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Living Discretely in a Continuous World. Trent Kull Winthrop University SCCTM Fall Conference October 23, 2009. The set of real numbers. Countability. All are infinite sets Naturals, whole, integers, and rationals are countable Irrationals are uncountable. Density. - PowerPoint PPT Presentation
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Living Discretely in a
Continuous World
Trent KullWinthrop University
SCCTM Fall ConferenceOctober 23, 2009
The set of real numbers ,3,2,1:Naturals
,3,2,1,0:Whole
2,-3,3,0,-1,1,-2,:Integers
0,: qq
pRationals
2,,:)( eelseallsIrrational
Countability
• All are infinite sets• Naturals, whole, integers, and rationals are
countable• Irrationals are uncountable
Density
• Every two rationals have another between them – the set is dense
• Irrationals are also dense, but “far greater in number”
Intuitive discreteness
• Discrete: “Spaces” between elements• Can be finite or infinite• Non-discrete: “No spaces,” “continuous”• Can be countable or uncountable
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.
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.
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
Extending the area problem
• Average value • Center of mass• Arc length• Volumes• Work
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
Binary relations
• A relation from a set to a set is a subset of the Cartesian product
• Simplistic domains, ranges, graphsYX
X Y
(4,5)(3,3),(2,3), R2
(4,4)(3,6),(3,3),(2,6),(2,4), R1
3,4,5,6,7Y ,2,3,4 X
R
Binary relations:Mathematica
Finite functions
• Vertical line test: “Every input has a single output”
• Example
• Mathematica
(4,4)(3,4),(2,1),(1,1), f
1,4 Y ,1,2,3,4 X
Composing finite functions
E)(d,B),(c,(b,6),A),(a, gf Z,Y :
(d,5)(c,3),(b,2),(a,1), g X,Y :g
E)(5,(4,7),B),(3,(2,6),A),(1, f Z,X :f
EB,7,A,6,Z ,dc,b,a,Y,1,2,3,4,5X
gf
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
(4,16)(3,9),(2,4),(1,1), f
1,4,9,16 Y , 1,2,3,4 X
An invertible finite function
(16,4)(9,3),(4,2),(1,1),
(4,16)(3,9),(2,4),(1,1),
1,4,9,16 , 1,2,3,4
1-
f
f
YX
xffxxff 11
ffff 11 (4,4)}(3,3),(2,2),{(1,1),
Transitioning (back) to continuous functions
• Mathematica
2
2
2
,|,
),,((1/2,1/4),(-3,9),(2,4),(-1,1),(1,1),
,,
xyandxyxf
f
xxxf
invertiblenotfnotf ,11
With a domain restriction
invertiblef
ontoandisf
xyandxyxf
xxxf
,011
0|,
,0,2
2
xffxxxxff
xf
1221
1
The sine function
,0),(,1),((0,0),xsin 2
xxx
uniquenessexistence
ityinvertibilensuresnrestrictioDomain
sinarcsinarcsinsin
)&(
1,1,11 ontoisbutnotClearly
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
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
?2,, oldyearseyouwereWhen
Discrete sports• Coarse discretizations sufficient• Baseball: 9 innings, 3 outs, 3 strikes, etc.• Golf: 18 holes, -1, par, +1, etc.
Discrete sports• Other times finer discretizations are necessary
Even finer• Track: World record 100m, 9.58
seconds• Closest finish in Nascar: .002
second separation
Digital media• Computer monitor: 1024 x 768 = 786,432
pixels• Digital television: 1920 x 1080 = 2,073,600
pixels • Camera: 5,240,000 pixels
Discrete color data
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."
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?
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.”
Flatland: A Journey of
Many Dimensions
• 2007 movie• Characters
– Square, Hex– Other geometric shapes
• Pursuit of knowlege
Flatland activity handouts• www.flatlandthemovie.com• Subdividing squares• Edge counts• Pattern recognition• Hypercubes
Handout: subdividing squaresnxn Vertices Edges Unit sq. V+E+S V-E+S
0x0 1 0 0 1 1
1x1 4 4 1 9 1
2x2 9 12 4 25 1
3x3 16 24 9 49 1
4x4 25 40 16 81 1
5x5 36 60 25 121 1
nxn (n+1)2 2n(n+1) n2 (2n+1) 2 1
Handout: Hypercubes
• Students work together• Sketch, analyze vertices & edges• Look for patterns• 0-cube, 1-cube, 2-cube, 3-cube, hypercubes
The 4th-dimension: DVD extra
ProfessorThomas Banchoff,Brown University
Finer discretizations of dimension:
Note that in this relationship:
D = log(N)/log(r)
Koch curve
Union of four copies of itself, each scaled
by a factor of 1/3.D = log(4)/log(3) ≈
1.262
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
Fractal dimensions:Menger Sponge
D= (log 20) / (log 3) ≈ 2.726833
Fractal dimensions:Sierpinski Carpet
D = log (8)/log(3) ≈ 1.8928
Common use of dimensionsin mathematics
• Multivariable calculus• Linear algebra• Mathematica
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
References
Slides, handouts, Mathematica file and references will be available at
http://faculty.winthrop.edu/kullt/ .
Thank you!