资源描述
单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。不能作为科学依据。,The Power of Good Question,Provide by:BME141,1/17,What,makes a good question?,If youre doing maths for fun,or are a professional mathematician,you answer is going to be different.,（,An easy question is boring,）,If you are a student facing exams,you might(understandably)say that good means easy.,2/17,Where,do these good questions come from?,Generalize,Simplify,and,vary,Look for,new tools,Take risks,3/17,Generalize,a,n,+b,n,=c,n,?,Innumber theory,Fermats Last Theorem,(sometimes called,Fermats conjecture,especially in older texts)states that no threepositive integers,a,b,and,c,satisfy the equation,a,n,+,b,n,=,c,n,for any integer value of,n,strictly greater than two.The cases,n,=1 and,n,=2 have been known to have infinitely many solutions since antiquity.,4/17,350 years!?,This theorem was firstconjectured byPierre de Fermatin 1637 in the margin of a copy ofArithmetica(,算术,)where he claimed he had a proof that was too large to fit in the margin.The first successful proofwas released in 1994 byAndrew Wiles,and formally published in 1995,after 358 years of effort by mathematicians.It is among the most notable theorems in thehistory of mathematics and prior to its proof,it was in the,Guinness Book of World Records,as the most difficult mathematical problem,one of the reasons being that it has the largest number of unsuccessful proofs.,5/17,万畅高清摄像机,万畅高清摄像机,万畅传输接入模块,万畅局端,模块和设备,视频枢纽,万局端模块,Fermats simple question turned out to be incredibly fruitful:it generated new mathematics,new insights and new ways of looking at things.Though hard,many mathematicians would regard this as a“good”question.,Together with,Ren Descartes,（笛卡尔）,Fermat,was one of the two leading mathematicians of the first half of the 17th century.,6/17,Simplify and vary,Gallery problem,A nice example is the art gallery problem:how many security guards do you need to be sure that together they can oversee the whole interior of an art gallery?,7/17,Answer,The first answer,given in 1978 five years afer the problem was posed.Using an ingenious line of attack,the mathematician S.Fisk proved that you never need more than 1/3 guards,where n is the number of vertices(corners)of the polygon.,8/17,30 years on,these problem is still going,What if the guards are not confined to the corners of the gallery?,Gallery problems,What if they are allowed to move around?,What if there are obstacles in the middle of the gallery that you cannot see through?,The walls are curved?,What if,instead of guarding a two-dimensional polygon,you are trying to guard a three-dimensional polyhedron?,9/17,Look for new tools,Calculus,There are also questions that are being asked,not by individuals,but by a whole age,，,crying out for new mathematical tools.Their answers can spawn something of a revolution.A great example is the invention of calculus in the seventeenth century.,10/17,Calculus,How can we describe continuous change?,A journey:speed is the rate of change of distance per time,so you simply divide the distance you traveled by the time it took to travel it.,(S/T),But of course,you didnt travel at that average speed at every moment of you journey.At some times you will have been going slower and at some times faster,with the speed varying continuously.To work out your exact speed at a particular moment in time,you have to calculate the instantaneous rate of change of distance with respect to time.,11/17,Applications of calculus,The methods for doing this were invented primarily by,Gottfried Leibniz,and,Isaac,Applications of integral calculus include computations involving area,volume,arc length,center of mass,work,and pressure.More advanced applications include power series and Fourier series.Calculus is also used to gain a more precise understanding of the nature of space,time,and motion.,Gottfried Leibniz(left),Isaac Newton(right),12/17,Take risks,Four colour theorem,Not all questions turn out to have interesting answers.Mathematicians simply have to accept the risk that a question they choose to work on may not be solved in their life time,or that it may turn out to have a boring answer.Its all part of the creative process.,A question that not be solved in their life time-Fermats Last Theorem,.,13/17,Four color theorem,It says that four colours are enough to colour a map drawn on the plane so that no two neighbouring countries have the same colour.The proof of this theorem,when it finally came in the 1970s after mathematicians had been wrestling with the theorem for over a century,was disappointing.It used a brute force approach involving a computer checking through a huge number of possibilities,making sure they did not provide a counter example to the theorem.The approach delivered no new insights at all.,A simple map coloured correctly with four colours.,14/17,万畅高清摄像机,万畅高清摄像机,万畅传输接入模块,万畅局端,模块和设备,视频枢纽,万局端模块,However,the part of their proof was actually done by a computer.No human being could in their lifetime ever actually read the entire proof to check that it was correct.Several mathematicians of the time complained that this meant that it wasnt really a proof at all!If nobody could check the proof,how could we ever know whether it was right or wrong?,Part of the world map,coloured in more than four colours.The minimum number of colours required is four.,15/17,万畅高清摄像机,万畅高清摄像机,万畅传输接入模块,万畅局端,模块和设备,视频枢纽,万局端模块,But mathematicians dont like giving up.Just like a cat keeps toying with a mouse,so mathematicians often keep toying with a problem if they dont like the answer that has already been found.They turn it this way and that,looking at it in different light and from different angles.Often its a new question,a new way of phrasing the problem,that leads to a satisfying answer and new directions in mathematics.,New question,New way,16/17,THATS All,THANK,YOU,17/17,
展开阅读全文
浙ICP备2021020529号-1 | 浙B2-20240490 
