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1、How do people produce scales?Throughout the ages people seem to be both excited and fearful about uncertain thingsAncestors living in the Yellow River Basin prayed to the river godsreverently for the peace of the river.When it turns clear afterraining,people look up at the sky hoping to see colorful
2、 clouds.When you climb a steep mountain path,you will be careful withwhere your step,for you do not know whether you will step on aloose stone next.With regard to uncertainty,mathematics reveals many definite facetsbehind uncertain things through continuous research.In contrast,everything in nature
3、seems to be more at ease.They just embark on anextremely smooth evolution road through natural selection.How do people produce scales?Trouble in Temperament TemperamentHow do these notes do、re、mi、fa、so、la、sicome?Common temperamentBe harmonious when the vibration frequency ratio between sounds was a
4、simple integer.Pitch and FrequencyABCWe find a string AC,and you will hear a note byplucking this string,and then you will hear arelatively high note by plucking half of the string,AB.the law of physics:the frequency of vibration of a stringis inversely proportional to its length.We heard two sounds
5、:the middle note is do,and the higher note is high do.The frequency of the latter is twice as many as the former one.Harmonious Sounds between F and 2F ABC1323ABC14Assume the frequency of the middle do:Fthe frequency of the high do:2F3F32F4FF34ContinueHold down 、of the string5161Only these 3 notes:F
6、,F,F?ABCABC3234How to do?DCAGIdea:let F be the tonic do.32F4989F Let F be the tonic do,then get F.1627Ideal:after a certain cycle,some tonic do is got that is,to fine n and N such thatNn223=C89The solutionno solution！approximate solutions：notice that 537.5927.59F can be considered to have looped to
7、the last tonic do.Drops this tonic by two 8 degrees to get .F243128Five-tone cadence Temperament Frequencies of 7 tonicsFFFFFFF98143272438643216128、corresponding to do,re,mi,fa,sol,la,and si.=328nN=322nn N+=32Emergence of the twelve-tone scaleFrom the perspective of solving equations,the diatonic sc
8、ale does not actually solve the problem of solving equations.322nN=After looping 5 times,there is a small error after all.The little discord between sounds cannot be heard clearly in a scale of 8 degrees.If it crosses over two or more octaves,this error will be continuously amplified.Another Solutio
9、nsC、#C(升 C)、D、#D、E、F、#F、G、#G、A、#A、B Corresponding letter names：Let cycle 12 times,then we have3275.129231212827=The twelve-tone scale:ModulationKey of C is modulated to flat ETwo ratios of the frequencies of adjacent sounds:，20482187243256The appearance of a ChineseZhu Zaiyu(1536-1611).A mathematici
10、an,historian and artist living in the Ming Dynasty.He had a royal lineage and was the ninth-generation descendant of Emperor Zhu Yuanzhang.His most famous work was to establish the twelve-tone equal temperament.A Chinese appeared,whose name was Zhu Zaiyu,a mathematician of the Ming Dynasty.He found
11、a solution at the same time as Westerners.Zhu Zaiyus MethodThe frequency ratio:the twelfth root of the ratio of 2:1.The frequency of the twelve-tone scale:F(C),1.059F(#CbD),1.122F(D),1.189F(#DbE),1.260F(E)、1.335F(F)、1.414F(#FbG),1.498F(G),1.587F(#GbA),1.682F(A),1.782F(#AbB),1.888F(B).Indeterminate E
12、quations322nN=Such an equation or system of equations,for which the numberof unknowns is greater than the number of equations,and theunknowns are subject to certain restrictions,such as rationalnumbers or integers.Diophantine equations:in memory of Diophantus outstanding contribution to indeterminat
13、e equations.Tombstone of DiophantusDiophantus(About A.D.246-330)was a Hellenistic mathematician.He was one of the founders of algebra and contributed to numbertheory and indeterminate equations.Therefore,in memory of him,people often call the indeterminate equations Diophantine equations.His tombsto
14、ne did not record his lifes achievements likeordinary people,but wrote a math problem.God gave him his boyhood one-sixth of his life,One twelfth more as youth while whiskers grew rife;And then yet one-seventh ere marriage begun;In five years there came a bouncing new son.Alas,the dear child of maste
