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Author Archives: mzspivey
Intuition for the Dual in Linear Programming
One of the most important theoretical results in linear programming is that every LP has a corresponding dual program. Where, exactly, this dual comes from can often seem mysterious. Several years ago I answered a question on a couple of … Continue reading
Posted in linear programming, optimization
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The Sum of Cubes is the Square of the Sum
It’s fairly wellknown, to those who know it, that . In other words, the square of the sum of the first n positive integers equals the sum of the cubes of the first n positive integers. It’s probably less wellknown … Continue reading
Posted in number theory
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Happy Birthday, Benoit Mandelbrot
Today’s Google doodle honors mathematician Benoit Mandelbrot. He would have been 96 today. If you’re interested in learning more about his life and work, the Google doodle link contains a short summary. If you want to go deeper, you can … Continue reading
Posted in complex numbers, fractals, people
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No Integer Solutions to a Mordell Equation
Equations of the form are called Mordell equations. In this post we’re going to prove that the equation has no integer solutions, using (with one exception) nothing more complicated than congruences. Theorem: There are no integer solutions to the equation … Continue reading
Posted in number theory
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Some Video Learning Suggestions
During this COVIDtide many of us have been seeking out online learning resources. I’ve done so quite a bit in the past few months, and I thought I would do a post to recommend some of these. They are all … Continue reading
Posted in education, teaching
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A Lesson on Converting Between Different Bases
We’re in the time of COVID19, and that has meant taking far more direct responsibility for my children’s learning than I ever have before. It’s been a lot of work, but it’s also been fun. In fact, I’ve been surprised … Continue reading
Posted in number theory
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A CoinFlipping Problem
One problem that I’ve assigned when discussing Markov chains is to calculate the expected number of flips required for a particular pattern to appear. (Here I mean a pattern such as heads, heads, heads, or HHH.) In this post I’m … Continue reading
Posted in coin flipping, Markov chains, probability
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Strong Induction Wasn’t Needed After All
Lately when I’ve taught the second principle of mathematical induction – also called “strong induction” – I’ve used the following example to illustrate why we need it. Prove that you can make any amount of postage of 12 cents or … Continue reading
Posted in number theory, proof techniques
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Arguments for 0.9999… Being Equal to 1
Recently I tried to explain to my 11yearold son why 0.9999… equals 1. The standard arguments for (at least the ones I’ve seen) assume more math background than he has. So I tried another couple of arguments, and they seemed … Continue reading
Posted in arithmetic, number theory
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