Patterns In Pascals Triangles

2326 Words5 Pages
bulb-icon

Recommended: Pascal triangle and combinations

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 ...

... middle of paper ...

... 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.

Show More
Related

  • The MIA System: A Short Study Of The Xmania System

    845 Words  | 2 Pages

    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.

    Read More

  • William Clifford And Blaise Pascal

    971 Words  | 2 Pages

    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”.

    Read More

  • Pascal's Wager

    851 Words  | 2 Pages

    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.

    Read More

  • Analysis of Blaise Pascal´s Book: Pensees

    1088 Words  | 3 Pages

    Blaise Pascal lived during a time when religion and science were clashing and challenging previous discoveries and ideas. Pascal lived from 1623 to 1662 due to his untimely death at the age of thirty nine. The scientific community grew enormously and Pascal was a great contributor to this growth. The growth in the scientific community is known as the Scientific Revolution. He lived in a time where an absolute monarch came into power, King Louis the XIV. Louis XIV was a believer in “one king, one law, and one faith” (Spielvogel, 2012). Pascal saw the destruction of protestant practices in France and the growth and acceptance of scientific discoveries. He used the scientific method to refine previous experiments that were thought to be logical but Pascal proved otherwise and eventually led to Pascal’s Law. He spent his life devoted to two loves: God and science. Within his book, “Pensees,” Pascal argues and shares his thoughts about God, science, and philosophy.

    Read More

  • College Admissions Essay: Expressing Creativity through Mathematics

    461 Words  | 1 Pages

    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.

    Read More

  • Numeracy Case Study

    1041 Words  | 3 Pages

    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.

    Read More

  • The Sierpinski Triangle

    1145 Words  | 3 Pages

    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.

    Read More

  • The Golden Mean in Anatomy

    957 Words  | 2 Pages

    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.

    Read More

  • My Love of Learning

    620 Words  | 2 Pages

    “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.

    Read More

  • Computer Engineer

    929 Words  | 2 Pages

    Computer engineering started about 5,000 years ago in China when they invented the abacus. The abacus is a manual calculator in which you move beads back and forth on rods to add or subtract. Other inventors of simple computers include Blaise Pascal who came up with the arithmetic machine for his father’s work. Also Charles Babbage produced the Analytical Engine, which combined math calculations from one problem and applied it to solve other complex problems. The Analytical Engine is similar to today’s computers.

    Read More

  • Recursive Sequence Essay

    1994 Words  | 4 Pages

    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).

    Read More

  • Leonardo Da Vinci's Life Without Fibonacci

    1610 Words  | 4 Pages

    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”.

    Read More

  • Rene Descartes Essay

    1509 Words  | 4 Pages

    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.

    Read More

  • Essay On Ada Lovelace

    756 Words  | 2 Pages

    Ada Lovelace was the daughter of famous poet at the time, Lord George Gordon Byron, and mother Anne Isabelle Milbanke, known as “the princess of parallelograms,” a mathematician. A few weeks after Ada Lovelace was born, her parents split. Her father left England and never returned. Women received inferior education that that of a man, but Isabelle Milbanke was more than able to give her daughter a superior education where she focused more on mathematics and science (Bellis). When Ada was 17, she was introduced to Mary Somerville, a Scottish astronomer and mathematician who’s party she heard Charles Babbage’s idea of the Analytic Engine, a new calculating engine (Toole). Charles Babbage, known as the father of computer invented the different calculators. Babbage became a mentor to Ada and helped her study advance math along with Augustus de Morgan, who was a professor at the University of London (Ada Lovelace Biography Mathematician, Computer Programmer (1815–1852)). In 1842, Charles Babbage presented in a seminar in Turin, his new developments on a new engine. Menabrea, an Italian, wrote a summary article of Babbage’s developments and published the article i...

    Read More

  • The History of Computers

    836 Words  | 2 Pages

    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.

    Read More

More about Patterns In Pascals Triangles

Open Document