15、r and sageAfter attaining half the measure of his fathers life chill fate took him.After consoling his fate by the science of numbers for four years,he ended his life.Let Diophantuss age be x,and the equation is 111154061272xxxxx+=The solution is84x=The story of Fermats Last TheoremWhat kind of myst
16、erious exploration of uncertain problemsdid the Pythagorean theorem evoke?Whether FermatsLast Theorem carries another pursuit of quantitativerelationship after the Pythagorean theorem?Fermats Last TheoremIt is impossible to write a cube of a number as a sum of two cubes,a fourthpower as a sum of fou
17、rth powers,and,in general,any power beyond thesecond as a sum of two same powers.:will have infinite solutions.:Pythagorean formula.For an integer n 2,nnnxyz+=has no positive integer solution.=1nxyz+=2n=222xyz+=Gousan gusi xianwu“Zhoubi Suanjing”records a passage from ShangGaos answer to the Duke of
18、 Zhou,“In a right triangle,gou(the short side)equals three,gu(the long side)equalsfour,then jingyu(the hypotenuse)equals five.”Pythagorean theorem:the Pythagorean school also discovered the fact of the Gougu theorem and proved it.Many ancient civilizations such as Babylon,Egypt,India,etc.,have disco
19、vered this fact successively.XianGouGuMysterious laws in algebra and geometryIn the book Euclids elements,Euclid made a generalizationthat described the relationship between sides and angles forany triangle,which is the cosine theorem.2222coscababC=+The Gougu theorem constitutes a basis for the rela
20、tionship of the sides and angles of triangles in Euclidean geometry,and thus becomes a natural starting point for people to understand the law of shapes in the universe.Using the Gougu theorem,ancient Chinese scientists proposed somemethods of calculating the square root,the cube root,and pi.Pythago
21、ras inGreece also found an irrational number using the Pythagorean theorem.The Searching of the Pythagorean NumbersVarious countries people were working on the search for the Pythagorean triples.Number theory.For uncertain equations,naturally we want to give a general expression of all solutionsTheo
22、rem.For the Pythagorean Numbers equation,if 2|y,all its solutions can be expressed as22xab=2yab=22zab=+=+Here .a and b are relatively prime,and one is odd,the other even.0ab,Chinas ancient mathematician Liu Wei gave several groups of Pythagoreantriples in The Nine Chapters on the Mathematical Art.(5
23、,12,13),(8,15,17),(7,24,25),(20,21,29)This search is a method of traversing natural numbers.After Fermat formulate the Fermats Last Theorem,he wrote a paragraph in the margin of this page,Prove of Fermats theoremFermats theorem:For an integer n which is greater than 2,an equation about x,y,and z,+=,
24、has no positive integer solution.“I have found a truly wonderful proof,but the margin is too small to contain it.”In this way,he wrote a few words lightly,but left hard work to his descendants to prove the theorem for more than 300 years.Prove of Fermats theoremFermats theorem:For an integer n which
25、 is greater than 2,an equation about x,y,and z,+=,has no positive integer solution.YearsMathematicianArchievement1770Eulern=3Fermatn=41823Dirichlet and Legendren=51839Lamn=71844Edward KummerProposed the concept of“ideal number”1922Mordellproposed the Mordell conjecture using the concept of genus1982
26、FaltingsProved the Mordell conjectureIn 1983,German mathematician Faltings proved the Mordell conjecture,and Faltings won the 1986 Fields Medal for this.Prove of Fermats theorem John Charles Fields(1863-1932)was a Canadian mathematicianand educator.He actively promoted the International Congress ofM
27、athematicians in Toronto in 1924 and advocated the establishmentof an international mathematics award by using the balance of theconference funds.Fields donated all his inheritance to the award before his death.In1932,the Ninth International Congress of Mathematicians in Zurichaccepted this proposal
28、 and named the award the Fields Medal,which rewards scientists who are under 40 years of age for theirmathematical excellence.ProgressesIn 1955:the Taniyama-Shimura conjecture.In 1985:the Frey curve was proposed.In 1986:the Frey curve was proved.In 1995:the Gushan-Shimura conjecture was proved,and t
