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Volume 2, Number 47

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24 November 1997                                   Vol. 2, No. 47


  CAI - Barker | Fractals - Bourke | Irrationals (MATH-TEACH)


A discussion of the use of the computer as a vehicle for 
supplementary mathematics instruction, and a brief history
of computer-aided instruction: its strengths and weaknesses,
the WWW, potential changes in teaching, and future trends.

Chris Barker is the author of two calculus@internet
Mathematica Track units that use a Web browser as the front 
end and Mathematica as the primary calculation engine.
The courses provide Mathematica basics: an overview, the
environment, arithmetic, using variables, working with
lists, algebraic operations, functions, graphics, vectors
and matrices, and calculus operations.

     - roots, domain and range
     - fitting exponential data
     - power functions
     - calculating velocities
     - the derivative function
     - Riemann sum pictures
     - calculating Riemann sums
     - symbolic derivatives
     - the first derivative and graphing
     - extrema
     - Newton's method.


   Learning the tools of the trade
     - creating slope fields
     - sketching graphical solutions
     - using differential equation, IVP, and numerical solvers
     - solving systems with the computer
   Applications of initial value problems
     - a suspended wire
     - the simple pendulum
     - population dynamics
     - electric circuits
   Applications of systems of differential equations
     - predator-prey problems
     - epidemiology problems
   Numerical methods
     - Euler's method


                      WHAT IS A FRACTAL? 


      (For faster access to this graphics-intensive
       page, load first without images.)
    "What is a fractal?

      B. Mandelbrot
        A rough or fragmented geometric shape that can be 
        subdivided in parts, each of which is (at least 
        approximately) a reduced-size copy of the whole.

        A set of points whose fractal dimension exceeds its 
        topological dimension."

Paul Bourke discusses and supplies detailed illustrations
for the Mandelbrot set, strange attractors, the Newton 
Raphson method, diffusion limited aggregation (DLA), fractal 
geometry (the von Koch snowflake), fractal landscapes, 
L-systems, and iterated function systems (IFS).

           Don't miss Paul Bourke's Fractal Site:



            A discussion from the MATH-TEACH list:

"I've been having a discussion with my student-teachers about 
 the conceptual difference between rationals and irrationals.  
 ... One student claims that there is no difference between
 constructing a 2 by 2 square and constructing a root(2) by 
 root(2) square. What do you think?" - Genevieve Boulet

    A conversation that ventures into a discussion of 
    irrationals in different bases, including base pi, 
    in order to ask whether the distinction between 
    rationals and irrationals has importance in physics 
    or engineering.

"When you go to straightedge-and-compass construction, the
 important issue is constructible and not constructible 
 lengths. So a 2x2 square and a sqrt(2)xsqrt(2) square 
 are pretty much the same; both can be constructed easily 
 enough given a unit segment. Now I'll be a bit heretical:  
 Irrationals are physically unnecessary (though of course
 they are mathematically necessary!)... the importance of 
 irrationals is in the realm of mathematics, not in the world 
 of construction." - Joshua Zucker

"The problem is that we have decided that topics must be
 pursued using the magic 4 (numerically, verbally.. etc.) 
 and so we HAVE to come up with some geometric demonstration 
 of irrational numbers. I think we should only do this when 
 it makes the concept clearer. In this case, I think 
 non-commensurable behavior is more subtle than other 
 approaches to irrational numbers. If the connection were 
 clearer, it is more likely that the Greeks would have 
 become more sophisticated in their conceptionalization of 
 irrational quantities. (But 2000 years later we still have 
 fun with Xeno's paradox.)..." - Ron Ferguson

" real life you never use irrationals, just rational 
 approximations. After all, within any small distance from 
 any irrational there are infinitely many rationals that 
 could be used to approximate it. This is true no matter how 
 small 'small' is. There is no measuring device that could 
 detect whether a number is rational or irrational. And, of 
 course, computers represent all real numbers as rationals 
 (decimal, binary, whatever) by rounding off to whatever 
 precision they can handle... I agree, irrationals are very 
 mysterious!" - Susan Addington

"I also agree. This morning, however, I walked four miles 
 from home to work in 56 minutes. The interesting thing is 
 that I was an irrational distance from home 56 minutes and 
 a rational distance from home 0 minutes." - Harry Sedinger

"Unless you insist that physics or engineering problems are
 not 'real life',... irrationals are very often used in
 'real life' and play a very important role in providing 
 meaningful scientific results." - Chi-Tien Hsu


The Math Forum's Epigone software archives, threads, and
searches the MATH-TEACH discussions:


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