Euler Identity in Home

[Han]

(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.

Quote:This is not a floating point issue. It's just continuous functions and significant digits' common sense. The best accurate and meaningful value your calc should give you is 0E-12, that is 2.06761537357E-13 rounded to 12, the rest is garbage. Yet people are not willing to get/believe it, go figure!

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.

Quote:HP, or their fans, have taken historically a very strange viewpoint in which having obvious accuracy in the display is not that important, but raw results are, leaving to the user the task of making sense of them, with awful consequences on the average, I'm sad to say.

The blind reliance on any calculating machine for the right answer is always going to result in awful consequences. To suggest that there is one, right approach to handling numerical calculations is also going to result in awful consequences.