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

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The set of real numbers

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

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

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

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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.

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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.

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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

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

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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

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

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Binary relations: Mathematica

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

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Composing finite functions

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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

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An invertible finite function

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Transitioning (back) to continuous functions Mathematica

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With a domain restriction

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The sine function

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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

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Common discretizations of 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

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

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Discrete sports Other times finer discretizations are necessary

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

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

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Discrete color data

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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."

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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?

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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.”

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

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

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Handout: subdividing squares nxnVerticesEdgesUnit sq.V+E+SV-E+S 0x010011 1x144191 2x29124251 3x316249491 4x4254016811 5x53660251211 nxn(n+1) 2 2n(n+1)n2n2 (2n+1) 2 1

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

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

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

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

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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

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

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

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Common use of dimensions in mathematics Multivariable calculus Linear algebra Mathematica

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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

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

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