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2014-06-21: Kurtosis, 4th order diffusion, and wave speed

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2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

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2014-03-05: Integration techniques: Fourier--Laplace Commutation

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2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

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2013-12-09: 2013 Flu Shots at PSU

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2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the area of a circle using elementary methods

Dual numbers are really just calculus infinitessimals

There is a nice page on the internet called Essential Math that has a nice collection of mathematics topics that are useful for the computational geometry and other hard aspects of computer game programming. It's interesting just to see how the emphasis of mathematical applications is changing the relative importance of various techniques.

One of the topics, which I'd never heard of before is something called dual numbers. Well, "that's something with an air of novelty", I thought, so I read up a little. Turns out, that while it does get some attention currently, it isn't too new an idea. In fact, it's just a special case of Liebnitz, Newton, and Archimede's ideas of infinitessimal arithmetic. I've written about infinitessimals some last year. For dual numbers, one is really just doing arithmetic with numbers of the form

\[ a(x) + b(x) dx \]

using the standard nilpotent infinitessimal product rule that

\[ dx^2 = 0.\]

So, while it seems a bit like magic at first pass, you really aren't getting any farther allong, than you are with standard calculus. Some people have said that this avoids the need for any symbolic differentiation, but any good graduate applied-math student should be able to point out that differentiation of polynomials doesn't require numerical differentation techniques anyway -- it can all be done with matrix operators.

That's not to say that infinitessimal arithmetic isn't going to be a useful trick sometimes -- I think it really is. But dual_numbers are just a new coat of paint on an old reliable work-horse.