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Mathematics 106

Mathematics 106. A View of Mathematics. 2009 semester 2. Carolyn Kennett Ross Moore Frank Valckenborgh Elena Vinogradova. Organisation of the unit Why mathematics? The number systems of mathematics. Organisation. Lecturers. Carolyn Kennett (unit convener): E7A308 Ross Moore: E7A419

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Mathematics 106

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  1. Mathematics 106 • A View of Mathematics 2009 semester 2 Carolyn Kennett Ross Moore Frank Valckenborgh Elena Vinogradova

  2. Organisation of the unit • Why mathematics? • The number systems of mathematics

  3. Organisation

  4. Lecturers • Carolyn Kennett (unit convener): E7A308 • Ross Moore: E7A419 • Frank Valckenborgh: E7A405 • Elena Vinogradova: E7A403

  5. Organisation • Lectures: 3 one-hour lectures each week • Tutorials: one each week, from week 2 • Assignments: 8 assignments • Test: one test (week 7) • Final exam http://www.maths.mq.edu.au/undergraduate/math106.html http://www.mathacademy.com/pr/minitext/anxiety/

  6. Content (tentative) • This is a unit about some of the concepts and ideas that are important in mathematics, not a unit in mathematics. • The number systems of mathematics • Basic geometry • A taste of calculus • Elementary graph theory • Aspects of probability theory • Symmetry in mathematics • Logic, paradoxes and notions of infinity

  7. Resources • Lectures and lecture slides • Library • Courant & Robbins. 1969. What is Mathematics? • Davis & Hersh. 1981. The Mathematical Experience. • Mac Lane. 1986. Mathematics: Form and Function • Internet • http://www.wikipedia.org/ • http://www.es.flinders.edu.au/~mattom/science+society/

  8. Why Mathematics?

  9. “A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the equivalent of: have you read a work of Shakespeare?” The Two Cultures (Charles Percy Snow, 1959)

  10. “A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the equivalent of: have you read a work of Shakespeare?” The Two Cultures (Charles Percy Snow, 1959) some very important mathematical concept or result

  11. Part of the cultural inheritance of the human species • reading & writing • counting (arithmetic) & geometry • the scientific method • Lingua franca of the physical sciences • Examples: Classical mechanics, electrodynamics, quantum physics, special & general relativity • Information security

  12. Construction of quantitative models of real-world phenomena • interesting predictions/questions • anticipation/reaction • Examples: Chemical kinetics, heat engines, population biology, epidemiology • Intrinsic esthetic properties: the beauty of mathematics

  13. La pensée ne doit jamais se soumettre,ni à un dogme,ni à un parti, ni à une passion, ni à un intérêt,ni à une idée préconçue,ni à quoi que ce soit,si ce n'est aux faits eux-mêmes,parce que, pour elle, se soumettre, ce serait cesser d'être. • Henri Poincaré (1854 - 1912)

  14. Thought may never submit,Either to a dogma,Or to a party,Or to a passion,Or to a vested interest,Or to a prejudice,Or to whatsoever,But only to the facts,Because to submit would meanThe end of all thought. • Henri Poincaré (1854 - 1912)

  15. What is Mathematics?

  16. "We have not succeeded in answering all of our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things." Source unknown "Mathematicians are like Frenchmen: whatever you say to them, they translate into their own language, and forthwith it is something entirely different!" Johann Wolfgang Goethe (1749 - 1832)

  17. Language for quantitative reasoning, developed to solve practical problems • Pure mathematics • Algebra • Analysis • Applied mathematics • Physical sciences • Engineering • Biology, humanities, etc. • The development of mathematics is driven by • intrinsic motives • the interplay between theory and applications

  18. Historical aspects • Prehistoric artefacts • Mesopotamia, Babylonian mathematics • numerical system: sexagesimal (base 60) • place-value system • arithmetic: multiplication tables, division • algebraic equations (linear, quadratic) • Babylonian astronomy Ishango bone (ca. 20000 BCE) Clay tablet YBC 7289 (ca. 1800–1600 BCE)

  19. Historical aspects • Ancient Egypt • numerical system: decimal (base 10) • arithmetic: multiplication, division • geometry: areas and volumes • number theory: prime numbers • sequences: arithmetic, geometric Rhind mathematical papyrus (ca. 1650 BCE)

