Euler Identity in Home

[Manolo Sobrino]

(04-15-2014 10:21 PM)Didier Lachieze Wrote:  Manolo, I'm confused, if fort SIN(3.14159265359) a calculator with 12 significant digits should return 0.00000000000, then for 2^(-40) should it also return 0.00000000000 and not 9.09494701773E-13 ?

Of course not, and that's the beauty of it. It has to do with the nature of the function you are considering (and where), for instance, is it bounded? How much does it change around this or that particular point?

You need more than 12 significant digits of Pi to specify a value different than 0.00000000000 for Pi rounded to 12 significant digits, but your 12 significant digits of 2, or 2.00000000000 are enough to specify any value with 12 significant digits of the exponential function.

In the first scenario you are mapping [0, 2 Pi) to [-1, 1]. In the second it's Reals* to Reals. Just think about the definition of function (eom style):

Let the sets X and Y be given. To each element x of X corresponds an element y of Y, which is denoted by f(x), then we say a function f is given on X.

Joe, just read my post of yesterday. It's not the same, and you will see why. Trailing zeroes can be significant figures. x=1.00000000000 means that I know that x is 1 with a precision of 11 decimals. You can specify a value of sin(1) with 11 decimals, and you can specify a value of sin(Pi) with 11 decimals as well, it happens to be 0.00000000000. Notice that if we calculate Sin(3.14159265359*10^-12) the right answer to 12 significant digits is 3.14159265359*10^-12.

As I wrote a few days ago, not worrying about what you can say or not according to your precision (and maiming it) yields "wrong" results, check the example sin(pi*(1+1E-11)). So it's not like there's any particular advantage to it. In the neighbourhood of Pi it gives correct results when rounded, but if the user doesn't know, and sometimes they can't know, it is misleading at best.

This reminds me of a thread of some time ago. The linear solver of the 50G spat results for an incompatible system and that's it, after some digging it turns out that it was producing the minimum norm Least Squares solution with no warning. How is that good for you? Do you have to check if the system has indeed a solution before trusting the solution? In an exam? Now, do you have to check with Mathematica wth is going on with the rounding and the real precision available for calculations in your calculator before you can understand the results for sin(), and then you have to interpret them? What's the point of a calculator I have to check elsewhere the things it shows me? And the results from a TI, that in this case are decent and accurate are bad for you, because it's a TI?? And as they get it right you suggest that they're doing tricks??? I just find all this extremely bizarre. I love RPN, but that has nothing to do with it: Amicus Plato, sed magis amica veritas.