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It's about not taking theorems for granted. I'm not a mathematician, but it was quite a realization for me when it hit me that a lot of the things we "found" in high school were... complete magic at …

Yeah, it can: Consider that polynomials of degree 4 or lower are said to be solvable (or solvable "in radicals" or whatever) whereas the higher-order ones are said to be unsolvable. But this is quit …

Two actually useful topics you could talk about are: Calculus of variations (if they're more mathy). Prove that the shortest path between two points is a straight line. Then generalize it to an arbi …

Every trapdoor function I've seen has been a discrete function. Do there exist continuous trapdoor functions? If so, what's an example of a continuous trapdoor function? And if not, why not?

Actually, I think you're wrong. This is not necessarily any indication of a problem. I have had instructors for the most challenging and interesting classes of my life themselves voluntarily state " …

Einstein summation convention is a self-explanatory example. Fourier transforms I feel the majority of people (myself included) abuse notation when describing Fourier transforms. For example, it's co …

Think of a mountain: far away it's nearly flat; near the top it's more steep. If you ask "How steep is it where I am standing right now?", you're asking exactly the same thing as "What is the rate of …

The asymptotic distribution of primes: nth prime ≈ n ln n as n → ∞ Try finding a formula for the nth prime in the non-asymptotic case!

Every single Machine Learning problem ever. Optimal statistical inference is generally much easier in the case of infinite data than finite data.

Here's a really cool application: Complex Step Differentiation The basic idea is that you normally compute a derivative as: $$f'(x) \approx \frac{f(x+h)-f(x)}{h}$$ This requires evaluating $f$ twice. …