  20. Ancient Greece and Hellenistic civilisation: μαϑημα (ca. 600 - 300 BCE) • Thales of Miletus (ca. 624 - 548 BCE) • Theorem of Thales: the angle inscribed in a semicircle is a right angle • Pythagorean School (ca. 580 - 500 BCE) • Logical systems for number theory and geometry • Theorem of Pythagoras • irrational numbers • Eudoxus (ca. 408 - 355 BCE) • Aristotle (ca. 384 - 322 BCE) • Euclid of Alexandria (fl. 300 BCE) • Elements: • logically coherent axiomatic system for geometry • method of formal proof, deductive reasoning • Hellenistic period: Archimedes, Hipparchos, Ptolemy Antikythera calculator (ca. 100 BCE)

  21. Islamic mathematics (ca. 600 - 1600 CE) • translations in Arabic of Greek texts • place-value decimal Hindu-Arabic numeral system • decimal point notation • applied mathematics • trigonometry, spherical geometry • some modern mathematical notations • al-Khwārizmī (ca. 780 - 850 CE) (“algorithm”) • algebra: systematic solution of linear and quadratic equations • arithmetic: Arabic numerals • Ibn al-Haytham (Alhazen) (ca. 965 - 1039 CE) • important contributions to geometry and number theory

  22. Chinese mathematics (from ca. 1600 BCE) • Indian mathematics (ca. 3000 BCE - 1600 CE) • Brahmagupta (598 - 668 CE) • decimal Hindu number system • trigonometric functions • differential calculus (1150 CE) • Bhaskara (1114 - 1185 CE) • Jyesthadeva (1500 - 1575 CE) • European mathematics • Renaissance (ca. 1300 - 1700 CE) • Enlightenment (ca. 1650 - 1800 CE) • modern times: explosion of mathematical knowledge

  23. “The road to wisdom? -- Well, it's plainand simple to express:Errand errend err againbut lessand lessand less.” Piet Hein (1905 - 1996) “Problems worthy of attack prove their worth by hitting back.” Piet Hein (1905 - 1996)

  24. The Number Systems of Mathematics

  25. Positive and non-negative integers • Integers • Rational numbers • Real numbers • The Pythagorean theorem • Complex numbers

  26. Aspects of the concept of number • The notion of a concrete quantity • one, two, three, four, many • Counting without numbers • construction of a one-to-one correspondence with a bag of • identical objects, e.g. pebbles (Latin: “calculi”), clay balls • objects of a different shape

  27. Development of an absolute value (non-positional) number system • decimal system (e.g. Egypt, Roman system) • duodecimal system • English: “eleven”, “twelve” • Dutch: “elf”, “twaalf” • vigesimal system (e.g. Aztec, Celts) • sexagesimal system (e.g. Sumerian number system) • Clock system: seconds, minutes

  28. Modern place-value or positional numeral systems • Babylonian sexagesimal number system (ca. 1800 BCE) • Chinese decimal system (200 BCE - 200 CE) • Indian decimal system (ca. 300 BCE - 450 CE) • Hindu - Arabic number system • the concept of an explicit “zero” • numbers as abstract concepts • Maya vigesimal system (ca. 500 CE) • Computer science: binary & hexadecimal systems • Approximations: Feynman contra the abacus

  29. Positional and non-positional number systems • Roman contra Hindu-Arabic

  30. Additional examples: CD + DC = M CXLII x IX = MCCVXXVIII Elementary calculations in a non-positional system quickly become complicated!

  31. 123 x 123 369 246 123 15129 42 x 42 84 168 1764 1 1 1 • Elementary calculations in a positional number system can be performed in a systematic way 123 + 456 579 1 1 1 88888 + 222 89110 123 x 123 = (123 x 3) + (123 x 20) + (123 x 100)

  32. 1231 - 788 443 1239 - 123 1116 102 12) 1231 12 03 0 31 24 7 125 4) 500 4 10 8 20 20 0 0 1 2 500 = 4 x 125 1231 = (12 x 102) + 7

  33. The positive integers • Origin: counting • Representation of the positive integers • decimal system • binary system • hexadecimal system • Prime numbers and composite numbers • Divisibility by positive integers • Sieve of Eratosthenes • Factorisation in prime numbers

  34. Representation of integers • Decimal system: base 10 • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Examples • [42]10 = (4 x 101) + 2 = 42 • [2009]10 = (2 x 103) + (0 x 102) + (0 x 101) + 9 = 2009 • [12345]10 = (1 x 104) + (2 x 103) + (3 x 102) + (4 x 101) + (5 x 100) = 12345 base

