Living Discretely in a Continuous World

1. Living Discretely in a Continuous World Trent Kull Winthrop University SCCTM Fall Conference October 23, 2009

2. The set of real numbers

3. Countability • All are infinite sets • Naturals, whole, integers, and rationals are countable • Irrationals are uncountable

4. Density • Every two rationals have another between them – the set is dense • Irrationals are also dense, but “far greater in number”

5. Intuitive discreteness • Discrete: “Spaces” between elements • Can be finite or infinite • Non-discrete: “No spaces,” “continuous” • Can be countable or uncountable

6. Important distinctions • Definitions can vary from text to text. • Texts on “finite mathematics” are often largely concerned with infinite sets. • Texts and courses dealing with discrete mathematics often have detailed (and useful) discussions with continuous sets.

7. Calculus • Calculus texts and courses need and use discrete mathematics. • In fact the two areas – discrete and continuous – can be used as educational enhancements of each other.

8. Discretization in calculus • Discrete sets coupled with limits • Notable discretizations: • Using tables to estimate limits • Using discrete points to estimate slopes of tangent lines with secant lines • Area estimations with rectangles & trapezoids

9. Extending the area problem • Average value • Center of mass • Arc length • Volumes • Work

10. Understanding discretization • Often seems tedious and unnecessary when shortcuts are revealed: • Limit definition of derivative • Infinite sums • Student complaints of “Why?” • Mathematical reality is the computational world largely relies on discrete approximations

11. Binary relations • A relation from a set to a set is a subset of the Cartesian product • Simplistic domains, ranges, graphs

12. Binary relations:Mathematica

13. Finite functions • Vertical line test: “Every input has a single output” • Example • Mathematica

14. Composing finite functions

15. Special types of functions • 1-1: “Every actual output has a single input” • Onto: “Every possible output has an input” • Invertible: “1-1 and onto” • Mathematica

16. An invertible finite function

17. Transitioning (back) to continuous functions • Mathematica

18. With a domain restriction

19. The sine function

20. Enhancing discrete mathematics • Early student familiarity with continuous mathematics • Refer to continuous examples when teaching subtleties of discrete math • Student learning may well benefit from dual discussion

21. Common discretizationsof continuous phenomena • Continuous time & growth • Ages: 1,13,18,21,40, etc. • Heights: 48”, 5’1”,6’ etc. • Irrational ages, heights? • Natural “obsession” with elements of certain discrete sets: a matter of simplicity

22. Discrete sports • Coarse discretizations sufficient • Baseball: 9 innings, 3 outs, 3 strikes, etc. • Golf: 18 holes, -1, par, +1, etc.

23. Discrete sports • Other times finer discretizations are necessary

24. Even finer • Track: World record 100m, 9.58 seconds • Closest finish in Nascar: .002 second separation

25. Digital media • Computer monitor: 1024 x 768 = 786,432 pixels • Digital television: 1920 x 1080 = 2,073,600 pixels • Camera: 5,240,000 pixels

26. Discrete color data

27. Science & engineering • Stephen Dick, the United States Naval Observatory's historian, points out that each nanosecond -- billionth of a second -- of error translates into a GPS error of one foot. A few nanoseconds of error, he points out, "may not seem like much, unless you are landing on an aircraft carrier, or targeting a missile."

28. Discrete dimensions • Dimensions are typically thought of in a discrete manner • Our physical 3 dimensional world: length, width, height • What if we lived in a zero, one, or two dimensional world?

29. Flatland: A Romance of Many Dimensions • 1884 novella • Author: Edwin A. Abbott • Pointland, Lineland, Flatland, Spaceland • “I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space.”

30. Flatland: A Journey of Many Dimensions • 2007 movie • Characters • Square, Hex • Other geometric shapes • Pursuit of knowlege

31. Flatland activity handouts • www.flatlandthemovie.com • Subdividing squares • Edge counts • Pattern recognition • Hypercubes

32. Handout: subdividing squares

33. Handout: Hypercubes • Students work together • Sketch, analyze vertices & edges • Look for patterns • 0-cube, 1-cube, 2-cube, 3-cube, hypercubes

34. The 4th-dimension: DVD extra Professor Thomas Banchoff, Brown University

35. Finer discretizations of dimension: Note that in this relationship: D = log(N)/log(r)

36. Koch curve Union of four copies of itself, each scaled by a factor of 1/3. D = log(4)/log(3) ≈ 1.262

37. Fractal dimensions:Sierpinski Triangle Union of three copies of itself, each scaled by a factor of 1/2. D = log(3)/log(2) ≈ 1.585

38. Fractal dimensions:Menger Sponge D= (log 20) / (log 3) ≈ 2.726833

39. Fractal dimensions:Sierpinski Carpet D = log (8)/log(3) ≈ 1.8928

40. Common use of dimensionsin mathematics • Multivariable calculus • Linear algebra • Mathematica

41. Summary • Study of discrete and continuous mathematics essential for young mathematicians • Digital approximations of our continuous world are well established and increasing in importance • The study of dimensions is both useful and interesting in mathematics and its applications

42. References Slides, handouts, Mathematica file and references will be available at http://faculty.winthrop.edu/kullt/ . Thank you!