Euler Identity in Home

[Han]

(04-14-2014 02:48 PM)ColinJDenman Wrote:  My attitude from the beginning has been a concern that observable error in a one-shot simple calculation questions what the errors are in a more complex one. And how to estimate them and rely on the results I see..

The issue that I see (based on the posts in this thread) is nowhere close to, say, a case of a calculator computing 2+3 and returning 5.8. I don't work for HP (nor do I claim to speak for them) but I feel pretty confident in saying that the error resulting from using the Home view in doing numerical calculations will very likely NOT balloon into something ridiculous. The error is somewhere out toward the 10th or 11th decimal place. That said, anyone using any sort of calculating tool will necessarily need to be aware of roundoff error. This goes for using the CAS view as well. The results you get, even if all the precision needed is there, will still need to be interpreted correctly. (E.g. I always tell my students to watch out for negative values that a calculator will correctly return, but will not make sense if they are solving for the length of a side of a box.)

If you are doing scientific work in which even the slightest bit of error may result in catastrophic consequences, a scientific calculator is hardly the tool -- whether designed by HP or any other company.

EDIT: To address your specific example of: \( \frac{\cos(\pi/2)}{\sin(\pi)} \) -- there is a singularity at \( \theta = \pi \) for \( \frac{\cos(\theta/2)}{\sin(\theta)} \) so any numerical answer your calculator returns is the wrong answer.

EDIT #2: For calculators (such as the TI-84), if I type in \( \cos(3.141592654) \) it spits out the answer as -1. This is mathematically wrong, as \( \cos( \pi ) = -1 \) and \( \pi \not= 3.141592654 \). So what happens if I need to actually compute the cosine of the rational number 3.141592654? The calculator gives me division by 0 if I do \( 1/ \cos( 3.141592654) \) even though I know the denominator is not 0. The answer is off by a magnitude of "infinity" if you will. Of course this entire example is a bit silly. Insisting that \( \cos ( 3.141592654 ) = -1 \) being "wrong" is just as silly as insisting that \( \cos (3.141592654 ) \not=-1 \) is "right" -- inside an environment that is limited in precision.

I recommend reading:

http://www.stewartcalculus.com/data/ESSE...mp_stu.pdf
http://apcentral.collegeboard.com/apc/me...11703.html
http://www2.edc.org/cme/showcase/KY/calcliesarticle.pdf

(The third link is extremely interesting -- the graph of \( \sin(46x)\) looks like \(-\sin(x)\).)