> It should be noted that the number e = 2.71828 ... does not have any importance in practice, its value just satisfies the curiosity to know it, but there is no need to use it in any application.
In calculations like compound financial interest, radioactive decay and population growth (and many others), e is either applied directly or derived implicitly.
> ... 2*pi is the most important transcendental number, not pi.
When using the exponential e^x or the natural logarithm, the number "e" is never used. Only ln 2 or its inverse are used inside the function evaluations, for argument range reduction.
In radioactive decay and population growth it is much simpler conceptually to use 2^x, not e^x, which is why this is done frequently even by people who are not aware that the computational cost of 2^x is lower and its accuracy is greater.
In compound financial interest using 2^x would also be much more natural than the use of e^x, but in financial applications tradition is usually more important than any actual technical arguments.
> When using the exponential e^x or the natural logarithm, the number "e" is never used. Only ln 2 or its inverse are used inside the function evaluations, for argument range reduction.
That is only true in the special case of computing a half-life. In the general case, e^x is required. When computing a large number of cases and to avoid confusion, e^x is the only valid operator. This is particularly true in compound interest calculations, which would fall apart entirely without the presence of e^x and ln(x).
> In radioactive decay and population growth it is much simpler conceptually to use 2^x, not e^x
See above -- it's only valid if a specific, narrow question is being posed.
> In compound financial interest using 2^x would also be much more natural than the use of e^x
That is only true to answer a specific question: How much time to double a compounded value? For all other cases, e^x is a requirement.
If your position were correct, if 2^x were a suitable replacement, then Euler's number would never have been invented. But that is not reality.
No, you did not try to understand what I have written.
The use of ln 2 for argument range reduction has nothing to do with half lives. It is needed in any computation of e^x or ln x, because the numbers are represented as binary numbers in computers and the functions are evaluated with approximation formulae that are valid only for a small range of input arguments.
The argument range reduction can be avoided only if you know before evaluation that the argument is close enough to 0 for an exponential or to 1 for a logarithm, so that an approximation formula can be applied directly. For a general-purpose library function you cannot know this.
Also the use of 2^x instead of e^x for radioactive decay, population growth or financial interest is not at all limited to the narrow cases of doublings or halvings. Those happen when x in an integer in 2^x, but 2^x accepts any real value as argument. There is no difference in the definition set between 2^x and e^x.
The only difference between using 2^x and e^x in those 3 applications is in a different constant in the exponent, which has the easier to understand meaning of being the doubling or halving time, when using 2^x and a less obvious meaning when using e^x. In fact, only doubling or halving times are directly measured for radioactive decay or population growth. When you want to use e^x, you must divide the measured values by ln 2, an extra step that brings no advantage whatsoever, because it must be implicitly reversed during every subsequent exponential evaluation when the argument range reduction is computed.
> The use of ln 2 for argument range reduction has nothing to do with half lives.
That is a false statement.
> In fact, only doubling or halving times are directly measured for radioactive decay or population growth.
That is a false statement -- in population studies, as just one example, the logistic function (https://en.wikipedia.org/wiki/Logistic_function) tracks the effect of population growth over time as environmental limits take hold. This is a detailed model that forms a cornerstone of population environmental studies. To be valid, it absolutely requires the presence of e^x in one or another form.
> ... because the numbers are represented as binary numbers in computers and the functions are evaluated with approximation formulae that are valid only for a small range of input arguments.
That is a spectacularly false statement.
> There is no difference in the definition set between 2^x and e^x.
That is absolutely false, and trivially so.
> No, you did not try to understand what I have written.
On the contrary, I understood it perfectly. From a mathematical standpoint, 2^x cannot substitute for e^x, anywhere, ever. They're not interchangeable.
I hope no math students read this conversation and acquire a distorted idea of the very important role played by Euler's number in many applied mathematical fields.
In calculations like compound financial interest, radioactive decay and population growth (and many others), e is either applied directly or derived implicitly.
> ... 2*pi is the most important transcendental number, not pi.
Gotta agree with this one.