Euler Identity in Home

[DavidM]

(04-15-2014 09:17 PM)Manolo Sobrino Wrote:  Han, what's so hard to understand that if you claim to work with 12 significant digits, and you round in every step to 12 significant digits, and you store numbers with 12 significant digits, you can't simply give back results calculated with 25 significant digits?

Perhaps my understanding of significant digits is different than yours, because I can't see where the 25 significant digits comes into play. As I was taught, -0.000000000000206761537357 still only has 12 significant digits. Where are you getting the 25 digit count from? I'm genuinely curious as to whether there is a different understanding of significant digits here. I suppose it's more likely that I just don't understand what you're trying to say.

I don't have a large collection of calculating/computing systems to compare, but I can say that I have checked the following systems to see what the result of "sin(3.14159265359)" is for each one:

Wolfram Alpha (on the web):
-2.0682310711021444E-13

MS Excel 2010 on my laptop:
-2.06823056944638E-13

HP 50g:
-2.06761537357E-13

HP 30b:
-2.0676153736E13

A compiled Delphi XE2 app using the default SIN function with extended (80-bit) reals (also on my laptop):
-2.06761528471349765E-13

The only TI calculator I have is a TI-34 (a 10-digit unit). Here's the results it gave:
a) 3.141592654 SIN: -4.1E-10
b) <pi> SIN: 0

Pressing the <pi> key results in the display showing the value 3.141592654 (the same as item a above). But clearly there's a difference. In an attempt to see that difference, I tried the following:
<pi> - 3.141592654 = -4.1E-10

It appears the TI-34 is carrying more digits than it displays for pi. But it is still rounding its result when using that value.

Of those systems, only the TI-34 rounded the final result to 0, and it only did that when I used the special key for pi. Granted, we are focused on calculators here. But the insistence that zero or 0.000000000000 is a more correct answer for sin(3.14159265359) than -2.068E-13 seems to be a TI-specific viewpoint which isn't shared by the other math libraries I am able to compare it to.

What's more interesting to me is the variance in the non-zero results.