4 ms·

You can always define angles in turns. But the problem is that it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is not so easily separated from the re

by scythe 3y ago

You can always define angles in turns. But the problem is that it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is not so easily separated from the rest of mathematics.

NotYourLawyer 3y ago
Yeah, you’re right. And radians are what make the trig identities involving derivatives work out nicely.
- adrian_b 3y ago
  The simpler derivation formula was important when such symbolic computation was done by hand. Now, except perhaps for school exercises, anything complicated is done with a computer and this advantage is much less important. The increased accuracy and simpler formulas in other places when measuring angles in cycles a.k.a. turns vastly outweigh the advantage of radians for differentiation. In real applications you almost always compute the derivative of sin(a*x), not of sin(x), so you have to carry a multiplicative constant through derivations anyway and the single advantage of the radian vanishes.
  - NotYourLawyer 3y ago
    If you’re using a computer for symbolic algebra or whatever, none of this matters anyway. The whole post is about simpler notation for the sake of making things less error-prone when working by hand.
    - adrian_b 3y ago
      For a computer it matters because it is possible to compute with higher accuracy and speed the trigonometric functions with the argument in cycles instead of radians (similarly for the binary exponential and logarithm vs. the "natural" exponential and logarithm). The main reason is that the reduction of the argument to the range where a polynomial approximation is valid becomes much simpler. Also, the primary inputs or the final outputs of any really complete computation are never in radians, because in physical devices it is not possible to realize radians with high accuracy, but only the angles that are in a rational relationship with the cycle. This is true both for geometric angles and for the phase angles of oscillations and waves.
adrian_b 3y ago
There is no need for that definition. It is possible to completely remove the e^x function from mathematics without losing anything. It is possible to express everything using a pair of functions, the real function 2^x and the complex function 1^x. Then the cosinus and the sinus are the real and imaginary parts of 1^x (where x is measured in cycles a.k.a. turns). The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant. In my opinion the simplifications that are introduced everywhere else are more important than this disadvantage. The real and complex function e^x was preferable in the 19th century, when numeric computations were avoided as too difficult and simple problems were solved by symbolic computation done with pen and paper. Now, when anything difficult is done with a computer, both the function e^x and associated units like the radian and the neper are obsolete. When programming computations on a computer, using 2^x and 1^x results in simpler and more accurate computations. Moreover, when doing real physical measurements it is possible to obtain highly accurate values when the units are cycle and octave, but not when they are radian and neper.
- kandel 3y ago
  Mind explaining how to express some simple functions? Im interested
  - adrian_b 3y ago
    All mathematical formulas can be inter-converted based on the identity: e^(x + i*y) = 2^(x/ln2) * 1^(y/2Pi) Most formulae from textbooks are written in such a way to be simpler with e^x and its inverse, but it is almost always possible to move the constants ln2 and 2Pi between various equations so that in the end they will disappear from most relations, with the exception of the derivation or integration formulae. In most applications, more equations are simplified than those which become more complicated. A very important advantage of 2^x and 1^x versus e^x is that for the former the reductions of the argument to the principal range where the function is approximated by a polynomial can be done with perfect accuracy and very quickly, unlike for the latter. Moreover, for the former it is easy to verify the accuracy of any approximation, because for any argument that is represented as a binary number the functions 2^x and 1^x can be computed with a finite number of sqrt invocations (based on the formulae for half angle) and sqrt can be computed with any number of desired digits. Computing e^x with an arbitrary precision is trickier, because it requires criteria for truncation of an infinite series. It should be noted that 1^1.0 = 1, 1^0.5 = -1, 1^0.25 = i, 1^0.75 = -i

by scythe 3y ago

You can always define angles in turns. But the problem is that it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is not so easily separated from the rest of mathematics.

NotYourLawyer 3y ago

Yeah, you’re right. And radians are what make the trig identities involving derivatives work out nicely.

adrian_b 3y ago
The simpler derivation formula was important when such symbolic computation was done by hand. Now, except perhaps for school exercises, anything complicated is done with a computer and this advantage is much less important. The increased accuracy and simpler formulas in other places when measuring angles in cycles a.k.a. turns vastly outweigh the advantage of radians for differentiation. In real applications you almost always compute the derivative of sin(a*x), not of sin(x), so you have to carry a multiplicative constant through derivations anyway and the single advantage of the radian vanishes.
- NotYourLawyer 3y ago
  If you’re using a computer for symbolic algebra or whatever, none of this matters anyway. The whole post is about simpler notation for the sake of making things less error-prone when working by hand.
  - adrian_b 3y ago
    For a computer it matters because it is possible to compute with higher accuracy and speed the trigonometric functions with the argument in cycles instead of radians (similarly for the binary exponential and logarithm vs. the "natural" exponential and logarithm). The main reason is that the reduction of the argument to the range where a polynomial approximation is valid becomes much simpler. Also, the primary inputs or the final outputs of any really complete computation are never in radians, because in physical devices it is not possible to realize radians with high accuracy, but only the angles that are in a rational relationship with the cycle. This is true both for geometric angles and for the phase angles of oscillations and waves.

adrian_b 3y ago

There is no need for that definition. It is possible to completely remove the e^x function from mathematics without losing anything. It is possible to express everything using a pair of functions, the real function 2^x and the complex function 1^x. Then the cosinus and the sinus are the real and imaginary parts of 1^x (where x is measured in cycles a.k.a. turns). The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant. In my opinion the simplifications that are introduced everywhere else are more important than this disadvantage. The real and complex function e^x was preferable in the 19th century, when numeric computations were avoided as too difficult and simple problems were solved by symbolic computation done with pen and paper. Now, when anything difficult is done with a computer, both the function e^x and associated units like the radian and the neper are obsolete. When programming computations on a computer, using 2^x and 1^x results in simpler and more accurate computations. Moreover, when doing real physical measurements it is possible to obtain highly accurate values when the units are cycle and octave, but not when they are radian and neper.

kandel 3y ago
Mind explaining how to express some simple functions? Im interested
- adrian_b 3y ago
  All mathematical formulas can be inter-converted based on the identity: e^(x + i*y) = 2^(x/ln2) * 1^(y/2Pi) Most formulae from textbooks are written in such a way to be simpler with e^x and its inverse, but it is almost always possible to move the constants ln2 and 2Pi between various equations so that in the end they will disappear from most relations, with the exception of the derivation or integration formulae. In most applications, more equations are simplified than those which become more complicated. A very important advantage of 2^x and 1^x versus e^x is that for the former the reductions of the argument to the principal range where the function is approximated by a polynomial can be done with perfect accuracy and very quickly, unlike for the latter. Moreover, for the former it is easy to verify the accuracy of any approximation, because for any argument that is represented as a binary number the functions 2^x and 1^x can be computed with a finite number of sqrt invocations (based on the formulae for half angle) and sqrt can be computed with any number of desired digits. Computing e^x with an arbitrary precision is trickier, because it requires criteria for truncation of an infinite series. It should be noted that 1^1.0 = 1, 1^0.5 = -1, 1^0.25 = i, 1^0.75 = -i