The Mathematics of Duelling

Duelling with pistols. If you were the one issuing the challenge, your dilemma was that custom dictated that your adversary be allowed to shoot first. Only then, if you were still able to shoot, would you be permitted to seek “satisfaction”.

How much of an advantage does the first shooter really have? In this article, we build a simple probability model, and implement a numerical model in a few lines of R code.

Two gentleman face off in the snow.  Convention dictates the challenged shoots first.

Two gentleman face off in the snow. Convention dictates the challenged shoots first.

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The Importance of Non-Technical Questions to Successful Mathematics Education

Students who are hard-working and otherwise successful, but whose peers, mentors, and home environment are mostly non-technical and disengaged from the ideas behind science and technology, are at substantially higher risk of disorientation, dissatisfaction, and disillusionment with mathematics and science.

In this article, I’ll develop this conjecture and suggest an approach that incorporates philosophical and humanistic elements into technical subjects. To reach and engage a broader popluation of students is critical if mathematics education is to directly contribute to the technical (& technological) literacy of a broader population of students.

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A Course in the Philosophy and Foundations of Mathematics

An examination of mathematical methods and the search for mathematical meaning.

During your studies of mathematics, physics and engineering, you may find yourself distracted or troubled by meta questions about mathematics — questions that fall outside the syllabi of most of the coursework that you’ll take.

For those for whom this itch is persistent, what follows is an outline and reading list for a Course in the Philosophy and Foundations of Mathematics. Among the topics included are the relation of mathematics to science, the examination of mathematical method, and the search for mathematical meaning.

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Finite Summation of Integer Powers (Part 3)

(Discrete Mathematics Techniques III)

We find a direct closed-form solution, i.e. one that does not require iteration, for the general case of the finite-summation-of-integer-powers problem S_p(N) = \sum_{k=1}^{N} k^p. Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the p+1 independent monomials a_j N^j (1 \leq j \leq p+1), where the a_j are unknown coefficients. Using the recurrence relation S_p(N+1) = S_p(N) + (N+1)^p, we obtain a linear combination of the monomials, which reduces to an easily solvable (p+1)-by-(p+1) triangular linear system in the unknown coefficients a_j of the closed-form polynomial solution. Maxima and Octave/Matlab codes for directly computing the closed-form solutions are included in the Appendices.

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Why Zero Raised to the Zero Power IS One

Updated! February 5, 2017

The value of zero raised to the zero power, (0^0), has been discussed since the time of Euler in the 18th century (1700s). There are three reasonable choices: 1,0, or “indeterminate”. Despite consensus amongst mathematicians that the correct answer is one, computing platforms seem to have reached a variety of conclusions: Google, R, Octave, Ruby, and Microsoft Calculator choose 1; Hexelon Max and TI-36 calculator choose 0; and Maxima and Excel throw an error (indeterminate). In this article, I’ll explain why, for discrete mathematics, the correct answer cannot be anything other than 0^0=1, for reasons that go beyond consistency with the Binomial Theorem (Knuth’s argument).
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