Pascal’s Triangle is a visual represenation a series of binomial expansions. The triangle emerges as a result of the function (x + y) ^n where n is an integer greater than or equal to zero. As n increases, the quantity of terms in the result increases:
1. (x + y)^0 = 1………………………………………………………………………………. one term
2. (x + y)^1 = x + y………………………………………………………………………… two terms
3. (x + y)^2 = x^2 + 2xy + y2……………………………………………………………. .three terms
Additionally, the integers represented on the triangle are found as the coefficients of the expansion. For instance, the third row of Pascal’s Triangle (0, 1, 2) is “1 2 1”, which corresponds with the coefficients of 3 above. The triangle itself is simply a visualization of this pattern.
The discovery of Pascal’s Triangle is widely considered to have taken place centuries prior to Pascal’s lifetime. However, Blaise Pascal was the first to publish the triangle, which he did in his 1654 work, “Traite du Triangle Arithmetique”. (Kazimir, 2014) Originally, interest in the concept was confined to gamblers as it provided a convenient method to calculate probabilities of an event such as a coin flip or game of dice. (Witchita, 2014) Since then, interest has tended toward the methods and applications of the triangle. The methods of use are essentially number theoretical and the applications are wide. Many fields such as algebra, probability, and combinatorics may find use in Pascal’s Triangle, and additional applications include identifying number sequences such as triangular and tetrahedral numbers.
Each application of Pascal’s Triangle can be solved using all methods available. The convenience of the triangle then is not necessarily in any particular method of examination, but in the variety of method...
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...head: head, tail; or tail, head. This yields a 2/4 or 50% chance. Using Pascal’s Triangle for 10 flips, we found the sum of the elements of the 10th row equals 1024. Using combinations, (10 C 5) = 252. Given these results, there are exactly 252 ways to have exactly 5 heads in 10 flips of a coin or (252/1024) = 24.6% chance. Our application of Pascal’s Triangle had given a result, which to some would be a priori counterintuitive.
It is clear to us that Pascal’s Triangle is a very interesting source of useful information. The study of binomial expansions has proven to be a fount of interesting patterns and its allure is the suggestion of more hidden within it. We hope that this cursory examination provides insight for those unfamiliar with Pascal’s Triangle and a renewed interest in those who have experience with it. That is the result which it had on these authors.
On the second day of class, the Professor Judit Kerekes developed a short chart of the Xmania system and briefly explained how students would experience a number problem. Professor Kerekes invented letters to name the quantities such as “A” for one box, “B” for two boxes. “C” is for three boxes, “D” is for four boxes and “E” is for five boxes. This chart confused me because I wasn’t too familiar with this system. One thing that generated a lot of excitement for me was when she used huge foam blocks shaped as dice. A student threw two blocks across the room and identified the symbol “0”, “A”, “B”, “C”, “D”, and “E.” To everyone’s amazement, we had fun practicing the Xmania system and learned as each table took turns trying to work out problems.
The 2's and 5's were arranged in such a way that one number formed a distinct shape in the midst of the jumble of the other number. A non-synesthetic would be incapable of distinguishing any pattern due to the close resemblance of the numbers. But, in 90% of the cases where people claimed to see colors they were easily able to discern the shape because it registered stood out for them as a completely different color.
Blaise Pascal was born on 19 June 1623 in Clermont Ferrand. He was a French mathematician, physicists, inventor, writer, and Christian philosopher. He was a child prodigy that was educated by his father. After a horrific accident, Pascal’s father was homebound. He and his sister were taken care of by a group called Jansenists and later converted to Jansenism. Later in 1650, the great philosopher decided to abandon his favorite pursuits of study religion. In one of his Pensees he referred to the abandonment as “contemplate the greatness and the misery of man”.
The “Blaise Pascaline,” as referred to in [3] would be considered today as an early version of a calculator. This project derived in part from helping out his father who had been promoted as a tax clerk, a job which required him to perform long calculations at work. Only one other mechanical device was known to add up figures before the Pascaline and that was known as the Schickard's calculating clock, created by German professor Wilhelm Schickard. Unlike Schickard device, Pascal’s calculator had a larger number of production and use despite the somewhat unreliability of the device. The device consisted of a wheel with eight movable parts for dialing and each part corresponding to a particular digit in a number. It worked by using gears and pins to add integers; addends were entered by hand and carriers from one column to the next were broadcast internally by falling weights lifted and dropped by the pins attached to the gears. It could even be manipulated to subtract, multiply and divide if one knew their way around the Pascaline. Subtraction was done by adding the nines complement version of the number being subtracted. Multiplication; accomplished by repeating additions and division performed by repeating subtractions. Balise Pascals went on to inspire directly inspired further work on calculating machines by other inventors such as Gottfried Leibniz and Samuel
Blaise Pascal was born on June 19, 1623. Pascal was a mathematician along with a Christian philosopher who wrote the Pensees which included his work called Pascal’s wager. The crucial outline of this wagers was that it cannot be proved or disprove that God does exists. There are four main parts to the wager that include his reasoning to that statement. It has been acknowledged that Pascal makes it clear that he is referring to the Christian God in his wager. This is the Christian God that promises his people will be rewarded with eternal life along with infinite bliss.
