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24 November 1997 Vol. 2, No. 47 THE MATH FORUM INTERNET NEWS CAI - Barker | Fractals - Bourke | Irrationals (MATH-TEACH) COMPUTER-AIDED INSTRUCTION - CHRISTOPHER BARKER http://calculus.sjdccd.cc.ca.us/Board/BoardHome.html 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. CALCULUS I http://calculus.sjdccd.cc.ca.us/CalcIMMA/CalcIMMA-h.html - 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. DIFFERENTIAL EQUATIONS http://ode.sjdccd.cc.ca.us/ODE/ODE-h.html 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? AN INTRODUCTION TO FRACTALS - PAUL BOURKE http://www.mhri.edu.au/~pdb/fractals/fracintro/ (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. Mathematical 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: http://www.mhri.edu.au/~pdb/fractals/ -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- IRRATIONALS A discussion from the MATH-TEACH list: http://mathforum.org/epigone/math-teach/werglilquen "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 "...in 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: http://mathforum.org/epigone/math-teach -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- CHECK OUT OUR WEB SITE: The Math Forum http://mathforum.org/ Ask Dr. Math http://mathforum.org/dr.math/ Problem of the Week http://mathforum.org/geopow/ Internet Resources http://mathforum.org/~steve/ Join the Math Forum http://mathforum.org/join.forum.html Send comments to the Math Forum Internet Newsletter editors _o \o_ __| \ / |__ o _ o/ \o/ __|- __/ \__/o \o | o/ o/__/ /\ /| | \ \ / \ / \ /o\ / \ / \ / | / \ / \ |

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