Even when you use the exponential e^x and the hyperbolic logarithm a.k.a. natural logarithm (which are useful only in symbolic computations and are inferior for any numeric computation), you never need to know the value of "e". The value itself is not needed for anything. When evaluating e^x or the hyperbolic logarithm you need only ln 2 or its inverse, in order to reduce the argument of the functions to a range where a polynomial approximation can be used to compute the function.
Moreover, you can replace any use of e^x with the use of 2^x, which inserts ln(2) constants in various places, (but removes ln 2 from the evaluations of exponentials and logarithms, which results in a net gain).
If you use only 2^x, you must know that its derivative is ln(2) * 2^x, and knowing this is enough to get rid of "e" anywhere. Even in derivation formulae, in actual applications most of the multiplications with ln 2 can be absorbed in multiplications with other constants, as you normally do not have 2^x expressions that are derived, but 2^(a*x), where you do ln(2)*a at compile time.
You start with the formula for the exponential of an imaginary argument, but there the use of "e" is just a conventional notation. The transcendental number "e" is never used in the evaluation of that formula and also none of the numbers produced by computing an exponential or logarithm of real numbers are involved in that formula.
The meaning of that formula is that if you take the expansion series of the exponential function and you replace in it the argument with an imaginary argument you obtain the expansion series for the corresponding trigonometric functions. The number "e" is nowhere involved in this.
Moreover, I consider that it is far more useful to write that formula in a different way, without any "e":
1^x = cos(2Pi*x) + i * sin(2Pi*x)
This gives the relation between the trigonometric functions with arguments measured in cycles and the unary exponential, whose argument is a real number and whose value is a complex number of absolute value equal to 1, and which describes the unit circle in the complex plane, for increasing arguments.
This formula appears more complex only because of using the traditional notation. If you call cos1 and sin1 the functions of period 1, then the formula becomes:
1^x = cos1(x) + i * sin1(x)
The unary exponential may appear weirder, but only because people are habituated from school with the exponential of imaginary arguments instead of it. None of these 2 functions is weirder than the other and the use of the unary exponential is frequently simpler than of the exponential of imaginary arguments, while also being more accurate (no rounding errors from argument range reduction) and faster to compute.
I want to add that any formula that contains exponentials of real arguments, e^x, and/or exponentials of imaginary arguments, e^(i*x), can be rewritten by using only binary exponentials, 2^x, and/or unary exponentials, 1^x, both having only real arguments.
With this substitution, some formulae become simpler and others become more complicated, but, when also considering the cost of the function evaluations, an overall greater simplicity is achieved.
In comparison with the "e" based exponentials, the binary exponential and the unary exponential and their inverses have the advantage that there are no rounding errors caused by argument range reduction, so they are preferable especially when the exponents can be very big or very small, while the "e" based exponentials can work fine for exponents guaranteed to be close to 0.
Moreover, you can replace any use of e^x with the use of 2^x, which inserts ln(2) constants in various places, (but removes ln 2 from the evaluations of exponentials and logarithms, which results in a net gain).
If you use only 2^x, you must know that its derivative is ln(2) * 2^x, and knowing this is enough to get rid of "e" anywhere. Even in derivation formulae, in actual applications most of the multiplications with ln 2 can be absorbed in multiplications with other constants, as you normally do not have 2^x expressions that are derived, but 2^(a*x), where you do ln(2)*a at compile time.
You start with the formula for the exponential of an imaginary argument, but there the use of "e" is just a conventional notation. The transcendental number "e" is never used in the evaluation of that formula and also none of the numbers produced by computing an exponential or logarithm of real numbers are involved in that formula.
The meaning of that formula is that if you take the expansion series of the exponential function and you replace in it the argument with an imaginary argument you obtain the expansion series for the corresponding trigonometric functions. The number "e" is nowhere involved in this.
Moreover, I consider that it is far more useful to write that formula in a different way, without any "e":
1^x = cos(2Pi*x) + i * sin(2Pi*x)
This gives the relation between the trigonometric functions with arguments measured in cycles and the unary exponential, whose argument is a real number and whose value is a complex number of absolute value equal to 1, and which describes the unit circle in the complex plane, for increasing arguments.
This formula appears more complex only because of using the traditional notation. If you call cos1 and sin1 the functions of period 1, then the formula becomes:
1^x = cos1(x) + i * sin1(x)
The unary exponential may appear weirder, but only because people are habituated from school with the exponential of imaginary arguments instead of it. None of these 2 functions is weirder than the other and the use of the unary exponential is frequently simpler than of the exponential of imaginary arguments, while also being more accurate (no rounding errors from argument range reduction) and faster to compute.