1. No worries, apology more than accepted :).

2. Good call. Math Wars is the longstanding problem, these skirmishes are the latest battle. Definitely not clear to the unfamiliar reader. I'll have to think how to reword it.

3. Thanks for the clarification. I write "Bad Teacher < Online Learning < Good Teacher" in the sense that a static video [or article] can't react to your confusion, explain a different way, etc. the same way an in-person teacher can. But Khan is in no way a bad teacher, and is probably in the 90th percentile of math teachers (probably the 99th percentile in terms of math background, after doing all the lessons for Khan Academy).

4. Awesome! You might like the following article on imaginary numbers, there was a great discussion on HN: http://news.ycombinator.com/item?id=2712575

Here is an even better discussion on the same topic, and the HN thread from last year: http://betterexplained.com/articles/a-visual-intuitive-guide... https://news.ycombinator.com/item?id=2712575

One great conclusion from this approach is how intuitive it becomes to understand the square root of i. I always thought you'd need another dimension to describe that, and another dimension for the square root of that unit, and so on.

But think of i as a 90° rotation, so applying it twice (squaring it) results in 180° which is -1. Then √i is a 45° rotation so applying it twice results in 90° which is i. Sure enough, this works out. The unit vector at 45° is 0.5√2 + 0.5√2 * i. Follow the rules of complex arithmetic to square that and you do indeed get i.

A rotation of -135° applied twice gives the same result. Square the unit vector of -0.5√2 + -0.5√2 * i and you also get i. We've arrived back at the axiom that all numbers have two square roots of opposite signs.

Last question: What's the cube root of i? Easy: a 30° rotation. The 30° unit vector is 0.5√3 + 0.5i, and cubing that does indeed get you i.