In: Science

Submitted By emily26

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Words 2213

Pages 9

Word Count: 1912

Euclidean geometry is the everyday “flat” or parabolic geometry which uses the axioms from Euclid’s book The Elements. Non-Euclidean geometry includes both hyperbolic and elliptical geometry [W5] and is a construction of shapes using a curved surface rather than an n-dimensional Euclidean space. The main difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. There has been much investigation into the first five of Euclid’s postulates; mainly into proving the formulation of the fifth one, the parallel postulate, is totally independent of the previous four. The parallel postulate states “that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” [W1] Many mathematicians have carried out extensive work into proving the parallel postulate and into the development of non-Euclidean geometry and the first to do so were the mathematicians Saccheri and Lambert. Lambert based most of his developments on previous results and conclusions by Saccheri. Saccheri looked at the three possibilities of the sum of the angles in a triangle. He found the first to be <180°, one being =180° and the final being >180°. He showed that the first, that the angles total <180°, was viable but only through the use of a new geometry system, non-Euclidean geometry therefore beginning the investigation into this. Although both Lambert and Saccheri contributed to this field of mathematics, the most famous…...

...say whether I was able to learn how to be a better teacher and what the teacher did that I could possibly use in the future. While analyzing and going through the process of this assignment it is helping realize how to become a better teacher as well. I would also like to get more comfortable and experience on using this template of the paper. Memories Of A Teacher My teacher, Mr. G, used many different instructional techniques and approaches to his lessons. Mr. G had taught me math for three years in a row, so I think that I have a good grasp on his approaches to the lessons that he would teach. He would assign many homework assignments, as well as in-class assignments, which helped me and other students understand and get practice with the lesson that we were learning. I think that with math having a lot of homework is a good thing. In my mind, the only way to learn how to do math is plenty of practice. The more you practice, the easier it will be. Mr. G would also have the students do some math problems on the chalk board or smart board to show the class and go over the corrections with the whole class so that everyone would understand the problem. Playing “racing” games also helped and added fun to the class. With the “racing” games, the students would get into groups and have to take turns doing problems on the chalk board and see who could get the correct answer first. It added fun and a little friendly competition to the class. It also helped the students want to......

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...Diana Garza 1-16-12 Reflection The ideas Stein presents on problem saving and just math in general are that everyone has a different way of saving their own math problems. For explains when you’re doing a math problem you submit all kinds of different numbers into a data or formula till something works or maybe it’s impossible to come up with a solution. For math in general he talks about how math is so big and its due in large measure to the wide variety of situations how it can sit for a long time without being unexamined. Waiting for someone comes along to find a totally unexpected use for it. Just like has work he couldn’t figure it out and someone else found a use for it and now everyone uses it for their banking account. For myself this made me think about how math isn’t always going to have a solution. To any math problem I come across have to come with a clear mind and ready to understand it carefully. If I don’t understand or having hard time taking a small break will help a lot. The guidelines for problem solving will help me a lot to take it step by step instead of trying to do it all at once. Just like the introduction said the impossible takes forever. The things that surprised me are that I didn’t realize how much math can be used in music and how someone who was trying to find something else came to the discovery that he find toe. What may people were trying to find before Feynmsn....

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...MATH 55 SOLUTION SET—SOLUTION SET #5 Note. Any typos or errors in this solution set should be reported to the GSI at isammis@math.berkeley.edu 4.1.8. How many diﬀerent three-letter initials with none of the letters repeated can people have. Solution. One has 26 choices for the ﬁrst initial, 25 for the second, and 24 for the third, for a total of (26)(25)(24) possible initials. 4.1.18. How many positive integers less than 1000 (a) are divisible by 7? (b) are divisible by 7 but not by 11? (c) are divisible by both 7 and 11? (d) are divisible by either 7 or 11? (e) are divisible by exactly one of 7 or 11? (f ) are divisible by neither 7 nor 11? (g) have distinct digits? (h) have distinct digits and are even? Solution. (a) Every 7th number is divisible by 7. Since 1000 = (7)(142) + 6, there are 142 multiples of seven less than 1000. (b) Every 77th number is divisible by 77. Since 1000 = (77)(12) + 76, there are 12 multiples of 77 less than 1000. We don’t want to count these, so there are 142 − 12 = 130 multiples of 7 but not 11 less than 1000. (c) We just ﬁgured this out to get (b)—there are 12. (d) Since 1000 = (11)(90) + 10, there are 90 multiples of 11 less than 1000. Now, if we add the 142 multiples of 7 to this, we get 232, but in doing this we’ve counted each multiple of 77 twice. We can correct for this by subtracting oﬀ the 12 items that we’ve counted twice. Thus, there are 232-12=220 positive integers less than 1000 divisible by 7 or 11. (e) If we want to exclude the......

