Comments by "disgruntledtoons" (@disgruntledtoons) on "3Blue1Brown" channel.

  1. When you're an engineer you use Bob's approach the first time you see a problem of a given type. If you see something similar, you remember the solution to the first instance, and try to see if it is adaptable to the present problem. Eventually you find a generalized solution. But you don't worry about generalized solutions until you have a valid reason for seek them (at least, not while you're on the clock). And having found the generalized solution, in every application you also take the time to see if the specifics create a special case that lends itself to an optimization.
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  2. TL;DR: LLMs work by doing a gob-ton of math to predict the next word.
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  3. Complex numbers have great practical use in the analysis of AC circuits. They turn calculus into algebra.
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  4. A 2x2 matrix, where the first row is [a, b] and the second row is [-b, a], emulates all of the behavior of the complex number a + bi. If you raise e to the matrix, you get the same result as for Euler's formula. And so on. I haven't checked to see what 4x4 matrix, if any, emulates a quaternion.
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  5. Of course if you're an engineer and don't need all that many digits of precision, you can (a) assume that there's a Taylor's expansion for the desired behavior over time, and (b) work out a function (with the initial conditions as the arguments of the function) that gives you each coefficient.
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  6. For polynomials higher than degree three, would a method based on quadratic approximations get you there faster? If the parabola intersects the x axis, the x value of the intersection is the next iteration of x; otherwise, the x-coordinate of the parabola's vertex is the next iteration.
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