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Words 8970

Pages 36

ON PROOF AND PROGRESS IN MATHEMATICS

WILLIAM P. THURSTON

This essay on the nature of proof and progress in mathematics was stimulated by the article of Jaﬀe and Quinn, “Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics”. Their article raises interesting issues that mathematicians should pay more attention to, but it also perpetuates some widely held beliefs and attitudes that need to be questioned and examined. The article had one paragraph portraying some of my work in a way that diverges from my experience, and it also diverges from the observations of people in the ﬁeld whom I’ve discussed it with as a reality check. After some reﬂection, it seemed to me that what Jaﬀe and Quinn wrote was an example of the phenomenon that people see what they are tuned to see. Their portrayal of my work resulted from projecting the sociology of mathematics onto a one-dimensional scale (speculation versus rigor) that ignores many basic phenomena. Responses to the Jaﬀe-Quinn article have been invited from a number of mathematicians, and I expect it to receive plenty of speciﬁc analysis and criticism from others. Therefore, I will concentrate in this essay on the positive rather than on the contranegative. I will describe my view of the process of mathematics, referring only occasionally to Jaﬀe and Quinn by way of comparison. In attempting to peel back layers of assumptions, it is important to try to begin with the right questions: 1. What is it that mathematicians accomplish? There are many issues buried in this question, which I have tried to phrase in a way that does not presuppose the nature of the answer. It would not be good to start, for example, with the question How do mathematicians prove theorems? This question…...

...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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...resource for? What will you use this resource for this term? | Academic Tutoring and Success Center | | | Everest Online Library | | | CARE | | | Smarthinking | | | Resources Available to Help You * Academic Tutoring and Success Center (Click on the “Tutoring” link under the Course Home) (http://sites.google.com/site/myacademicsuccesscenter/Home) * Request Live Webcam Tutoring and tutoring for multiple courses * Request a Success Coach * Watch helpful webinars * Smarthinking has a variety of services (Click on the “Tutoring” link under the Course Home) * Access live tutors or schedule a personal tutoring session 48 hours in advance for: * Writing (paragraph development),Reading Comprehension, Math, Accounting, Finance, Medical Assisting, Medical Terminology, and Microsoft Office Programs * Access Everest University’s Online Library from the course by clicking “Everest Online Library” under the Course Home. * This is the one stop access point to useful resources for research, writing and citing papers, getting help with a class or finding a job. Your quest for information begins here. Please take some time to review all of the available resources. * CARE – FREE Student Assistance Program - available 24/7, 365 days-a-year * CALL: (888) 852-6238 or EMAIL: info@EverestCARES.com * The CARE Student Assistance Program will work together with them to find a solution * The CARE helpline can assist you......

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...Math History Essay Math is a subject I enjoy doing. Every day when I come home from school, it is the very first thing I do. Math is by far my favorite subject. It is more exciting than memorizing facts or writing an essay. Math involves thinking and solving, and I like to think through and solve problems. Even though math is not my strongest suit, I still enjoy doing anything related to it. Interest in taking Pre-Cal began during my freshman year. Two years ago, I took Geometry and fell in love with it. I enjoyed everything from proofs to finding the area of a shape. I remember people complaining how difficult Geometry was and thinking the exact opposite. By the end of freshman year, I was sad that Geometry was coming to an end. However, I was informed that Pre-Cal is similar to Geometry and is offered my junior year. Ever since then, I could not wait for Algebra 2 to be over. I also decided to take this class because it will build the foundation needed for Calculus the following year. To be successful in Pre-Cal this year, I will try my best to spare time to review notes I took the day before. I will also spend time with a study group so I can get advice by my peers. Whenever I have unanswered questions and cannot contact a teacher, I can rely on my friends who can explain and guide me along the right answer. As a teacher, you can walk around the class when there is extra time to do homework to easily answer questions that I may have regarding the assignment.......

