Levelling Up

We enjoy doing comfortable things. We don’t want to stretch ourselves too much, because when you stretch, there’s a possibility of overstretching. We fear this potential negative, so we buck the other way, contenting ourselves with doing activities and pursuing projects that aren’t too risky. It’s not that we don’t want to do better work and overcome new challenges. Ask anybody around you if they want to overcome challenges, and their answer will be “yes”.…

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The Rational Roots Theorem

One of the skills students learn in secondary school is to factor quadratic expressions. In particular, they learn how to solve equations like x2+2x+1=0. There are a slew of techniques one can use to deal with quadratics, and they mostly rely on the fact that questions asked in assignments and tests have “nice” factorizations. Most expressions have integer solutions, or at worst rational ones. This makes it straightforward to factor. Of course, this might take…

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Mathematics Isn’t Just Numbers

We often equate mathematics with numbers, as if mathematics doesn’t extend further than doing arithmetic. Each time this happens, I have to restrain myself from going on a rant. I want to grab the person by the collar and exclaim, “There’s so much more to mathematics than just numbers! It’s like saying that running is just a bunch of one-legged hops. While that might be technically true, it’s not the way most people would describe…

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Degeneracy of the Quantum Harmonic Oscillator

Note: This post uses MathJax for rendering, so I would recommend going to the site for the best experience. I just love being able to find neat ways to solve problems. In particular, there’s something about a combinatorial problem that is so satisfying when solved. The problem may initially look difficult, but a slight shift in perspective can bring the solution right into focus. This is the case with this problem, which is why I’m…

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