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...This article is about the study of topics, such as quantity and structure. For other uses, see Mathematics (disambiguation). "Math" redirects here. For other uses, see Math (disambiguation). Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1] Mathematics is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8] Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become......

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...Alternatives to Euclidean Geometry and Its Applications Negations to Euclid’s fifth postulate, known as the parallel postulate, give rise to the emergence of other types of geometries. Its existence stands in the respective models which their originators have imagined and designed them to be. The development of these geometries and its eventual recognition give humans some mathematical systems as alternative to Euclidean geometry. The controversial Euclid’s fifth postulate is phrased in this manner, to wit: “If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which is the angles less than the two right angles.” which has been rephrased, and what is known as the parallel postulate as follows: “Given a line L and an external point P not on L, there exists a unique line m passing through P and parallel to L.” With the sphere as its model, is spherical (also called reimannian or elliptic) geometry being advanced by German mathematician, Bernhard Riemann who proposes the absence of a parallel line with Euclid’s fifth postulate as reference. His proposition is as follows: “ If L is any line and P is any point not on L, then there are no lines through P that are parallel to L” It contradicts Euclid’s fifth postulate mainly because no matter how careful one in constructing a line with a straightedge- as straight as it is- that......

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...can see how to split up the original equation into its factor pair, this is the quickest and allows you to solve the problem in one step. Week 9 capstone part 1 Has the content in this course allowed you to think of math as a useful tool? If so, how? What concepts investigated in this course can apply to your personal and professional life? In the course, I have learned about polynomials, rational expressions, radical equations, and quadratic equations. Quadratic equations seem to have the most real life applications -- in things such as ticket sales, bike repairs, and modeling. Rational expressions are also important, if I know how long it takes me to clean my sons room, and know how long it takes him to clean his own room. I can use rational expressions to determine how long it will take the two of us working together to clean his room. The Math lab site was useful in some ways, since it allowed me to check my answers to the problems immediately. However, especially in math 117, it was too sensitive to formatting of the equations and answers. I sometimes put an answer into the math lab that I knew was right, but it marked it wrong because of the math lab expecting slightly different formatting Week 9 capstone part 2 I really didn't use center for math excellence because i found that MML was more convenient for me. I think that MML reassures you that you’re doing the problem correctly. MML is extra support because it carefully walks you through the problem visually......

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...later study some math models for population, demand of product, speed of an object, ... Some typical functions used for models are (1) Polynomial function P (x) = anxn + an−1xn−1 + · · · + a1x + a0 where ai are coeﬃcients. If an = 0, then the degree of P (x) is n. (a) Linear model f (x) = mx + b where m is slope, b is y-intercept. ex. The relationship between Fahrenheit (F) and Celsius (C) is F = 9 C + 32. 5 (b) Quadratic model f (x) = ax2 + bx + c. (c) Cubic model A polynomial with degree 3. (2) Power function f (x) = xa where a is constant. Ex. Sketch the graph of power function if a is (1) positive integer; (2) reciprocal of positive integer; (3) -1; 6 (3) Rational function P (x) Q(x) where P, Q are polynomials. ex. ﬁnd domain of f (x) = 2−3x . x2 −4 f (x) = (4) Algebraic function It is a function constructed by poly√ nomials using algebraic operations (such as +, −, ×, ÷, n ). √ ex. ﬁnd domain and symmetry of f (x) = x2 +1 x3 (5) Trigonometric functions The common trigonometric functions are sin, cos, tan, cot. See more details on Reference Page 2 of the Text. (6) Exponential functions f (x) = ax where the base a = 1 is a positive constant. ex. Sketch the graph of y = ax when a is a constant satisfying (1) a < 1 (2) a > 1. 7 (7) Logarithmic functions f (x) = loga x where a = 1 is a positive constant. ex. Sketch the graph of y = loga x when a is a constant satisfying (1) a < 1 (2) a > 1. (8) Transcendental functions It is a non-algebraic......