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...Variable | Range of Values Found | Values selected for project | Speed of Bullet | 1055-3054 Feet Per Second** | 3054 feet per second | Power of locomotive | 3000-6000horse power | 6000 horse power | Height of a very tall building | 859-2716ft | 2716 feet | Chose the highest value for each area because I figured it would be best to compare the top amounts. Speed of bullet – 4122.9 watts Power of locomotive – 4474200 watts Height of building – 2716 feet **Bullet speed varies greatly depending on size/shape/weight of bullet and distance from muzzle (wind resistance and gravity). Volume for water = ((3.1417 * Radius^2) *Depth) * 7.47 = gallons *8.35lb = weight 97,001,173.49175lb’s E=10^(1.5(9)+4.8) = 10^(18.3) = 1.995262315x10^18jouls 1515.428229m/second .941643154 miles per hour 5500 miles My bullet goes 930 meters per second. The power is astronomically higher then that of the locomotive. The height of the building I chose is 827 meters which is a lot larger then 10 meters however when you compare it to the 15 mile dept while in the ocean iits much smaller. Hawks, Chuck. "Rifle Ballistics Summary." Rifle Ballistics Summary. N.p., 2003. Web. 6 May 2013. <http://www.chuckhawks.com/rifle_ballistics_table.htm>. "Power of a Train." Power of a Train. Ed. Glenn Elert. N.p., 2001. Web. 6 May 2013. <http://hypertextbook.com/facts/2001/RadmilaIlyayeva.shtml>. "World's Tallest Buildings1." N.p., 2013. Web. 6 May 2013.......

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...STAT2011 Statistical Models sydney.edu.au/science/maths/stat2011 Semester 1, 2014 Computer Exercise Weeks 1 Due by the end of your week 2 session Last compiled: March 11, 2014 Username: mac 1. Below appears the code to generate a single sample of size 4000 from the population {1, 2, 3, 4, 5, 6}. form it into a 1000-by-4 matrix and then ﬁnd the minimum of each row: > rolls1 table(rolls1) rolls1 1 2 3 4 5 6 703 625 679 662 672 659 2. Next we form this 4000-long vector into a 1000-by-4 matrix: > four.rolls=matrix(rolls1,ncol=4,nrow=1000) 3. Next we ﬁnd the minimum of each row: > min.roll=apply(four.rolls,1,min) 4. Finally we count how many times the minimum of the 4 rolls was a 1: > sum(min.roll==1) [1] 549 5. (a) First simulate 48,000 rolls: > rolls2=sample(x=c(1,2,3,4,5,6),size=48000,replace=TRUE) > table(rolls2) rolls2 1 2 3 4 5 6 8166 8027 8068 7868 7912 7959 (b) Next we form this into a 2-column matrix (thus with 24,000 rows): > two.rolls=matrix(rolls2,nrow=24000,ncol=2) (c) Here we compute the sum of each (2-roll) row: > sum.rolls=apply(two.rolls,1,sum) > table(sum.rolls) sum.rolls 2 3 4 5 6 7 8 9 10 11 742 1339 2006 2570 3409 4013 3423 2651 1913 1291 1 12 643 Note table() gives us the frequency table for the 24,000 row sums. (d) Next we form the vector of sums into a 24-row matrix (thus with 1,000 columns): > twodozen=matrix(sum.rolls,nrow=24,ncol=1000,byrow=TRUE) (e) To ﬁnd the 1,000 column minima use > min.pair=apply(twodozen,2,min) (f) Finally compute......

