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In seventh grade algebra, I learned how to derive the equation for a line. In this context, a “line” implicitly means a two-way-infinite line, shooting off to infinity in either direction. What isn’t taught in school is the much more difficult and subtle question of how to derive the equation of a finite [...]

Think of me as the Frank Gilbreth of university mathematics: always looking for ways to make it more efficient. There’s a perpetual struggle in mathematics education about how many facts and formulas should be presented vs. how much power they grant. It’s nice to know two ways to solve a task, but in [...]

Continuing the discussion about the limits of computation, which I started with the recent article on The Halting Problem, let me tell you about one of the most fascinating number sequences ever discovered, the sequence of Busy Beaver Numbers.
As you know, every computer program is ultimately stored as just a finite sequence of 0’s and [...]

The Halting Problem is one of the most profound discoveries in computer science. It gives us an idea about the limits of what can be computed and what cannot. In this article, I’ll explain this cunning problem using standard computer terminology, rather than the very formal mathematical objects which are usually used. [...]

When people ask me, “What math should I study so I can (fill in the blank)”, the answer I give them isn’t quite what they expect. The best answer to this question is: whichever mathematics you think is the most fun and interesting. This answer doesn’t depend at all on what (blank) [...]

Starting tonight, I’m doing a 30-day challenge: study logic for one or two hours a day for a month. Why am I suddenly doing this challenge when I’m already a grad student in mathematics? Because the time is approaching for me to take the general exam. This test, also sometimes called the [...]

Although mathematics is famous for its precision and clarity, there are some times when ambiguity creeps in. Here are a few examples. I’ll update this list whenever I encounter new examples.
1. Inequalities
Consider the two inequalities concerning real numbers: x≠x+1 and x≠2x. Both are “true”, but in different senses. The [...]

As a math teacher, I’m always wishing I had more free time with my students, where I could show them some of the advanced real-world applications of higher math. I think students get bored with a lot of the “standard” application examples– measuring bridges, designing skyscrapers; even sending a spaceship to the moon, which we [...]

Once in sixth grade mathclass, I misplaced a graded quiz on fractions. Mrs. Locatelli called me to her desk while we were doing work on our own, and started discussing what we were gonna do, because by sheer misfortune, she had managed to delete the grade as well. The situation looked difficult, until I reminded [...]

The number one murderer of truth is ideology. Ideology: we expect it in politics and religion, government and propaganda. We do not expect ideology in mathematics. Mathematics is supposed to be the one last refuge for absolute truth in a world battered by storms of confusion and uncertainty, right? At least that’s the standard soundbite. [...]