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... Alternatives to Euclidean Geometry Student name: Institution: Alternatives to Euclidean Geometry According to Johnson (2013) Euclidean Geometry , commonly known as high school geometry, is a mathematical study of geometry based on undefined terms such as points, lines and or planes; definitions and other theories of a mathematician known as Euclid (330 B.C.) While a number of Euclid’s research findings had been earlier stated by Greek Mathematicians, Euclid has received a lot of recognition for developing the very first comprehensive deductive systems. Euclid’s approach to mathematical geometry involved providing all the theorems from a finite number of axioms (postulates). Euclidean Geometry is basically a study of flat surfaces. Geometrical concepts can easily be illustrated by drawings on a chalkboard or a piece of paper. A number of concepts are known in a flat surface. These concepts include, the shortest distance between points, which is known to be one unique straight line, the angle sum of a triangle, which adds up to 180 degrees and the concept of perpendicular to any line.( Johnson, 2013, p.45) In his text, Mr. Euclid detailed his fifth axiom, the famous parallel axiom, in this manner: If a straight line traversing any two straight lines forms interior angles on one side less than two right angles, the two straight lines, if indefinitely extrapolated, will meet on that same side where the angles smaller than the two right angles. In......

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...and solve problems in everyday life”. In my everyday life I have to keep the balance in my check book, pay bills, take care of kids, run my house, cook, clean etc. With cooking I am using math, measuring how much food to make for four people (I still haven’t mastered that one). With bills I am using math, how much each company gets, to how much money I have to spare (which these days is not much). In my everyday life I do use some form of a math. It might not be how I was taught, but I have learned to adapt to my surroundings and do math how I know it be used, the basic ways, none of that fancy stuff. For my weakest ability I would say I fall into “Confidence with Mathematics”. Math has never been one of my favorite subjects to learn. It is like my brain knows I have to learn it, but it puts up a wall and doesn’t allow the information to stay in there. The handout “The Case for Quantitative Literacy” states I should be at ease with applying quantitative methods, and comfortable with quantitative ideas. To be honest this class scares the crap out of me, and I am worried I won’t do well in this class. The handout also says confidence is the opposite of “Math Anxiety”, well I can assure you I have plenty of anxiety right now with this class. I have never been a confident person with math, I guess I doubt my abilities, because once I get over my fears and anxiety I do fine. I just have to mentally get myself there and usually it’s towards the end of the class. There are......

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... | | |Location | | |on-line | | | | | |Times | | |on-line, or in Maier Hall , Math Lab, Peninsula College | | | | | |Start Date | | |Sept. 21, 2015 End Date Dec. 9, 2015 | | | | | |Course Credits ...

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...Difference Between Euclidean and Spherical Trigonometry 1 Non-Euclidean geometry is geometry that is not based on the postulates of Euclidean geometry. The five postulates of Euclidean geometry are: 1. Two points determine one line segment. 2. A line segment can be extended infinitely. 3. A center and radius determine a circle. 4. All right angles are congruent. 5. Given a line and a point not on the line, there exists exactly one line containing the given point parallel to the given line. The fifth postulate is sometimes called the parallel postulate. It determines the curvature of the geometry’s space. If there is one line parallel to the given line (like in Euclidean geometry), it has no curvature. If there are at least two lines parallel to the given line, it has a negative curvature. If there are no lines parallel to the given line, it has a positive curvature. The most important non-Euclidean geometries are hyperbolic geometry and spherical geometry. Hyperbolic geometry is the geometry on a hyperbolic surface. A hyperbolic surface has a negative curvature. Thus, the fifth postulate of hyperbolic geometry is that there are at least two lines parallel to the given line through the given point. 2 Spherical geometry is the geometry on the surface of a sphere. The five postulates of spherical geometry are: 1. Two points determine one line segment, unless the points are antipodal (the endpoints of a diameter of the sphere), in which case ...

