living discretely in a continuous world

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!