"Vladimir Arnold forcefully stated in one of his books that it is wrong to think about finite difference equations as approximations of differential equations. It is the differential equation which approximates finite difference laws of physics; it is the result of taking an asymptotic limit at zero. Being an approximation, it is easier to solve and study."

"In summary, one way to compute log factorial is to pre-compute log(n!) for n = 1, 2, 3, … 256 and store the results in an array. For values of n ≤ 256, look up the result from the table. For n > 256, return

(x – 1/2) log(x) – x + (1/2) log(2 π) + 1/(12 x)

with x = n + 1."

"Not only is the error in sin(θ) ≈ θ approximately θ3/6, it is in fact bounded by θ3/6 for small, positive θ. This comes from the alternating series theorem. When you make an approximation from an alternating Taylor series, the error is bounded by the first term you leave out."

"You can get a surprisingly good approximation by simply fitting a line to the first and last points. This is known as “Bancroft’s rule.”"

"When we use np as our approximation, we’re ignoring the terms involving p2 and higher powers of p. When p is small, higher powers of p are very small and can be ignored. ... Now how small do p and n have to be? If you calculate the approximation np and get a small answer, then it’s a good answer. Why? The error in the np approximation is roughly n(n-1)p2/2, which is less than (np)2. And if np is small, (np)2 is very small."

"Roll five dice and use the sum to simulate samples from a normal distribution." ... "The variance of a uniform[0,1] random variable is 1/12. So adding 12 together makes the variance 1. That’s why his trick produces a standard random sample."

"This problem of large relative error is not limited to the t distribution but is typical of normal approximations in general. The normal distribution has very thin tails, and in most applications the normal will be used to approximate a distribution with thicker tails. In that case the relative error in the normal approximation will be large in the tails."

"The real problem with most approximation algorithms is not that the guarantees are too weak or that claimed bounds don't match empirical performance; it's that they suck compared even to easy heuristics, precisely because they're designed for the worst case."

"The art of guessing results and solving problems without doing a proof
or an exact calculation. Techniques include extreme-cases reasoning,
dimensional analysis, successive approximation, discretization,
generalization, and pictorial analysis. Applica