principle of transmission of fluid-pressure that was proposed by Blaise Pascal. According to Bloomfield, the law is a principle in fluid mechanics that states that for a particular position within a fluid at rest, the pressure is the same in all directions.
Pascal's principle is defined as
A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid
Reference List
Bloomfield, Louis (2006). How Things Work: The Physics of Everyday Life (Third Edition). John Wiley & Sons.
This principle is stated mathematically as:
is the hydrostatic pressure (given in pascals in the SI system), or the difference in pressure at two points within a fluid column, due to the weight of the fluid;
ρ is the fluid density (in kilograms per cubic meter in the SI system);
g is acceleration due to gravity (normally using the sea level acceleration due to Earth's gravity in metres per second squared);
is the height of fluid above the point of measurement, or the difference in elevation between the two points within the fluid column (in metres in SI).
The intuitive explanation of this formula is that the change in...

...Blaise Pascal
“There are two types of minds - the mathematical, and what might be called the intuitive. The former arrives at its views slowly, but they are firm and rigid; the latter is endowed with greater flexibility and applies itself simultaneously to the dive.” From childhood he was a scientific prodigy. Just from this quote of his you can tell that even his mind in itself can fathom things that none of us even think about on a daily basis. BlaisePascal was born June 19, 1623 in Clermont, France. He was third born out of four children and was Etienne Pascal’s, the father, only son. But at only three years old, Blaise’s mother died, leaving the four children up to Etienne Pascal. In 1632 the Pascal family moved to Paris, France. Blaise’s father had unorthodox educational views and decided to teach his son himself. He said that Blaise was not allowed to study math or science before the age of fifteen. But of course it was impossible to keep his son’s mind away from those two subjects. At just age twelve he started to work on geometry by himself, and before long he realized that the sum of the angles of a triangle are two right angles. When his father found out about this, he gave in and allowed Blaise to have a copy of Euclid.
At age fourteen, Blaise started to attend his father’s meetings. While there he met Girard Desargues and at age fifteen came to admire his work. In June of 1639, Blaise...