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...37 -$25,000 = $27,563.37 b) After another year passes, you wish to pay off the loan. How much do you need to pay to pay it off? M = A(1+r/n)nt A = $27,563 , r = 0.05 , n = 360 , t = 1 M = $ 27,563.37(1+0.05/360)360 = $ 28,976.47 Essay (15 points) 4. While everyone dreams of high interest rates for investments, usually high interest rates come with other disadvantages. Using the interest or other sources, research and write an essay on the advantages and disadvantages of higher interest rates on investments. Look at factors like risk, reward, and possible other things that would change to balance out the higher interest rates. Requirements for essay * Write your essay in this document – do not save it in a separate file. * You must clearly state your position with well-structured paragraphs using proper grammar, spelling, and sentence structure. * This is not an “opinion” question – you must offer evidence to support your position, using properly-cited sources. * Your answer must be between ¾-1 page in length. * You must cite and reference at least one source (book, website, periodical) using APA format. The required website counts as one source. You may submit your Assignments to the Math Center for review. Tutors will not grade or correct the Assignment, but they will provide guidance for improvement. Tutors will not, however, help you find web sites for the Assignment. Be sure to submit Assignments early enough to......

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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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...Dec 4 | 11.5: Alternating Series | | 12 | Dec 7 – Dec 11 | 11.6: Absolute Convergence and the Ratio and Root Tests Review for Midterm Exam 2Midterm Exam 2 | Exam 2 : Wed, Dec 10, 5:30-7:00pm Sections: 10.1-10.4, 11.1-11.5 | 13 | Dec 14 – Dec 18 | 11.8: Power Series11.9: Representation of Functions as Power Series | | 14 | Jan 4 – Jan 8 | 11.10: Taylor and Maclaurin Series 11.11: Applications of Taylor PolynomialsComplex Numbers | | 15 | Jan 11 – Jan 15 | Review for Final Exam | Final Exam (comprehensive) | Math Learning Center (NAB239) The Department of Mathematics and Statistics offers a Math Learning Center in NAB239. The goal of this free of charge tutoring service is to provide students with a supportive atmosphere where they have access to assistance and resources outside the classroom. No need to make an appointment-just walk in. Your questions or concerns are welcome to Dr. Saadia Khouyibaba at skhouyibaba@aus.edu or cas-mlc@aus.edu Math 104 Suggested Problems TEXTBOOK: Calculus Early Transcendentals, 7th edition by James Stewart Section | Page | Exercises | 7.1 | 468 | 3, 4, 7, 9, 10, 11, 13, 15, 18, 24, 26, 32, 33, 42 | 7.2 | 476 | 3, 7, 10, 13, 15, 19, 22, 25, 28, 29, 34, 39, 41, 55 | 7.3 | 483 | 1, 2, 3, 5, 8, 9, 13, 15, 23, 24, 26, 27 | 7.4 | 492 | 1, 3, 6, 7, 9, 11, 15, 17, 22, 23, 31, 43, 45, 47, 49, 54 | 7.5 | 499 | 3, 7, 8, 15, 17, 33, 37, 41, 42, 44, 45, 49, 58, 70, 73, 76, 80 | 7.8 | 527 | 1, 2, 5, 7, 10, 12, 13, 17, 19, 21, 25,......

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...objects – what do the students already know? Are there any misconceptions?- Using large versions of various 3D shapes, identify each object. Discuss the features of each shape e.g. faces, edges etc. - As a class, place the objects into groups based on similar features. Ensure students use reasoning for placing shapes into a certain group | - Students are already familiar with recognising and describing 3D shapes from stage 1 | Working mathematically MA2-1WM,MA2-3WM | EnglishEN2-1A | Visual Auditory/ linguistic | 2 | Measurement and GeometryMA2-14MG | - Discuss features of 3D shapes describing similarities and differences – focus on language e.g. faces, vertex, base, side, flat/curved surface- Students draw/sketch a rectangular prism in maths books and label as much as they can e.g. sides, edges, faces, vertex etc. Students continue to draw as many 3D shapes as they can - Discuss 3D object in a variety of contexts e.g. buildings, packaging | - Students are familiar with the names and some features of 3D shapes | N/A | EnglishEN2-1A | LinguisticVisual/ spatial | 3 | Measurement and GeometryMA2-14MG Working mathematicallyMA2-3WM | - Discuss different views of 3D objects in the room using language such as top, side, front etc. Draw examples on board- Students are each given a small whiteboard, whiteboard marker and eraser and sit in a circle. Teacher holds up 3D shapes and asks students to sketch different views (holding up answers to teacher each time). - Discuss correct......

