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What's So Great About Polynomials?

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American math education is notorious for introducing mathematical concepts long before it introduces the reason to care about them. Case in point: polynomials. You know them. They're the functions that look like this:   $$f(x) = x^2 - x$$ or like this $$f(x) = -3x^3 + x^2 - \frac{1}{2} x - \pi.$$ You probably learned about these when you took Algebra. If you're like most students, you weren't immediately given a very good reason to care! In this post, I'll try to patch this hole by motivating polynomials for a student at a low level. They're not as unnatural as they might first appear; there is at least one good reason to care about polynomials!   Writing Down Functions is Hard In your math education, you've probably spent a lot of time staring at weird functions like this:   or this:     or this:   Drawing weird squiggly functions is easy. But what if you actually wanted to write one of these down ? Like, think about the function in the last picture he...
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Probabilities Are Less Real Than You Think

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 "Sir, the possibility of successfully navigating an asteroid field is approximately three thousand seven hundred and twenty to one!" -- C-3PO Probability statements tend to carry some of the gravitas and authority of mathematics. I claim that we tend to take these statements too seriously. Real world events do not have intrinsic probabilities. Let's start with an anecdote. A Story About A Big Number I once worked as a counselor at the Ross program, which is a summer camp for mathematically inclined high school kids. (If you are such a high school kid,  you should apply !) During the afternoons, the counselors would often collect around tables in the library to "grade," meaning we would all collaborate to distract each other from our grading duties. One "grading" afternoon, a counselor pulled a random book from the library shelf behind him. He found a checkout receipt inside with a transaction number printed on it. He laid it in the middle of th...
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