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
However, one of the most important parts of Geometry is the Pythagorean Theorem. The Pythagorean Theorem is the most important because it is one of the most commonly used theorems in Geometry and in all of math. The Pythagorean Theorem is as old as Geometry itself. It was developed by the Egyptians and Chinese and finalized by a Greek philosopher Pythagoras. It can also help solve many real world problems. The Pythagorean Theorem is fascinating because of who developed it, how it has affected influential
When I am studying the Law of Cosines, this theorem, which describes the relationship of the three sides of the triangles, reminds me of another special theorem using right triangles that I learned in primary school, which is the Pythagorean Theorem, as the Law of Cosines is the generalization of the Pythagorean Theorem. Surprisingly, I found myself not familiar with the proving of the Pythagorean Theorem. I knew what the Pythagorean Theorem was, but never paid attention to how to prove it. Also
The History of the Pythagorean Theorem The Pythagorean Theorem is a named after a Greek mathematician and philosopher, named Pythagoras, but is he really the one who discovered the theorem? It?s kind of like the Crossover Dribble in basketball. Most people attribute the move to Allen Iverson, but when you look into the history of the crossover some say it began in a street basketball game by a guy named Richard Kirkland, some attribute it to Oscar Robertson in the 60?s, but it was definitely
Bayes' Theorem I first became interested in Bayes' Theorem after reading Blind Man's Bluff, Sontag (1998). The book made mention how Bayes' Theorem was used to locate a missing thermonuclear bomb in Spain in 1966. Furthermore, it was again used by the military to locate the missing submarine USS Scorpion (Sontag, pg. 97) that had imploded when it sank several years later. I was intrigued by the nature of the theory and wanted to know more about it. When I was reading our textbook for 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 that
Since hundred years ago, when people started to make maps to show distinct regions, such as states or countries, the four color theorem has been well known among many mapmakers. Because a mapmaker who can plan very well, will only need four colors to color the map that he makes. The basic rule of coloring a map is that if two regions are next to each other, the mapmaker has to use two different colors to color the adjacent regions. The reason is because when two regions share one boundary can never
Prior to construction the company offers residents $5,000 each to sign a waiver saying they will not complain about excessive noise from the turning wind turbines [Yardley, 2010]. Analyze this offer in terms of the Coase Theorem. According to the terms of the Coase Theorem implies that once property rights are established, government intervention is not required to deal with externalities. In this case, if the Caithness Energy can come to a bargaining agreement with all the residents that’s affected
Graph Theory: The Four Coloring Theorem "Every planar map is four colorable," seems like a pretty basic and easily provable statement. However, this simple concept took over one hundred years and involved more than a dozen mathematicians to finally prove it. Throughout the century that many men pondered this idea, many other problems, solutions, and mathematical concepts were created. I find the Four Coloring Theorem to be very interesting because of it's apparent simplicity paired with it's
hundred years ago, a Greek mathematician, Pythagoras, invented the Pythagorean Theorem. The Theorem was related to the length of each side of a right-angled triangle. In a right-angled triangle, the square on the hypotenuse, the side opposite to the right angle, equals to the sum of the squares on the other two sides. (148, Poskitt) To know more about this famous theorem, we can look at the other forms of the Pythagorean Theorem, such as it can also be written as c^2-a^2=b^2 which is for reverse operations
In Chapter 2 of Journey Through Genius, titled “Euclid’s Proof of the Pythagorean Theorem,” the author, William Dunham begins by introducing the Greek contributions to mathematics. The first figure introduced, Plato, brought enthusiasm to the subject. He was not an actual mathematician; he was a philosopher. His main contribution to math was establishing the Academy, a center devoted to “learning and contemplation for talented scholars.” The Academy was mainly focused on mathematics and produced
Sometimes a theorem is so important that it becomes known as a fundamental theorem in mathematics. This is the case for the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra states that every polynomial equation of degree n, greater than or equal to one, has exactly n complex zeros. In fact, there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. The Fundamental Theorem of Algebra can
Fermat’s Last Theorem The year is 1637. Pierre de Fermat sits in his library, huddled over a copy of Arithmetica written by the Greek mathematician Diaphantus in the third century A. D. Turning the page, Fermat comes across the Pythagorean equation: x 2 + y 2 = z 2. He leans back in his chair to think and wonders if this property is limited to the power of two only. He bends over the book again, scanning ahead through the pages to look for any clues. Suddenly, he begins writing intensely
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
Math IA Four Color Theorem Matt Reed Four Color Theorem I. Introduction Ever since the beginning of travel and exploration, maps have helped people record the specifics of new and unexplored regions of the earth. The earliest maps were crudely drawn by hand and were rough estimates of geographic area based on interpretation of the land. Once people began coloring maps, to designate partitions within regions, the problem arose regarding the necessary number of colors it would take to color a map
What is the significance of Gödel’s Incompleteness Theorems for the philosophy of Mathematics? Gödel’s incompleteness theorems, established in the first half of the twentieth century, have transformed the way many mathematicians, philosophers and even computer scientists have thought about mathematics. Although throughout the entirety of his work, he is neither concise nor always clear; it is obvious Gödel's theorems unearth a series of restrictions of an axiomatic and mechanical view of mathematics
YTHAGOREAN THEOREM 3 Pythagorean Theorem Introduction The purpose of this paper is for grade 9 and grade 10 students in Ontario high schools to know a brief history about the development of the Pythagorean theorem, write a proof of the theorem, solve problems using the Pythagorean theorem, and calculate primitive Pythagorean triples with one odd and one even number. The theorem is called by di erent names: Pythagoras' theorem, the hypotenuse theorem or Euclid I 47, so called because it is listed
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
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
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