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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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...Sample Exam 2 - MATH 321 Problem 1. Change the order of integration and evaluate. (a) (b) 2 0 1 0 1 (x y/2 + y)2 dxdy. + y 3 x) dxdy. 1 0 0 x 0 y 1 (x2 y 1/2 Problem 2. (a) Sketch the region for the integral f (x, y, z) dzdydx. (b) Write the integral with the integration order dxdydz. THE FUNCTION f IS NOT GIVEN, SO THAT NO EVALUATION IS REQUIRED. Problem 3. Evaluate e−x −y dxdy, where B consists of points B (x, y) satisfying x2 + y 2 ≤ 1 and y ≤ 0. − Problem 4. (a) Compute the integral of f along the path → if c − f (x, y, z) = x + y + yz and →(t) = (sin t, cos t, t), 0 ≤ t ≤ 2π. c → − → − → − (b) Find the work done by the force F (x, y) = (x2 − y 2 ) i + 2xy j in moving a particle counterclockwise around the square with corners (0, 0), (a, 0), (a, a), (0, a), a > 0. Problem 5. (a) Compute the integral of z 2 over the surface of the unit sphere. → → − − → − → − − F · d S , where F (x, y, z) = (x, y, −y) and S is → (b) Calculate S the cylindrical surface deﬁned by x2 + y 2 = 1, 0 ≤ z ≤ 1, with normal pointing out of the cylinder. → − Problem 6. Let S be an oriented surface and C a closed curve → − bounding S . Verify the equality → − → − → → − − ( × F ) · dS = F ·ds − → → − if F is a gradient ﬁeld. S C 2 2 1 ...

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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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...50 Excellent Extended Essays Alhazen’s Billiard Problem © International Baccalaureate Organization 2008 1 50 Excellent Extended Essays Alhazen’s Billiard Problem 2 © International Baccalaureate Organization 2008 50 Excellent Extended Essays Alhazen’s Billiard Problem © International Baccalaureate Organization 2008 3 50 Excellent Extended Essays Alhazen’s Billiard Problem 4 © International Baccalaureate Organization 2008 50 Excellent Extended Essays Alhazen’s Billiard Problem © International Baccalaureate Organization 2008 5 50 Excellent Extended Essays Alhazen’s Billiard Problem 6 © International Baccalaureate Organization 2008 50 Excellent Extended Essays Alhazen’s Billiard Problem © International Baccalaureate Organization 2008 7 50 Excellent Extended Essays Alhazen’s Billiard Problem 8 © International Baccalaureate Organization 2008 50 Excellent Extended Essays Alhazen’s Billiard Problem © International Baccalaureate Organization 2008 9 50 Excellent Extended Essays Alhazen’s Billiard Problem 10 © International Baccalaureate Organization 2008 50 Excellent Extended Essays Alhazen’s Billiard Problem © International Baccalaureate Organization 2008 11 50 Excellent Extended Essays Alhazen’s Billiard Problem 12 © International Baccalaureate Organization 2008 50 Excellent Extended Essays Alhazen’s Billiard Problem © International Baccalaureate......

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...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 law, so a “preponderance of the evidence” or “beyond any reasonable doubt” isn’t good enough. Geometry teaches students like myself how to think critically and prove things in many ways using a step by step process. And if we didn’t use logics and reasonings then we would be jumping to conclusions and making up things a principle posited in the fourteenth century (by William of Occam (1288 C.E.–1348 C.E.)) includes that your proof system should have the smallest possible set of axioms and logical rules. Proofs are important to math since they allow us to think the way we do, analysis the reasonings for things logically.It's also, in real life since everything you do you have to have a reason and logic. Furthermore, if we didn't have proofs we wouldn’t believe in what we do and we wouldn’t be able to question and argue with others. Geometry has influenced architecture and art by finding out how much material is needed for a job by using area formulas and while brick laying a mason must......

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