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 algebraic, which contains binomial functions, arithmetic sequences and sigma notation. I chose the Binomial Theorem because of its complexity, yet simplicity. Its efficiency fascinates me and I would like to share this theorem that can be utilized to solve things in the Mathematical world
Exploring the Binomial Expansion Theorem Introduction In algebra binomial expansion is the expansion of powers of a binomial. A binomial expansion is an expression in which it contains two terms eg, (a+b). This expression could also have a power on the outside of the brackets. Aim To generate a formula for finding the general expanded form of binomial expressions of the form (a+b)n. (Source The Sheet) Basic Binomial Expansions (a+b)1 = a+b (a+b)2 = a2+2ab+ b2 (a+b)3 = a3+ 3a2b + 3ab2 + b3 (a+b)4
contributed to the fields of mathematics, physical science and computers in countless ways. 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
Blaise Pascal Blaise Pascal was born at Clermont, Auvergne, France on June 19, 1628. He was 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
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 is always room in mathematics, however, for imagination, for expansion of previous concepts. In the case of Pascal’s Triangle, a two-dimensional pattern, it can be extended into a third dimension, forming a
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
10x^2 + 10x^3 + 5x^4 + x^5 ..... If you just look at the coefficients of the polynomials that you get, you'll see Pascal's Triangle! Because of this connection, the entries in Pascal's Triangle are called the binomial coefficients.There's a pretty simple formula for figuring out the binomial coefficients (Dr. Math, 4): n! [n:k] = -------- k! (n-k)! 6 * 5 * 4 * 3 * 2 * 1 For example, [6:3] = ------------------------ = 20.
Permutation of Letters EMMA is investigating the amount of different arrangements of letters in her name; she does the same with her friend LUCY. LUCY has twice as many arrangements as EMMA, they are curious as to why this is and decide to investigate other names and find reasons for their answers. EMMA - emma, eamm, emam, aemm, amme, amem, meam, maem, mame, mema, mmea, mmea, LUCY - lucy, luyc, lycu, lyuc, lcyu, lcuy, ulcy, ulyc, uylc,
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 1/2base*height. But I will first need to find the height. To do this I will use Pythagoras theorem which is a2 + b2 = h2. [IMAGE] [IMAGE] First I will half the triangle so I get a right angle triangle with the base as 100m and the hypotenuse as 400m. Now I will find the height: a2 + b2= h2 a2 + 1002 = 4002 a2 = 4002 -
A Critique of Berger's Uncertainty Reduction Theory How do people get to know each other? Bugs Bunny likes to open up every conversation with the question, "What's up Doc? Why does he do this? Is Bugs Bunny "uncertain"? Let's explore this idea of uncertainty. Shifting focus now to college students. As many other college students at Ohio University, I am put into situations that make me uncertain of my surroundings almost every time I go to a class for the first time, a group meeting, or social
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 202500-2500=b² 200000=b² Ö200000=b
1795, he continued his mathematical studies at the University of Gö ttingen. In 1799, he obtained his doctorate in absentia from the University of Helmstedt, for providing the first reasonably complete proof of what is now called the fundamental theorem of algebra. He stated that: Any polynomial with real coefficients can be factored into the product of real linear and/or real quadratic factors. At the age of 24, he published Disquisitiones arithmeticae, in which he formulated systematic and widely
provide a solution to this problem (Thoen and Lefebvre, 2001). 2 Origin of segmental reporting Four theorems that are characterized by an accounting or a financial background can be considered as factors that created a need for the segmentation of information. In the following paragraphs, a brief description of these theorems will be given. 2.1 The fineness-theorem This theorem states that “given two sets containing the same information, if one is broken down more finely, it will be
parameters the sample must be large enough. [IMAGE] According to the Central Limit Theorem: n If the sample size is large enough, the distribution of the sample mean is approximately Normal. n The variance of the distribution of the sample mean is equal to the variance of the sample mean divided by the sample size. These are true whatever the distribution of the parent population. The Central Limit Theorem allows predictions to be made about the distribution of the sample mean without
Fermat’s Last Theorem--which states that an + bn = cn is untrue for any circumstance in which a, b, c are not three positive integers and n is an integer greater than two—has long resided with the collection of other seemingly impossible proofs. Such a characterization seems distant and ill-informed, seeing as today’s smartphones and gadgets have far surpassed the computing capabilities of even the most powerful computers some decades ago. This renaissance of technology has not, however, eased this
geometry book Theorem 1-1 Vertical Angles Theorem Vertical angles are congruent. Theorem 1-2 Congruent Supplements Theorem If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent. Theorem 1-3 Congruent Complements Theorem If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent. Theorem 2-1 Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle is 180. Theorem 2-2 Exterior
right angle triangle. Pythagoras Theorem is a² + b² = c². 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side of a right angled triangle. So the (smallest number)² + (middle number)² = (largest number)² The number 3, 4 and 5 satisfy this condition 3² + 4² = 5² because 3² = 3 x 3 = 9 4² = 4 x 4 = 16 5² = 5 x 5 = 25 and so 3² + 4² = 9 + 16 = 25 = 5² The numbers 5,12, 13 and 7,24,25 also work for this theorem 5² + 12² = 13² because 5²
Language plays a crucial role in helping a poet get his point across and this can be seen used be all the poems to help them explore the theme of death with the reader. This includes the formal, brutal and emotive language that Chinua Achebe uses in “mother in a refugee camp.” This can be seen when Achebe says, “The air was heavy with odor of diarrhea, of unwashed children with washed out ribs” this is very brutal and the is no holding back with the use of a euphemism or a simile as seen in the other
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Pierre-Simon Laplace was born on March 23, 1749 in France (Pierre-Simon Laplace, 2000). He was a mathematician and astronomer who made great findings that contributed to mathematical astronomy and probability (Pierre-Simon Laplace, 2000). Not much is known about Laplace’s childhood because he rarely ever talked about his early days (Marquis de laplace, 2013). However, it is known that his family was middle-class and rich neighbors paid for him to attend school when they realized how talented the