Wigner, Eugene P. 1960. The Unreasonable Effectiveness of Mathematics. Communications on Pure and Applied Mathematics 13: 1-14.
The Fibonacci sequence was introduced as a problem involving population growth based on assumptions. Fibonacci got the idea from early Indian and Arabian Mathematics. He grew the theory and introduced it to the western world. The sequence is explained by starting at 1, 2 then adding the two t...
The great field of mathematics stretches back in history some 8 millennia to the age of primitive man, who learned to count to ten on his fingers. This led to the development of the decimal scale, the numeric scale of base ten (Hooper 4). Mathematics has grown greatly since those primitive times, in the present day there are literally thousands of laws, theorems, and equations which govern the use of ten simple symbols representing the ten base numbers. The field of mathematics is ever changing, and therefor, there is a great demand for mathematicians to keep improving our skills in utilizing the numeric system. Many great people, both past and present, have made great contributions to the field. Among the most famous are Archimedes, Euclid, Ptolemy, and Pythagoras, all of which are men. This seems to be a common trend in mathematics, for almost all classical mathematicians were male.
Blaise Pascal has contributed to the field of mathematics in countless ways imaginable. His focal contribution to mathematics is the Pascal Triangle. Made to show binomial coefficients, it was probably found by mathematicians in Greece and India but they never received the credit. To build the triangle you put a 1 at the top and then continue placing numbers below it in a triangular pattern. Each number is the two numbers above it added together (except for the numbers on the edges which are all ‘1’). There are patterns within the triangle such as odds and evens, horizontal sums, exponents of 11, squares, Fibonacci sequence, and the triangle is symmetrical. The many uses of Pascal’s triangles range from probability (heads and tails), combinations, and there is a formula for working out any missing value in the Pascal Triangle: . It can also be used to find coefficients in binomial expressions (put citation). Another staple of Pascal’s contributions to projective geometry is a proof called Pascal’s theore...
In mathematics, Pascal’s triangle is taught everywhere throughout schools. He also started probability theory that many if not all mathematicians today use. Pascal even changed science by his experiments on atmospheric pressure and later had units of pressure named after him for his study. Pascal also, has a law in physics named after him. His inventions were just as impactful. Pascal created one of the first digital calculators. Pascal also invented the core principles of the roulette machine when study a perpetual motion theory.
One of Polya’s most noted problem solving techniques can be found in “How to Solve it”, 2nd ed., Princeton University Press, 1957.
The Bernoulli family had eight significant and important mathematicians, starting with Jacob Bernoulli, born in 1654. Though there was a great deal of hatred and jealousy between the Bernuollis, they made many remarkable contributions in mathematics and science and helped progress mathematics to become what it is today. For example, Daniel discovered a way to measure blood pressure that was used for 170 years, which advanced the medical field. Daniel’s way of measuring pressure is still used today to measure the air speed of a plane. Without the Bernoulli family’s contributions and advancements to calculus, probability, and other areas of mathematics and science, mathematics would not be where it is now.
Named after the Polish mathematician, Waclaw Sierpinski, the Sierpinski Triangle has been the topic of much study since Sierpinski first discovered it in the early twentieth century. Although it appears simple, the Sierpinski Triangle is actually a complex and intriguing fractal. Fractals have been studied since 1905, when the Mandelbrot Set was discovered, and since then have been used in many ways. One important aspect of fractals is their self-similarity, the idea that if you zoom in on any patch of the fractal, you will see an image that is similar to the original. Because of this, fractals are infinitely detailed and have many interesting properties. Fractals also have a practical use: they can be used to measure the length of coastlines. Because fractals are broken into infinitely small, similar pieces, they prove useful when measuring the length of irregularly shaped objects. Fractals also make beautiful art.
The history of the computer dates back all the way to the prehistoric times. The first step towards the development of the computer, the abacus, was developed in Babylonia in 500 B.C. and functioned as a simple counting tool. It was not until thousands of years later that the first calculator was produced. In 1623, the first mechanical calculator was invented by Wilhelm Schikard, the “Calculating Clock,” as it was often referred to as, “performed it’s operations by wheels, which worked similar to a car’s odometer” (Evolution, 1). Still, there had not yet been anything invented that could even be characterized as a computer. Finally, in 1625 the slide rule was created becoming “the first analog computer of the modern ages” (Evolution, 1). One of the biggest breakthroughs came from by Blaise Pascal in 1642, who invented a mechanical calculator whose main function was adding and subtracting numbers. Years later, Gottfried Leibnez improved Pascal’s model by allowing it to also perform such operations as multiplying, dividing, taking the square root.
A triangle is a shape with a total of three sides. The triangle to me looks like one side of “ The Great Pyramid”. A triangle is a two dimensional figure. In a three-dimensional form, it is a pyramid. I strongly believe that the triangle is the most unique shape of all of the shapes.