By Assad Ebrahim, on January 21st, 2018 (52,099 views) |
Topic: Maths--Technical
2nd ed. January 21, 2018; 1st ed. Feb 8th, 2010
Abstract
This three part paper explores solving the sum of powers problem  using discrete maths techniques (recurrence relations, matrix systems) to obtain a solution polynomials whose coefficients turn out to be exactly the Bernoulli numbers  .
Part 1 (this paper) solves the problem using recurrence relations in a way which a high school student could emulate for small  . In Part 2, we develop a general recursive solution that works for arbitrary , from which we can build a table of values to assist in finding the coefficients of the solution polynomial, coefficients that are precisely the Bernoulli numbers discovered in 1713. In Part 3, we show how by transforming the problem into a linear system, we may obtain a direct (non-recursive) solution which directly calculates the Bernoulli number for any power . Source code is provided for all solutions.
Readers who are interested in this topic are referred also to lovely paper by Bearden (March 1996, American Mathematical Monthly), which tells the mathematical story and fills in the history (thanks to a reader for this great reference).
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By Assad Ebrahim, on January 20th, 2018 (21,726 views) |
Topic: Maths--Technical
By Assad Ebrahim, on January 19th, 2018 (12,688 views) |
Topic: Mathematics, Maths--Technical
(Discrete Mathematics Techniques III)
1st ed. Apr 2nd, 2010
Abstract
This is the last in the 3-part series of articles on finding for oneself the solution to the sum of integer power problem, and in the process discovering the Bernoulli numbers. In Part 3 (this paper), 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  . Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the  independent monomials  ( ), where the  are unknown coefficients. Using the recurrence relation  , we obtain a linear combination of the monomials, which reduces to an easily solvable  -by- triangular linear system in the unknown coefficients of the closed-form polynomial solution. Maxima and Octave/Matlab codes for directly computing the closed-form solutions are included in the Appendices.
A lovely paper by Bearden (March 1996, American Mathematical Monthly), which was shared with me by a reader, tells the mathematical story nicely, with much of the history filled in.
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By Assad Ebrahim, on February 25th, 2010 (50,136 views) |
Topic: Mathematics, Maths--Technical
Updated! February 5, 2017
The value of zero raised to the zero power,  , 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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By Assad Ebrahim, on February 22nd, 2010 (63,824 views) |
Topic: Maths--Tools
If you haven’t done so already, you may want to start by reading the Preface to the Computing Series: Software as a Force Multiplier, Sections 1-3.
Maxima: a Computer Algebra System (CAS) for symbolic computation
Last updated: Feb 19, 2023 (fixed links). Nov 11, 2022 (added omega-math’s excellent web interface, and generating function calculation of the partition of integers problem).
Maxima is a computer algebra system (CAS) for symbolic computation that is free, open source, runs on multiple operating systems (Win,Mac,Linux), and covers a wide range of mathematical capabilities and graphical capabilities. These include algebraic simplification, polynomials, methods from calculus, matrix equations, differential equations, number theory, combinatorics, hypergeometric functions, tensors, gravitational physics, PDEs, nonlinear systems, plus including 2-D/3-D plotting and animation. With a large and responsive user community, there is plenty of help to get up the learning curve, and with its active developer base, Maxima and its ecosystem continue to gain capability, including a fantastic web interface by Omega-Math/Vroom-Labs (see the screenshot below, r0*0). The result is a free, versatile, powerful mathematical computing package for engineers, scientists, mathematicians, programmers, and students. This article will help you get started with Maxima and set you up with resources to flatten the learning curve.
 Omega-Math’s web interface to Maxima. Used here to calculate the first 10 elements of p(n), the number of ways to partition integer n, using a generating function comprising a truncated series of polynomials up to degree n=10
By Assad Ebrahim, on January 31st, 2010 (8,430 views) |
Topic: Education, Maths--Philosophy
The mathematician Alfred North Whitehead observed that “[Advancement occurs] by extending the number of important operations which we can perform without thinking of them.” (Introduction to Mathematics, 1911 ) This is certainly true in mathematics where the development of judicious notation, accompanied by good mathematical technique, extends the capability to perform chains of complex reasoning accurately and efficiently. Through proper problem formulation (tractable yet generalizable), one can sometimes pass from a single insight to the solution of a family of problems, and in some cases, to the solution to the general question itself.
Here, mathematical computing can provide a useful benefit: helping to efficiently explore conjectures, dispatch with false directions, and save time during the development, error-checking and validation stages of obtaining general results. In industry, where specific or semi-general results are needed fast, such tools allow rapidly working up the required material and providing the necessary certainty before the fully general results or complete proof are ready.
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, the coefficients of whose solution are the famous Bernoulli numbers (1716). In this paper we show to how obtain a 
for any given 