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from /.: Trigonometry Redefined without Sines and Cosines
Basically, this guy has replaced distance with "quadrance" (squared distance) and angle with "spread" (squared sine) and worked out trig on this basis. Apparently he thinks that distance is confusing because it involves taking a square root, and angle is confusing because "defining an angle correctly requires calculus" [emphasis his] and thus "cannot be fundamental".
Yes, calculating the precise value of an angle given the two lines that define it (in Cartesian coordinates) is mathematically problematic. But angle itself is pretty intuitive, in a way that I have trouble imagining "spread" to be. Also, it is occasionally useful to talk about angles > 180 degrees...which "spread" cannot represent.
As for "quadrance"...certainly I've used that in place of distance when all I needed to do was compare two distances (i.e., I didn't need to know the magnitude, just the relative magnitudes). But the notion that quadrance is more fundamental than distance because the mathematical calculation of quadrance doesn't involve a square root (this guy avoids roots like the plague) seems, well, sort of silly.
Fundamentally, I think he needs to get out more often. Then he might realize that distance and angle are used because they are easy to _measure_ (not calculate) and thus more susceptible to intuition.
I'd like to ask this guy to write down navigation directions using quadrance and spread as his vocabulary. ;)
This all reminds me of the representation of numbers (for computational purposes) as a vector of their prime factors (so 36 would be represented by [2 2 0 0 ... 0] and 11 would be represented by [0 0 0 0 1 0 ... 0]. This turns out to be (a) really handy for multiplication (just add the vectors), (b) really a pain for addition, and (c) completely unintuitive to most people.
Basically, this guy has replaced distance with "quadrance" (squared distance) and angle with "spread" (squared sine) and worked out trig on this basis. Apparently he thinks that distance is confusing because it involves taking a square root, and angle is confusing because "defining an angle correctly requires calculus" [emphasis his] and thus "cannot be fundamental".
Yes, calculating the precise value of an angle given the two lines that define it (in Cartesian coordinates) is mathematically problematic. But angle itself is pretty intuitive, in a way that I have trouble imagining "spread" to be. Also, it is occasionally useful to talk about angles > 180 degrees...which "spread" cannot represent.
As for "quadrance"...certainly I've used that in place of distance when all I needed to do was compare two distances (i.e., I didn't need to know the magnitude, just the relative magnitudes). But the notion that quadrance is more fundamental than distance because the mathematical calculation of quadrance doesn't involve a square root (this guy avoids roots like the plague) seems, well, sort of silly.
Fundamentally, I think he needs to get out more often. Then he might realize that distance and angle are used because they are easy to _measure_ (not calculate) and thus more susceptible to intuition.
I'd like to ask this guy to write down navigation directions using quadrance and spread as his vocabulary. ;)
This all reminds me of the representation of numbers (for computational purposes) as a vector of their prime factors (so 36 would be represented by [2 2 0 0 ... 0] and 11 would be represented by [0 0 0 0 1 0 ... 0]. This turns out to be (a) really handy for multiplication (just add the vectors), (b) really a pain for addition, and (c) completely unintuitive to most people.