I agree -- I'm appreciative that he's trying to explain math to a wider audience, but realistically, reading through textbook definitions at 100wpm just confuse

I agree -- I'm appreciative that he's trying to explain math to a wider audience, but realistically, reading through textbook definitions at 100wpm just confuses people. (Shameless plug starting now). I think we need to focus on the intuition behind _why_ things happen, not repeating the textbook definitions that confused us the first time. Why are radians more natural than degrees? Because they are from the point of view of the mover, not the observer. Would you tell a runner to run 3 laps or run X degrees around the track? (http://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/ http://betterexplained.com/articles/intuitive-guide-to-angle...) What does e mean? It's the result of 100% continuous growth (http://betterexplained.com/articles/developing-your-intuition-for-math/ http://betterexplained.com/articles/developing-your-intuitio...). The infinite Taylor series can be seen, intuitively, as your principal (1), the interest that principal earns (100% of 1 = 1), the interest your interest earns (1/2), the interest that 2nd level interest earns (1/6) and so on. e has simpler definitions and the Taylor series was only used to "explain" sine and cosine. i can be seen as a rotation (http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ http://betterexplained.com/articles/a-visual-intuitive-guide...). But more importantly, it's an "operation which rotates". When you write 3 * i you're saying "take 3 and transform it in some way -- rotate it". When you start combining these ideas (http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-...) something neat arises. You can guess, intuitively, that if e^x means "100% interest for x years" then e^ix means ("100% rotated interest for x years"). What's rotated interest? It's change that pulls you sideways (90 degrees) and not forward (which compounds and makes you go further along the number line). The neat thing about constant, 90-degree growth is that it moves you in a circle (imagine a stick on the ground with a firecracker mounted perpendicular). That's how gravity, etc. work too -- your velocity is always tangent to your position when orbiting. The intuitive reading of "e^ipi = -1" (which is more clear... why the +1 = 0? Oh yes, we wanted to rearrange it so another constant appeared to make it more mystical) is "Start at 1 and intend to grow at 100% interest. Whoops, you are getting rotated interest (because of i). Go for pi seconds... this takes you 100% * pi units around the circle, because rotated interest does not compound (you don't spin faster and faster). Halfway around the circle is -1." In general, sine/cosine give you vertical and horizontal position of an angle, so we can generalize even more, but this is confusing at first. If you focus on the real meaning of the operations, not the DNA-splicing of the Taylor series, it's a lot easier to see what's happening. I didn't make sense of it until years after college because I kept looking at it from a formal mathematical viewpoint, instead of focusing on what clicked for me. Bonus question: if someone claims e^i*pi + 1 = 0 is their favorite formula, ask them what i^i is. If they struggle, the probably only understand the mechanics and not the meaning of it. Even better, have them explain in words why i^i would be a real number.