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.
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.
The function 1^x is constant everywhere. You can write (-1)^x to get the kind of effect you are looking for, but doing this with 1^x is madness.
>The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant.
There is more to it than that. The exponential function is not really e^x, it is lim_{N->∞} (1 + x/N)^N. The advantage of this definition is that, since it is valid everywhere on the complex plane, it allows the rigorous definition of real powers without recourse to inverse functions. We still teach students to prove things.
But also, isn't this a little contradictory? The goal of the original blog post was to simplify various differential equations. If you aren't simplifying symbolic differentiation, then what are you simplifying?
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
You have misunderstood my point. I have not said anything about polynomials.
I have said that the so called "natural" exponential, normally written as e^x or exp(x) of either real or complex argument and its inverse, the hyperbolic a.k.a. natural logarithm, and any other functions derived from it are neither needed nor useful when computations are done by computers, as opposed to computations done with pen and paper.
In all traditional formulae where the "natural" exponential function or functions derived from it occur, all occurrences can be replaced using a pair of functions of real argument, the function 2^x with real value and the function 1^x with complex value, either directly or with functions derived from this pair, e.g. the binary logarithm.
In computer programs this substitution results in both higher accuracy and higher speed and it has as a side effect that the units radian and neper are never needed.
It should be noted that even in the 19th century, when the "natural" exponential and logarithm and the trigonometric functions with argument in radians were useful for symbolic computations done by hand, they were never used for practical numeric computations.
All practical numeric computations were done using the function 10^x and the trigonometric functions with argument in degrees and their inverses, by using mathematical tables where the values of these functions were tabulated (or equivalently, by using slide rules).
The use of the "natural" exponential and logarithm and of the trigonometric functions with argument in radians for practical computations has become widespread only after the development of the electronic computers, after programming languages like Fortran have included them as standard functions.
I consider that this has been a mistake, similar to the use of decimal numbers in some computers. Both the use of decimal numbers and the use of the "natural" exponential and logarithm and of the trigonometric functions with argument in radians are sub-optimal in all their possible applications.
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.
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.
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.
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.