Report

What is Math? Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 1 1 Math = Language Syntax / grammar Vocabulary Communication between mathematicians, student/teacher, ... Math > Language Math < Language You can make statements that are difficult or impossible to make in natural languages. Truth-values of statements can be decided within the system. You can’t say “I love you” Niels Østergård, Birkerød Gymnasium, May 2006 Mathematical statements do not refer to the real World. ...or do they? Lesson 1 2 Scale of abstraction Crafts Engineering Exact Sciences Concrete Specific Synthesis Matter Reality Niels Østergård, Birkerød Gymnasium, May 2006 Applied Math Lesson 1 Pure Math Meta Math Abstract General Analysis Ideas Model 3 Mathematical modelling Reality Model Ferryboat Plastic boat Paper on desktop Classical geometry Classical geometry Coordinate geometry All sets with three elements Number three Union of disjoint sets Addition 3 3+2=5 Abstraction Lesson 1 Niels Østergård, Birkerød Gymnasium, May 2006 4 Math is a formal system Whenever you ask “why” and get an answer, you can ask “but why that”? This leads to infinite regress. You have to stop somewhere, take something on faith, as rules of the game. In math, besides the laws of logic, we have axioms or postulates, i.e. simple (primitive) properties of simple undefined concepts. The axioms are postulated, not proved. New theorems (more complicated properties of the concepts) are proved by applying deductive logic to axioms and to already proven theorems. Mathematical theorems are true in an absolute sense different from scientific truth (i.e. empirical or inductive truthsLesson like 1“The sun will rise tomorrow”). Niels Østergård, Birkerød Gymnasium, May 2006 5 Euclidean geometry Fundamental concepts point distance circle line angle parallel Axioms / postulates 1. Unique line segment between any two points 2. Unique indefinite extension of line segment 3. Unique circle with any centre and radius 4. All right angles are equal 5. Parallel postulate, e.g.: If two lines are drawn which intersect a third 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 enough. Or: Given any line and any point not on the line, one and only one line exists through the point that doesn't intersect the first line. Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 1 6 Euclidean plane geometry vs. spherical geometry Concept Plane Sphere "point" point pair of opposite points "line" "circle" Parallel postulate straight line circle great circle minor circle Valid Invalid Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 1 7 Sum of angles Proof of angular sum A+B+C = 180° for arbitrary triangle: This proof uses the parallel postulate, so it is not valid for spherical triangles! Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 1 8 Division of circles Problem: A number of points are marked on the perimeter of a circle, and all the lines connecting pairs of these points are drawn. Into how many regions can the lines cut the circle for a certain number of points? Notation: Let us call the number of points n, and let us call the largest possible number of regions for a given number of points R(n). Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 1 9 Solution: Let us consider the first few cases: n Figures R(n) n Figures R(n) 1 1 4 8 2 2 5 16 3 4 6 ? Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 1 10 The case n = 6 2 9 3 11 12 13 10 26 14 27 15 16 1 8 25 28 31 18 29 7 24 23 17 4 19 30 22 21 20 5 6 R(6) = 31 Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 1 11 Number of subsets Problem: How many different subsets does a set with n elements have? Note: The empty set (Ø) and the set itself are always counted as subsets. Notation: The number of subsets is called S(n). N 0 1 2 3 4 Set Subsets Ø Ø {a} Ø, {a} {a,b} Ø, {a}, {b}, {a,b} {a,b,c} Ø, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c} abcd Ø, a, b, c, d, ab, ac, ad, bc, bd, cd, abc, abd, acd, bcd, abcd 5 abcde Ø, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde, abcd, abce, abde, acde, bcde, abcde 6 abcdef Niels Østergård, Birkerød Gymnasium, May 2006 S(n) 1 2 4 8 16 32 Lesson 1 12 Better approach Listing the subsets in a different order, a pattern emerges: n Set Subsets S(n) 0 Ø 1 a 2 ab 3 abc 4 abcd 5 abcde 1 Ø Ø 2 Ø {a} = {a} Ø a Ø {b} = {b} {a} {b} = {a,b} 4 Ø {a} {b} {a,b} Ø {c} = {c} {a} {c} = {a,c} {b} {c} = {b,c} {a,b} {c} = {a,b,c} Ø a b ab c ac bc abc Ød = d ad = ad bd = bd ... = abd cd acd bcd abcd 8 16 32 Mathematical induction For n = 0 For any n>0 Obviously S(0)=1. Given the S(n-1) subsets of the set with n-1 elements, the S(n) subsets of the set with n elements are: The same S(n-1) sets, plus the same S(n-1) sets with the new element added to each of them. Thus S(n) = 2·S(n-1). n As this argumentBirkerød works for any valueMay of n,2006 the doubling sequence Niels Østergård, Gymnasium, Lesson 1 (given by the formula S(n) = 2 ) will go on for ever. 13 What is Good Math? Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 2 14 Tennis tournament A tennis club arranges a knockout tournament in which the winner takes it all. There are 843 participants. Round Matches No. 1st In the first round of the tournament, the 843 players are matched; one odd player is left out in this round. This means that 842/2=421 matches are played. 421 2nd In the second round, the 421 winners plus the odd player are matched. Thus, 422/2=211 matches are played. 211 3rd In the third round, the 211 winners from the second round are matched; again one odd player must be left out. 105 and so on Last In the last round, only two winners are left so only one match is played – the final! Problem: How many matches were played before the final winner was found? (A relevant question if you have to pay the ballboys!) Solution: 421+211+105+ . . . +1 = Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 2 1 Sum ? 15 Better solution At the end of the tournament, all players except the final winner will have lost exactly one match. In each match there is exactly one looser. Therefore, the number of matches equals the number of players excluding the final winner. Suppose the tournament has n participants. Then, n-1 matches will be played before the winner is found! What qualities make this a better solution? Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 2 16 Qualities Beautiful Elegant Eye opener Aha! General Productive Practical Useful Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 2 17 You need math Most occupations today involve math Numbers and calculations, e.g. related to economy Formal systems, e.g. computer programs and instruction manuals Many news stories involve math (and very often get it wrong) Economy, large numbers Science and technology Graphs, diagrams and tables Statistics and percentages In a democracy, as many citizens as possible should understand as much as possible about society. Your everyday life involves math Interests, compound interests and mortgage Percentages, e.g. moms (VAT) and discounts Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 2 Measurements for interior decoration... 18 You need math Most occupations today involve math Numbers and calculations, e.g. related to economy Formal systems, e.g. computer programs and instruction manuals Many news stories involve math (and very often get it wrong) Economy, large numbers Science and technology Graphs, diagrams and tables Statistics and percentages In a democracy, as many citizens as possible should understand as much as possible about society. Your everyday life involves math Interests, compound interests and mortgage Percentages, e.g. moms (VAT) and discounts Niels Østergård, Birkerød Gymnasium, May 2006 Lesson 2 Measurements for interior decoration... 19