29、henWiles won the special honor of the International Congress ofMathematicians.Han Xin Counting Soldiers and Chinese Remainder TheoremWhat isDayanqiuyishu theChineseRemainderTheorem?What is the modern representation used by theancient Chinese mathematical text on the multiple solutionsof uncertain eq
30、uations?Why do indeterminate equationsconceal the advanced thinking of cryptography?Remainder ProblemN:the number of things,x,y,and z:the first,second,and third frequencies.+=+=+=273523zNyNxNIndeterminate Equations：Now there are an unknown number of things.If we count by threes,there is a remainder
31、2;if we count by fives,there is a remainder 3;if we count by sevens,there is a remainder 2.Find the number of things.Concise number theory:2(mod3)3(mod5)2(mod7)N Chinese remainder theorem“the Dayan-Seeking-unity Method”Let N=bi(mod ai),i=1,2,n.If aiis relatively prime,thenThe Chinese mathematician Q
32、in Jiushao of the Song Dynasty introduced a method of solving polynomial equations in Shushu jiuzhang.1 122nnNbkb kb kPM=+where12,1(mod),niiiiMMa aa kuaa=P is an integer and is a positive integer.+=+=+=273523zNyNxN23105215321235321=+=PuuuN,1,2,2321=uuuP123,?u u uP=TheAncient Story of SecrecySun Tzus
33、 art of war:General must be able to mystify his men,and thus keep them in total ignorance.”Han Xin kept secrets to his enemies,even to his own people.Confidentiality method of Han XinLegendary folk story:Han Xin Counting SoldiersIn Huaian,Jiangsu Province,there is a legendaryfolk story of Han Xin Co
34、unting Soldiers.At one time,Han Xin took 1,500 soldiers to fight and four to fivehundred soldiers were lost.Han Xin asked the soldiers to stand in rows of three,and two soldiers were left;to stand in rows of five,four left;to stand in rows of seven,six left.Han Xin immediately knew the number of sol
35、diers now,1049.SolutionThe actual number of soldiers is 1049？Hanxin：completely confidential.123105,35 221 415 6105 104MNuuuP=+=1232,2,1Puuu=SetThen the solution isTransmission of InformationModern cryptography:I can tell the informationto anyone,but people still dont know what theinformation is.Cryp
36、tography Principles everyone could knowunknownXiaofang has one hundred yuan,and her mothertemporarily borrows four yuan from her,so she hasXiaofang asks her mother for four yuan,to restoreher moneyPositive process:subtractioninverse process:additionA message A,through a mapping f,was transformed int
37、o B=f(A),that is fABfAB 1Decrypt.To solve the inverse transformation of f,then we can know the original informationAfB=1()100496964100Real Solutions60、90、120?Answer:the smallest or smaller solution that satisfies the conditions.What is the number that is divisible by 2,3,and 5?30The answer to“Han Xi
38、n Counting Soldiers”problem:with,1049N=1P=18u=26u=31u=Han Xins secret codeWhy cant you remember your mobile number?What kind of disaster would the temperament face ifonly the indeterminate equations are to be solved simply?Facinguncertainty,whyisnaturalselectionmoreimportant?Approximate SolutionsThe
39、 temperament equation：322nN=Diatonic scale：do、re、mi、fa、so、la、ti。Many approximate solutions537.592Twelve-tone scale75.1292312Relative Errors8 :5.1%537.592128 :1.37%2147483648 :0.2%Much smallerMore precise solutions：5332151972563.222231275.1292312Laws of PsychologyAn American psychologist Miller found
40、 that the capacity that people can remember things at a time in short-term memory is very limited the number of items human could instantaneously perceive and memorize is 7 2.Psychological lawThe hundred-mark system:often divided into five grades Excellent for scores over 90,Good for 80-90,Fair for
41、70-80,Pass for 60-70,and Failure for scores below 60.Easy to remember:four-digit number of your house3578A bit difficult:eleven-digit number of your mobile phone39510733521Examination score:five-point scale No problem:seven-digit telephone number of your home3578Disaster of Temperament The twelve-tone scaleMore harmonious,but harder to master.The diatonic scaleHarmonious and easy.The 53-tone scale Nearly harmonious,but useless.Natural Selection An ancient Chinese temperamentdo、re、mi、so、laThe method of natural selection shows the great wisdom of nature?