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Prosthaphaeresis works because an angle measure inherently is a type of logarithm. It’s the logarithm of a rotation, instead of the logarithm of a scale. Like o

Prosthaphaeresis works because an angle measure inherently is a type of logarithm. It’s the logarithm of a rotation, instead of the logarithm of a scale. Like other kinds of logarithms, this converts multiplication (composition of rotations) to addition (addition of angle measures). If you want to multiply two numbers, you can treat them as rotations, take the logarithm of each (i.e. find the angle measure), add the two, then take the exponential. cf. https://en.wikipedia.org/wiki/Complex_logarithm https://en.wikipedia.org/wiki/Complex_logarithm

HPsquared 6y ago
Is there a more general definition of logarithms than being the inverse of the exponential function? Mathworld says Prosthapharesis formulas "convert a product of functions into a sum or difference" [0], which resembles a property of logarithms (i.e. log(ab) == log(a) + log(b)). However I don't quite get how angle measurement itself is a type of logarithm. Maybe something to do with Euler's formula? [0] https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html
- eigenket 6y ago
  Starting for Euler's formula you can write cos(x) = (exp(ix) + exp(-ix))/2 so the cosine is in some sense a "type of exponential". If the cosine is a type of exponential then its inverse (the function that takes you to the angle measurement) is inherently a "type of logarithm". This is a very handwavey statement. I would say a more precise statement is that the proper exponential and log maps are the "correct" thing to use to map multiplication to adidtion and vice-versa. They are exactly the maps that do this properly. The fact that cosine is a sum of exponentials means that you can write the inverse of cosine as a (slightly ugly) formula in terms of logarithms (arcos(z) = -i log(sqrt(1-z^2) + z). This is the sense in which the map to angle measures is a "type of log".
  - jacobolus 6y ago
    The angle measure i acos(x) is the logarithm of the rotation x + i√(1 − x²). If you want to compute the product of two rotations, you can take logarithms and rewrite it as a sum of angle measures: [x + i√(1 − x²)][y + i√(1 − y²)] = exp(i acos(x) + i acos(y)) [x + i√(1 − x²)][y − i√(1 − y²)] = exp(i acos(x) − i acos(y)) Now if you just look at the real parts, you can write: xy − √(1 − x²)√(1 − y²) = cos(acos(x) + acos(y)) xy + √(1 − x²)√(1 − y²) = cos(acos(x) − acos(y)) Taking the sum on each side: 2xy = cos(acos(x) + acos(y)) + cos(acos(x) − acos(y)) This method is rather more cumbersome (due to rescaling and then converting the factors to rotations as an intermediate step) but fundamentally based on the same concept as: xy = exp(log(x) + log(y))
- scythe 6y ago
  There is an axiomatic approach that defines a logarithm as any function f(x) such that f(x•y) = f(x) + f(y). But it's easy to prove that this reduces to the usual definition for logarithms on the real numbers.

HPsquared 6y ago

Is there a more general definition of logarithms than being the inverse of the exponential function? Mathworld says Prosthapharesis formulas "convert a product of functions into a sum or difference" [0], which resembles a property of logarithms (i.e. log(ab) == log(a) + log(b)). However I don't quite get how angle measurement itself is a type of logarithm. Maybe something to do with Euler's formula? [0] https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

eigenket 6y ago
Starting for Euler's formula you can write cos(x) = (exp(ix) + exp(-ix))/2 so the cosine is in some sense a "type of exponential". If the cosine is a type of exponential then its inverse (the function that takes you to the angle measurement) is inherently a "type of logarithm". This is a very handwavey statement. I would say a more precise statement is that the proper exponential and log maps are the "correct" thing to use to map multiplication to adidtion and vice-versa. They are exactly the maps that do this properly. The fact that cosine is a sum of exponentials means that you can write the inverse of cosine as a (slightly ugly) formula in terms of logarithms (arcos(z) = -i log(sqrt(1-z^2) + z). This is the sense in which the map to angle measures is a "type of log".
- jacobolus 6y ago
  The angle measure i acos(x) is the logarithm of the rotation x + i√(1 − x²). If you want to compute the product of two rotations, you can take logarithms and rewrite it as a sum of angle measures: [x + i√(1 − x²)][y + i√(1 − y²)] = exp(i acos(x) + i acos(y)) [x + i√(1 − x²)][y − i√(1 − y²)] = exp(i acos(x) − i acos(y)) Now if you just look at the real parts, you can write: xy − √(1 − x²)√(1 − y²) = cos(acos(x) + acos(y)) xy + √(1 − x²)√(1 − y²) = cos(acos(x) − acos(y)) Taking the sum on each side: 2xy = cos(acos(x) + acos(y)) + cos(acos(x) − acos(y)) This method is rather more cumbersome (due to rescaling and then converting the factors to rotations as an intermediate step) but fundamentally based on the same concept as: xy = exp(log(x) + log(y))
scythe 6y ago
There is an axiomatic approach that defines a logarithm as any function f(x) such that f(x•y) = f(x) + f(y). But it's easy to prove that this reduces to the usual definition for logarithms on the real numbers.