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Pascal triangle and combinations
Pascals triangle and combinations
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Candidate name: Sakariye Abdirizak
Subject: Mathematics SL
Candidate number: 000511-0073
School: Hvitfeldtska Gymnasiet
Patterns in Pascal’s Triangles
I decided to do my math exploration internal assessment on the different patterns in the Pascal triangle. My aim is to discover and elaborate on the many different patterns exhibited in the Pascal’s triangle. One of the main reasons this choice of topic spoke to me is because it relates to a lot of things we do in math class such as Pascal’s triangle, probability, sequences and series, binomial theory, and negative coefficients. Another reason I choose this topic because I am very interested in patterns and I find myself intrigued by patterns and puzzles. It will be interesting to see what I discover.
Blaise Pascal was a French philosopher and mathematician who lived in the 1600s. He is known for inventing the calculator and it is he whom the Pascal’s triangle was named after.
What is the Pascal’s triangle?
The top of the triangle is the number 1 and each new row below contains a number more than the line above. The additional numbers determined by the sum of the numbers to the left and right of the row above. If there is a not figure both to the left and right in the line above then the number the same as the one 's to the left or right in the line above. This means that each line starts and ends with the number 1. As shown in figure 1 .
Figure1
So this is how a Pascal’s triangle looks like. It is mainly used for algebra but what is unique about the Pascal’s triangle is its diversity. As you can see it begins with a 1 at the top and then after shows the coefficients for (a+b)n for all n ≥ 1. You could also say that the ...
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... row. Although the mathematics of it could be easily grasped I think it’s strange the occurrence of this and the pattern that is present. Lastly the Pascal’s triangle could be used as a helpful indicator to how many segments, triangles and such there are in a circle with nth points. This is useful as you can just use the triangle to reach the conclusions, in which otherwise you would have to draw the circle and find out the longer way. To conclude, I feel like this exploration developed my mathematical curiosity, and I appreciate the usefulness and beauty of mathematics. I realized the multi-dimensions of the Pascal’s triangle and its historical perspectives. Lastly, it taught me to explore and analyze patterns and problems and I think that will be useful in both school and real life-situations, because math is a big part of the world we live in.
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.
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”.
I also learned that mathematics was more than merely an intellectual activity: it was a necessary tool for getting a grip on all sorts of problems in science and engineering. Without mathematics there is no progress. However, mathematics could also show its nasty face during periods in which problems that seemed so simple at first sight refused to be solved for a long time. Every math student will recognize these periods of frustration and helplessness.
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.
It is amazing to see how mathematics has such an influence on the world and the evidence it creates. The world is affected by numbers and mathematics all the time and this mysterious number known as the golden number has proven to be the center of everything.
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...
“Class,” I announced, “today I will teach you a simpler method to find the greatest common factor and the least common multiple of a set of numbers.” In fifth grade, my teacher asked if anyone had any other methods to find the greatest common factor of two numbers. I volunteered, and soon the entire class, and teacher, was using my method to solve problems. Teaching my class as a fifth grader inspired me to teach others how important math and science is. These days, I enjoy helping my friends with their math homework, knowing that I am helping them understand the concept and improve their grades.
Leonhard Paul Euler was born the son of a pastor on April 15, 1707 in Basel, Switzerland. Soon after he was born, his family moved to Riehen, where Leonhard would spend most of his childhood. Leonhard’s father, Paul, was good friends with the Bernoulli family, whose patriarch, Johann Bernoulli, was then viewed as Europe’s leading mathematician. Bernoulli would eventually become a great influence on Leonhard’s life. When Leonhard was thirteen, he was sent to live with his maternal grandmother in Basel, where he enrolled in the University of Basel and eventually earned his Master’s in Philosophy, and wrote his dissertation comparing the philosophies of Newton and Descartes. Euler was following in his father’s footsteps, studying theology, Greek, and Hebrew, and was determined to become a pastor. However, Johann Bernoulli was convinced Euler was destined to become a great mathematician, and talked Paul Euler into letting his son pursue his own passio...
To investigate the notion of numeracy, I approach seven people to give their view of numeracy and how it relates to mathematics. The following is a discussion of two responses I receive from this short survey. I shall briefly discuss their views of numeracy and how it relates to mathematics in the light of the Australian Curriculum as well as the 21st Century Numeracy Model (Goos 2007). Note: see appendix 1 for their responses.
Therefore the sequence 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, is called a recursive sequence. When the recursive numbers are arranged in a certain way, this sequence creates a spiral pattern and this pattern is reflected in various places in real life (nature).
Fibonacci was born in approximately 1175 AD with the birth name of Leonardo in Pisa, Italy. During his life he went by many names, but Leonardo was the one constant. Very little is known of his early life, and what is known is only found through his works. Leonardo’s history begins with his father’s reassignment to North Africa, and that is where Fibonacci’s mathematical journey begins. His father, Guilielmo, was an Italian man who worked as a secretary for the Republic of Pisa. When reassigned to Algeria in about 1192, he took his son Leonardo with him. This is where Leonardo first learned of arithmetic, and was interested in the “Hindu-Arabic” numerical style (St. Andrews, Biography). In 1200 Leonardo ended his travels around the Mediterranean and returned to Pisa. Two years later he published his first book. Liber Abaci, meaning “The Book of Calculations”.
If you have ever heard the phrase, “I think; therefore I am.” Then you might not know who said that famous quote. The author behind those famous words is none other than Rene Descartes. He was a 17th century philosopher, mathematician, and writer. As a mathematician, he is credited with being the creator of techniques for algebraic geometry. As a philosopher, he created views of the world that is still seen as fact today. Such as how the world is made of matter and some fundamental properties for matter. Descartes is also a co-creator of the law of refraction, which is used for rainbows. In his day, Descartes was an innovative mathematician who developed many theories and properties for math and science. He was a writer who had many works that explained his ideas. His most famous work was Meditations on First Philosophy. This book was mostly about his ideas about science, but he had books about mathematics too. Descartes’ Dream: The World According to Mathematics is a collection of essays talking about his views of algebra and geometry.
Deep within the realm of fractal math lies a fascinating triangle filled with unique properties and intriguing patterns. This is the Sierpinski Triangle, a fractal of triangles with an area of zero and an infinitely long perimeter. There are many ways to create this triangle and many areas of study in which it appears.
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.