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...Distinguishing factors between Euclidean and non- Euclidean spaces: The space we inhabit cannot solely be determined by a priori Hassanah Smith Professor Mandik Philosophy of space and time There are a plethora of ways to distinguish the differences between Euclidean and non- Euclidean geometries. Understanding both geometries can help one determine our physical space rather than inferring because of past experiences, or in this instance postulates of geometry. Euclidean geometry studies planes and solid figures based on a number of axioms and theories. This is explained using flat spaces, hence the usage of paper, and dry erase boards in classrooms, and other flat planes to illustrate these geometrical standards. Some of Euclid’s concepts are 1. The shortest distances between two points is a straight line. 2. The sum of all angles in a triangle equals one hundred eighty degrees. 3. Perpendicular lines are associated with forming right angles. 4. All right angles are equal 5. Circles can be constructed when the point for the center and a distance of the radius is given. But Euclid is mostly recognized for the parallel postulate. This states that through a point not on a line, there is no more than one line parallel through the line. (Roberts, 2012) These geometries went unchallenged for decades until other forms of geometry was introduced in the early nineteen hundreds, because Euclid’s geometry could not be applied to explain all......

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...principles of what is now called Euclidean Geometry. Euclidean Geometry is a mathematical system and consists of in a small set of appealing postulates that are accepted as true. In fact, Euclid was able to come up with a great portion of plane geometry from five postulates. These postulates include: A straight line segment can be drawn joining any two points, to extend a finite straight line continuously in a straight line, given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center, all right angles are congruent, and if two lines are drawn which intersect a third line in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended infinitely. Also, known as the Parallel Postulate. The elements also, included the five common notions that include: Things that are equal to the same thing are also equal to one another (Transitive property of equality). If equals are added to equals, then the wholes are equal (Addition property of equality). If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality). Things that coincide with one another are equal to one another (Reflexive Property). The whole is greater than the part. Mathematical proofs is an argument, a justification, which convinces other people that something is true. Math isn’t a court of......

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...rewarded when correct solution is achieved. “The harder the struggle, the sweeter the victory is a common expression. This study was brought about when Mathematics, especially problem solving processes, Mathematics interest and Mathematics aptitude. Problem solving, which the National Council for Teacher in Mathematics (NCTM) 1980’s widely heralded statement in its agenda for action and problem solving has been the theme of the council. Knowledge and skills of Mathematics problem solving is believed to help school goers solve problems in their day to day of existence. The ancient claimed “Mathematics is the queen of Knowledge,” hence it is only right to say that Mathematics enhanced students understanding of the important principles in math, that is, as a cooperative and never ending process. Mathematics also made them exert more effort in improving their achievement and inspire them in relating Mathematics and applications for the solution of human problems. There are many ways of solving problems. The one presented by Polya – a Mathematician of the 18th Century. According to Polya, to solve problems in Mathematics, a student must follow certain steps of processes. Polya’s steps for solving Mathematics problems are; (a) Understand the Problem, (b) Devise a Plan, (c) Carry out the Plan, and (d) Look Back (McCoy 1994). Due to this vision, the Department of Education continues their research on how to improve the quality of the Education. Also new methods, strategies and......

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...Math is used everyday – adding the cost of the groceries before checkout, totaling up the monthly bills, estimating the distance and time a car ride is to a place a person has not been. The problems worked this week have showed how math works in the real world. This paper will show how two math problems from chapter five real world applications numbers 35 and 37 worked out. Number 35 A person hired a firm to build a CB radio tower. The firm charges $100 for labor for the first 10 feet. After that, the cost of labor for each succeeding 10 feet is $25 more than the preceding 10 feet. That is, the nest 10 feet will cost $125; the next 10 feet will cost $150, etc. How much will it cost to build a 90-foot tower? Solving this problem involves the arithmetic sequence. The arithmetic sequence is a sequence of numbers in which each succeeding term differs from the preceding term by the same amount (Bluman, 2011). n = number of terms altogether n = 9 d = the common differences d = 25 ª1 = first term ª1 = 100 ªn = last term ª2 = ª9 The formula used to solve this problem came from the book page 222. ªn = ª1 + (n -1)d ª9 = 100 + (9-1)25 ª9 = 100 + (8)25 ...

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