  35. Representation of integers • Binary system: base 2 • Digits: 0, 1 • Examples • [8]10 = (1 x 23) + (0 x 22) + (0 x 21) + 0 = 1000 • [21]10 = (1 x 24) + (0 x 23) + (1 x 22) + (0 x 21) + 1 = 10101 • [42]10 = (1 x 25) + (0 x 24) + (1 x 23) + (0 x 22) + (1 x 21) + 0 = 101010 base 10

  36. Representation of integers • Hexadecimal system: base 16 • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F • Examples • [8]10 = 8 x 160 = 8 • [21]10 = (1 x 161) + (5 x 160) = 15 • [42]10 = (2 x 161) + (10 x 160) = 2A • [12347]10 = (3 x 163) + 59 = (3 x 163) + (3 x 161) + (11 x 160) = 303B • [AA]16 = (10 x 161) + (10 x 160) = 160 + 10 = [170]10

  37. Prime numbers and composite numbers • Some positive numbers divide other positive numbers, in the sense that we obtain again an integer after the division • Examples: • 5 divides 10, since 10 = 2 x 5 • 3 divides 27, since 27 = 3 x 9 • 2 does not divide 5, since 5 = (2 x 2) + 1 • 33 divides 99, since 99 = 3 x 33 • Prime numbers are the primitive building blocks for the positive integer numbers

  38. Prime numbers and composite numbers • Tricks with arithmetic: • A number is divisible by 2 if and only if its decimal representation is even. • A number is divisible by 3 if and only if the sum of the digits in its decimal representation is divisible by 3. • A number is divisible by 5 if and only if the last digit of its decimal representation is a 0 or a 5. • A number is divisible by 6 if and only if it is divisible by 2 and 3. • A number is divisible by 9 if and only if the sum of the digits in its decimal representation is divisible by 9. • A number is divisible by 11 if and only if the alternating sum of the digits in its decimal representation is divisible by 11.

  39. Prime numbers and composite numbers • Examples: • 315 is divisible by 3, since 3+1+5 = 9 is divisible by 3 • 315 is divisible by 9, since 3+1+5 = 9 is divisible by 9 • 315 is divisible by 5, since its last digit equals 5 • 315 is not divisible by 6, since it is an odd number • 315 is not divisible by 11, since 3-1+5 = 7 is not divisible by 11 • 315315 is divisible by 11, since 3-1+5-3+1-5 = 0 is divisible by 11 • All these results follow from the properties of congruence relations in number theory

  40. 315 = (3 x 100) + (1 x 10) + 5 315 - (3 + 1 + 5) = (3 x 99) + (1 x 9) + (5 - 5) is always divisible by 9 is divisible by 9 in this case Consequently, 315 is divisible by 9 Prime numbers and composite numbers • Worked example:

  41. Prime numbers and composite numbers • Prime number: a positive number for which the only positive integers are 1 and the number itself • Examples: 2, 3, 5, 7, 11 • Remark: the number 1 is, by convention, not considered a prime! • Composite number: a positive number which is not a prime • Examples: • 4, since 4 is divisible by 1, 2 and 4 • 30, since 30 is divisible by 1, 2, 3, 5, 6, 10, 15 and 30

  42. Prime numbers and composite numbers • The sieve of Eratosthenes (ca. 276 - 195 BCE): • Algorithm to obtain all primes up to a specified integer

  43. Prime numbers and composite numbers • Fundamental theorem of arithmetic: • Each positive integer can be expressed as a product of prime numbers. • This decomposition is unique. • Finding this decomposition, is quite hard. • Examples: • 98 = 2 x 49 = 2 x 7 x 7 • 123 = 3 x 41 • 315315 = 3 x 3 x 5 x 11 x 637 = 3 x 3 x 5 x 11 x 7 x 91 = 3 x 3 x 5 x 11 x 7 x 7 x 13

  44. Prime numbers and composite numbers • There are infinitely many primes • Argument: By contradiction! We assume that there exist only a finite number of primes, and so we can enumerate them. There is a first one, say p1, and a last one, say pN; if there would be 66 primes, then N=66; if there would be one million, then N=1000000; the exact number is not important for the argument. The product of this finite number of primes, is positive. Consider the next integer, x = (p1 x . . . x pN) + 1. This integer is not divisible by any of the primes p1, p2, . . ., pN, since the remainder is always 1. For example, • x = p1 x (p2 x . . . x pN) + 1 • and so x is not divisible by the first prime. Now x is strictly larger than all primes, and so x is not a prime. Consequently, it is a composite, hence it should be divisible by a prime, and we obtain a contradiction.

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