Euler Identity in Home

[Manolo Sobrino]

(04-14-2014 07:31 PM)Han Wrote:  

(04-14-2014 07:10 PM)Manolo Sobrino Wrote:  Which value did the calc pick? The sixth? Why? But they are all good! in fact any value in (-3.20665*10^-12, 4.79318*10^-12) would be a good answer.

The calc picked the sixth case because it rounds \( \pi \) to 3.141592653590 and not any of the other values. And while all the other values are just as good, it only makes sense to map to one of them -- namely the one corresponding to an input of 3.141592653590.

No, it doesn't.

The consistent answer for 12 significant digits is 0.000000000000, and that is not zero, those zeroes have a meaning.

Quote:I agree and disagree. I agree in terms of significant digits because we are doing numerical calculations and know that there will be roundoff error. I disagree in terms of pure mathematical results. Take for example:

\[ (10^{15}+1.2-10^{15})*100 \]

Most calculators will give you -- even Maple -- will give you a result of 0. This is a valid answer in practical terms. It is, of course, the wrong answer mathematically.

Well, in such a case that famous answer for sin (pi) is also wrong. You can't have your cake and eat it. Either you are consistent or you aren't.

Maple is able to do arbitrary precision calculations, you should try that.

Quote:The blind reliance on any calculating machine for the right answer is always going to result in awful consequences.

I fully agree on that

Quote:To suggest that there is one, right approach to handling numerical calculations is also going to result in awful consequences.

That's what professional Mathematicians do, and HP has taken advantage of it in the past. Kahan might still be available. If not, I'd suggest Higham.

You know, this thing Maths is unluckily for some not a matter of opinion, either you get it right or you don't. In the real world of Logic, Math departments and peer reviewed journals your preferences don't matter. If you keep on being wrong for the sake of it you're out.