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, graphs
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 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
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
Common use of dimensions in 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/http://faculty.winthrop.edu/kullt/. Thank you!