Pascal's Triangle Pascal's Triangle Blasé Pacal was born in France in 1623. He was a child prodigy and was fascinated by mathematics. When Pascal was 19 he invented the first calculating machine that actually worked. Many other people had tried to do the same but did not succeed. One of the topics that deeply interested him was the likelihood of an event happening (probability). This interest came to Pascal from a gambler who asked him to help him make a better guess so he could make an educated
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
Nikko S.A. Gammad P2C Scicomm PASCAL’S PYRAMID AND ITS APPLICATIONS IN STATISTICS Addition, especially of small numbers, is a process that can be done over many repetitions. Sometimes, it produces interesting patterns. One such pattern is in Pascal’s Triangle, where each row can be constructed by adding the numbers on the row above. This particular pattern is significant in that, among other things, it shows a representation of the coefficients of a binomial expansion to a particular power. There
Combinations in Pascal’s Triangle Pascal’s Triangle is a relatively simple picture to create, but the patterns that can be found within it are seemingly endless. Pascal’s Triangle is formed by adding the closest two numbers from the previous row to form the next number in the row directly below, starting with the number 1 at the very tip. This 1 is said to be in the zeroth row. After this you can imagine that the entire triangle is surrounded by 0s. This allows us to say that the next row (row
than Blaise Pascal. Born in 1623 in Clermont, France, he was born into a family of respected mathematicians. Being the childhood prodigy that he was, he came up with a theory at the age of three that was Euclid’s book on the sum of the interior of triangles. At the age of sixteen, he was brought by his father Etienne to discuss about math with the greatest minds at the time. He spent his life working with math but also came up with a plethora of new discoveries in the physical sciences, religion, computers
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
Even the smallest tasks can impact the world in a significant way. Math, despite its trivial appearance, is large in grandeur that governs our world from the inside and the outside. The many twists and turns that exist in Mathematics make its versatility unparalleled and continues to awe the many Mathematicians today and the many more to come. The Binomial Theorem is one such phenomenon, which was founded by the combined efforts of Blaise Pascal, Isaac Newton and many others. This theorem is mainly
the son of Étienne Pascal, his father, and Antoinette Bégone, his mother who died when Blaise was only four years old. After her death, his only family was his father and his two sisters, Gilberte, and Jacqueline, both of whom played key roles in Pascal's life. When Blaise was seven he moved from Clermont with his father and sisters to Paris. It was at this time that his father began to school his son. Though being strong intellectually, Blaise had a pathetic physique. Things went quite well
Clermont-Ferrand France and died at the age of 39 of tuberculosis on the 19th August 1662 in Paris, but the bulk of his career, his success and life achievement began in his early years. As a young boy, Pascal’s lost his mother and soon afterward his father moved the family, Blaise and his two sisters to Paris. Pascal’s father, Étienne Pascal was a mathematician himself and taught Pascal Latin and Greek, which at the time was considered
terms in each expansion plus one. The coefficients in each binomial expansion is a line on Pascal’s triangle. The nth power is the number line the coefficients are on. (a+b)1 = (1)a+b (a+b)2 = (1)a2 + (2)ab + (1)b2 (a+b)3 = (1)a3+ (3)a2b + (3)ab2 + (1)b3 (a+b)4 = (1)a4+ (4)a3b + (6)a2b2+ (4)ab3+ (1)b4 If one was to predict the coefficients in (a+b)7 it would be the seventh line on Pascal’s triangle. The coefficients are 1, 7, 21, 35, 35, 21, 7, 1. The indices on a and b both have their
be made by the application of trigonometry. (Company, 2009) Since its discovery, triangulation has lent the world many beneficial advantages. Polygons exist as multi-sided shapes. These shapes can be subdivided into many non-overlapping triangles. These triangles connect corners to corners, creating diagonals that section it off. These sections are called convex hulls. The many uses of algorithms allow anyone to calculate these triangular hulls. The mathematics behind triangulating polygons was originated
Beyond Pythagoras - Mathematical Investigation 1) Do both 5, 12, 13 and 7, 24, 25 satisfy a similar condition of : (Smallest number)² + (Middle Number)² = (Largest Number) ² ? 5, 12, 13 Smallest number 5² = 5 x 5 = 25 Middle Number 12² = 12 x 12 = 144+ 169 Largest Number 13² = 13 x 13 = 169 7, 24, 25 Smallest number 7² = 7 x 7 = 49 Middle Number 24² = 24 x 24 = 576+ 625 Largest Number 25² = 25 x 25 = 625 Yes, each set of numbers does satisfy the condition.
what a perfect being is, than God must be a sovereign being. Similar to his triangle theory that it is not a necessity to imagine a triangle. It is not a necessity to imagine a perfect being rather a thought that has run through our mind. The triangle as imagined and conceived has three sides and a hundred and eighty degree angles as always. It is imperative that these characteristics are always attributed to the triangle, likewise the attributes of a perfect being are placed on God. In order to prove
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
with them. 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
How far does imaginary numbers go back in history? First must know that an imaginary number is a number that is expressed in terms of the square root of a negative number. This fact took several centuries of convincing for certain mathematicians to believe, but imaginary numbers have been used all the back to the first century, and is now being widely used by people all around the world to this day. It is thanks to people like Heron of Alexandria, Girolamo Cardano, Rafael Bombelli, and other mathematician’s
find the area of irregular triangles and a regular triangle, irregular quadrilaterals and a regular square, this will prove whether irregular polygons are larger that regular polygons. Area of an isosceles irregular triangle: ======================================== (Note: I found there is not a right angle triangle with the perimeter of exactly 1000m, the closest I got to it is on the results table below.) To find the area of an isosceles triangle I will need to use the formula
fitting 4 triangles inside each triangular surface of an icosahedron; which is one of the five solids created by the ancient Greeks. When considering a icosahedron, or any regular polyhedral for that matter, we have the following formulas to consider: 1. V = 10υ2 + 2 2. F = 20υ2 3. E =
Using Tangrams To Explore Mathematical Concepts Representations have always been a very important part of teaching mathematics. The visuals and hands on experiences help to aide the teachers by assisting them in relaying important topics and concepts to the students. By having a representation, the students are more likely to remember what they have learned, and recall the lesson when it comes time to take a test or do their homework. Within mathematics, many different manipulatives are
area when using 1000 meters of fencing, was a square with the measurement of 250m x 250m and the area=62500m² Isosceles Triangles I am now going to look at different size Isosceles triangles to find which one has the biggest area. I am going to use Pythagoras Theorem to find the height of the triangle. Pythagoras Theorem: a²=b²+c² Formula To Find A Triangles Area: ½ x base x height 1. Base=100m Sides=450m [IMAGE] [IMAGE] a²=b²+c² 450²=b²+50² 202500=b²+2500